distribution of rational points on algebraic varieties · at courant, especially andrea, chaitu,...
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Distribution of rational points on algebraic varieties
by
Sho Tanimoto
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Mathematics
New York University
May 2012
Professor Yuri Tschinkel
To my wife, Mei.
iii
Acknowledgements
First and foremost it is my pleasure to thank my advisor Yuri Tschinkel for his pa-
tience and guidance throughout my Ph.D.. He has influenced my development from
an ordinary math student to a professional researcher in mathematics. I learned
from him, his patience, work ethic, creativity, and broad outlook in mathematics,
and these have shaped me as a professional mathematician.
I would also like to thank my second advisor Fedor Bogomolov. His passion,
intuition, creativity, and the view of mathematics have inspired me in my research.
His generosity has supported my study and life, and without it, I would not have
completed this dissertation.
I would like to thank Sylvain Cappell for his warmest support which he has
given me since I was a first year student at Courant. Brendan Hassett is a great
inspiration for my research, and I also benefited from conversations and discus-
sions with Benjamin Bakker, David Harvey, Sonal Jain, Ilya Karzhemanov, Brian
Lehmann, and Tony Varilly.
Many thanks to my fellow students in our algebraic geometry and number
theory group, Michael Burr, Edgar Costa, Lukas Koehler, Fedor Soloviev, Jordan
Thomas, and Pankaj Vishe, for their insights, questions, help, and friendship.
In particular, Michael and Jordan helped me to improve the exposition of this
thesis, and Lukas did a fantastic job to activate our algebraic geometry seminar.
I would also like to thank friends with whom I shared time as a graduate student
at Courant, especially Andrea, Chaitu, Dingyu, Felix, Hantaek, Ivan, Jong Ho,
Jungwoon, Kela, Nengli, Shoshana, and Sukbin. I owe a special debt of gratitude
to Tamar Arnon for her warmest support throughout my Ph.D..
Going back to Japan, I would like to thank my teachers at my college, Takao
iv
Fujita, Shihoko Ishii, and Nobushige Kurokawa for their support and encourage-
ment. I learned a great deal of algebraic geometry from my advisor Takao Fujita
when I was an undergraduate student at Tokyo Institute of Technology. Ryoichi
Aoki, my physics teacher at my high school, is one who let me consider a career
path in academia. He always kept his door open for me, and he shared time with
me to discuss my questions in physics.
Finally, I thank my family: my parents, Shin and Mariko, my brother Ryo, and
our lovely cats, Chao and Mie have supported me via skype for the last five years
and provided me an ideal atmosphere when I was in Japan.
And my wife Mei, for me, she is everything. She has supported and encouraged
me all the time with her love via skype, phone, and shipments, and without her
support, I could not have achieved any accomplishment which today I have. I
thank her with all my love, and I wish the best luck for our continuing journey.
v
Abstract
We study the asymptotic formulas for the number of rational points of bounded
height on equivariant compactifications of solvable linear algebraic groups over a
number field, and we introduce the notion of balanced line bundles.
In the first part, we discuss Height zeta functions of equivariant compactifica-
tions of solvable linear algebraic groups and show that the main term of a Height
zeta function coincides with the prediction of Manin’s conjecture by using geomet-
ric integration techniques developed by Chambert-Loir and Tschinkel. Then we
study Height zeta functions of equivariant compactifications of the ax + b-group
and prove Manin’s conjecture under some mild geometric conditions.
In the second part, we introduce the notion of balanced line bundles and give
a systematic study of this notion in terms of birational geometry. We identify
balanced big line bundles for many cases, e.g., del Pezzo surfaces, Fano 3-folds of
Picard rank one, flag varieties, toric varieties, and equivariant compactifications
of homogeneous spaces. The last result has important arithmetic applications in
[CLT02] and [GTBT11].
vi
Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I Height zeta functions 8
1 Height functions 9
1.1 Metrics and Heights . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Manin’s conjecture and Height zeta functions 22
2.1 Manin’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Height zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Height zeta functions of equivariant compactifications of semi-
direct products of algebraic groups 37
3.1 Geometry of equivariant compactifications . . . . . . . . . . . . . . 38
3.2 The main terms of Height zeta functions of equivariant compactifi-
cations of solvable algebraic groups . . . . . . . . . . . . . . . . . . 46
vii
3.3 Height zeta functions of equivariant compactifications of the ax+ b-
group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
II Balanced line bundles 99
4 Balanced line bundles 100
4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2 Balanced line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5 Fano varieties 115
5.1 Del Pezzo surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Del Pezzo surface fibrations . . . . . . . . . . . . . . . . . . . . . . 120
5.3 Fano threefolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6 Equivariant geometry 125
6.1 Generalized flag varieties . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2 Equivariant compactifications of homogeneous spaces . . . . . . . . 129
6.3 Toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Bibliography 146
viii
Introduction
One of the classical problems in mathematics is to study integral or rational
solutions to Diophantine equations, i.e., integral or rational solutions to polynomial
equations with integer coefficients. The most famous Diophantine equations are
the Fermat equations :
xn + yn = zn,
where n is a positive integer. There are many traditional questions concerning
Diophantine equations, which include:
1. (Existence) Do integral or rational solutions exist?
2. (Finiteness) If solutions exist, are there finitely or infinitely many?
3. (Distribution) If there are infinitely many solutions, how are they distributed?
For example, one could determine their density in various topologies or
asymptotic formulae for the counting functions of solutions with respect to
appropriate size functions.
4. (Construction) Construct some or all solutions effectively.
The solutions to each of these problems have lead to intensive and long studies,
and they are still active areas in modern mathematics. Although there are many
elementary and ad hoc approaches to these questions, modern research in this field
focuses on intrinsic properties of Diophantine equations, e.g., their geometry.
1
Diophantine geometry
Note that answers to the above questions are stable under changes of coordi-
nates. Perhaps this is the first observation to support the philosophy that “Geom-
etry determines Arithmetic”, appearing in [HS00]. The subject which studies the
phenomena realizing this philosophy is called Diophantine geometry, and is one of
main topics in current mathematics.
One of the most important invariants in Diophantine geometry is the canonical
line bundle KX of a smooth projective variety X, and we believe that this geometric
invariant roughly determines the behavior of the distribution of rational points, at
least after taking a finite extension. To illustrate this idea, consider the case of
curves: Let F be a number field and C a smooth projective curve defined over F .
The behavior of the distribution of rational points is determined by one geometric
invariant, the genus g of C:
• When g = 0, −KC is ample and C(F ) is either empty or infinite;
• When g = 1, KC is trivial and C(F ) is either empty or a finitely generated
abelian group. C(F ) is infinite if and only if the Mordell-Weil rank, the rank
of C(F ), is positive.
• When g ≥ 2, KC is ample and C(F ) is finite. (Mordell-Faltings theorem)
In particular, after taking a finite extension of F , C(F ) is infinite when −KX is
ample or trivial. Based on this evidence, we believe in the following principle which
is the most important guiding principle in Diophantine geometry:
The positivity of the canonical line bundle (or the anticanonical line bundle)
determines the distribution of rational points on X.
2
We hope that the principle applies to higher dimensional cases as well, and
thus we consider the following classification of smooth projective varieties:
• (Fano-type): −KX is ample, e.g., Pn, del Pezzo surfaces, and Fano manifolds;
• (Intermediate type): KX is trivial, e.g., abelian varieties and K3 surfaces;
• (General type): KX is ample, e.g., projective manifolds of general type.
My research interest lies in Fano and intermediate type varieties which are expected
to have infinitely many rational points, at least after taking a finite extension. More
precisely, I am interested in the distribution of rational points on these varieties.
Distribution of rational points
There are many problems involving the distribution of rational points on al-
gebraic varieties, but here we would like to mention two problems: Let X be a
smooth projective variety defined over a number field F . One can ask the following
two questions:
• (Potential density): Is X(F ) dense in the Zariski topology, possibly after
taking a finite extension of F?
• (Asymptotic formulae): Let L = (L, ‖ · ‖) be a big, adelically metrized
line bundle on X and HL the associated height (see Definitions 4 and 6 in
Subsection 1.1.2). What is the asymptotic formula of the counting function
of rational points of bounded height? In other words, determine
N(X,L,B) := #x ∈ X(F ) | HL(x) 6 B ∼ ???
as B→∞, for a suitable Zariski open subset X.
3
The answer to the potential density problem is believed to be affirmative for
Fano and Intermediate type varieties. It is possible that X(F ) is empty due to the
arithmetic obstructions, e.g., the Hasse principle and the Brauer-Manin obstruc-
tions (see, for example, [Man86]). To focus on the effect of geometry, it is essential
to allow one to take a finite extension. [HT00] and [BT99] provided the first results
in this direction, and [Cam04] and [Abr09] describe a general framework of this
problem.
An answer for the second question should be viewed as a strong, quantitative
version of the density of rational points, and this thesis mainly discusses this
problem.
Manin’s conjecture
Next, we describe the second problem in more detail. Let X ⊂ Pn be a smooth
projective variety defined over the field of rational numbers Q with given embed-
ding into a projective space. A height function H on Pn is defined as follows:
H : Pn(Q) = Zn+1prim/± 1 3 (x0 : · · · : xn) 7→
√x2
0 + · · ·+ x2n ∈ R>0, (1)
where (x0, · · · , xn) is a primitive integral vector, i.e., xi ∈ Z and gcdi(xi) = 1.
Consider the following function which counts the number of rational points of
bounded height:
N(X,H,B) = #P ∈ X(Q) | H(P ) ≤ B .
where X ⊂ X is a Zariski open subset and B is a positive real number. Manin’s
program, initiated in [FMT89], relates the asymptotics of N(X,H,B), for a suitable
4
Zariski open subset X, to certain global geometric invariants of the underlying
variety X. The following is the proposed conjecture:
Conjecture 1 (Manin’s conjecture). Let X be a smooth Fano variety defined over
a number field F . Let L be a big, adelically metrized line bundle on X. Then,
after taking a finite extension, there exist a Zariski open subset X ⊂ X and a
positive constant c(X,L) > 0 such that
N(X,L,B) = c(X,L)Ba(X,L) log(B)b(X,L)−1(1 + o(1)), B→∞,
Here a(X,L) and b(X,L) are geometric constants which were introduced in this
context in [FMT89] and [BM90] (and recalled in Chapter 2, Subsection 2.1.1) and
c(X,L) is a Tamagawa-type number defined in [Pey95], [BT98b].
Although Manin’s conjecture admits counterexamples, found by Batyrev and
Tschinkel in [BT96b] (see Example 19 in Subsection 2.1.2), it has turned out to be
quite influential. Over the last 20 years, it lead to many important theorems and
stimulated the development of several new research directions and methods. The
following methods have been intensively studied by various researchers:
• universal torsor method combined with lattice point counts;
• variants of the circle method;
• ergodic theory and mixing;
• and Height zeta functions and spectral theory on adelic groups.
The universal torsor method has been successful in the treatment of singular del
Pezzo surfaces over Q. The circle method was used to prove Manin’s conjecture
5
for low degree hypersurfaces in high dimensional projective spaces. Ergodic theory
and mixing settled many new cases of equivariant compactifications of reductive
groups. The Height zeta functions method has the advantage that it is able to
analyze the second term of the counting function. For recent surveys highlighting
these methods, see [Tsc09], [Bro07], [Bro09], [CL10], and [Oh10]. In this thesis,
we discuss two different aspects of Manin’s program: One is the theory of Height
zeta functions, and another is the notion of balanced line bundles.
Height zeta functions
In part I, we discuss the theory of Height zeta functions. A Height zeta function
is a certain Dirichlet series defined by
Z(X,L, s) =∑
P∈X(F )
H(P )−s, s ∈ C.
This series absolutely and uniformly converges when <(s) 0, for a suitable
Zariski open subset X ⊂ X, so that Z(X,L, s) defines a holomorphic function
for <(s) 0. The connection between Manin’s conjecture and Height zeta func-
tions is given by the Tauberian theorem (see Theorem 27). Hence, our task here is
to obtain the meromorphic continuation of the Height zeta function and to identify
the location of the pole and its order. In general, the study of meromorphic con-
tinuations of Height zeta functions is extremely difficult, and it has been studied
intensively when X is an equivariant compactification of a linear algebraic group
G by using Spectral theory on an adelic space G(AF ). In this part, first we review
the basic background of heights, Manin’s conjecture, and Height zeta functions,
then we discuss results in [TT12], e.g., Height zeta functions of equivariant com-
6
pactifications of semi-direct products. We discuss the main term of a Height zeta
function of an equivariant compactification of a solvable group, and we confirm
that the main term coincides with the prediction of Manin’s conjecture. Then we
study meromorphic continuations of Height zeta functions of the ax+b-group, and
produce new examples of rational surfaces satisfying Manin’s conjecture.
Balanced line bundles
In part II, we introduce and study the notion of balanced line bundles. This
part consists of results from [HTT12]. Implicitly, Manin’s conjecture asserts that
in the case when L = −KX , any proper subvariety Y of X, which is not contained
in the exceptional set X \ X, does not contribute a positive proportion to the
main term of the asymptotic formula, i.e.,
N(Y ,L|Y ,B)
N(X,L,B)→ 0 as B→∞.
Internal consistency would imply that for all such Y we have
(a(Y, L|Y )), b(Y, L|Y )) < (a(X,L), b(X,L)),
in the lexicographic ordering. We say that L is balanced with respect to Y when
this strict inequality holds. One can ask the following question: “How can this
inequality be checked by geometry?” This question arose from the study of ergodic
theory in [GTBT11], where they needed a version of this statement to prove that
the asymptotic volume of a height ball approximates the number of rational points
of bounded height. In this part, we give a systematic study of balanced line bundles
in terms of birational geometry, and explore the geometric meaning of this notion.
7
Part I
Height zeta functions
8
Chapter 1
Height functions
In this chapter, we review the theory of height functions. Height functions
measure arithmetic and geometric complexities of solutions of Diophantine equa-
tions, and they provide us appropriate quantification of the size of solutions. This
enables us to count the number of rational points on projective varieties with in-
finitely many rational points. Height functions were initially developed by Andre
Weil and D. G. Northcott, and a very nice exposition of the theory of the Weil
height machine can be found in [HS00]. Here we discuss height theory which was
developed in the context of Manin’s conjecture. The main references of this chapter
are [CLT10], [Sal98], and [HS00]. We closely follow the exposition of [CLT10]
1.1 Metrics and Heights
Let F be a number field and oF its ring of integers. Let Val(F ) be the set
of equivalence classes of absolute values of F . For each v ∈ Val(F ), we say v is
non-archimedean if v defines the p-adic topology on Q, and v is archimedean if it
defines the real topology on Q. We denote the corresponding completion of F with
9
respect to v by Fv and the ring of integers by ov when v is non-archimedean. If
v is non-archimedean, we identify v with the discrete valuation of ov. Let | · |v be
the associated norm on Fv, normalized by
|x|v =
ordinal absolute value |x| if Fv = R,
square of ordinal absolute value |x|2 if Fv = C,
(Npv)−v(x) if v is non-archimedean,
where pv is the unique maximal ideal of ov and Npv is the size of the finite residue
field ov/pv. With these norms, any additive Haar measure µ on the local field Fv
satisfies
µ(xΩ) = |x|vµ(Ω). (1.1)
Furthermore, the product formula
∏v∈Val(F )
|x|v = 1, (1.2)
holds for any x ∈ F× := F \ 0.
1.1.1 Metrics on local fields
Let F be a local field, i.e., R, C or a finite extension of Qp. Let X be a smooth
variety defined over F . The set of F -valued points X(F ) naturally becomes a
F -analytic manifold (see [Sal98, Section1] for definitions). Let U ⊂ X(F ) be an
analytic open set. A complex valued function f : U → C is smooth if it is C∞
when F = R or C and it is locally constant when F is a p-adic field. For any
non-vanishing analytic function f , P 7→ |f(P )| is a smooth function.
10
Let L be a line bundle on X. L(F ) is a F -analytic line bundle on X(F ). We
denote a fiber of L at P ∈ X by LP :
Definition 2 (Smooth metrics). A smooth metric on L is a collection of functions
LP (F )→ R, for all P ∈ X(F ), denoted by l 7→ ‖l‖ such that
• for any l ∈ LP (F ) \ 0, ‖l‖ > 0;
• for any l ∈ LP (F ) and x ∈ F , ‖xl‖ = |x|‖l‖;
• for any analytic open subset U ⊂ X(F ) and any non-vanishing analytic
section s ∈ Γ(U,L(F )), a function U 3 P 7→ ‖s(P )‖ ∈ R is smooth.
Note that ‖ · ‖ is smooth if and only if there exists a Zariski open covering Ui of
X and non-vanishing sections si ∈ Γ(Ui, L) such that Ui(F ) 3 P 7→ ‖si(P )‖ ∈ R is
smooth. The trivial line bundle OX admits a canonical metric defined by ‖1‖ = 1.
Let L,M be two metrized line bundles on X. We can define smooth metrics on
L⊗M and L−1 by
‖l ⊗m‖ = ‖l‖‖m‖, ‖φ‖ =|φ(l)|‖l‖
,
where P ∈ X(F ), l ∈ LP (F ), m ∈MP (F ), and φ ∈ L−1P (F ).
An important construction of metrics is given by an integral model of X. Here
we assume that F is non-archimedean. Let oF be a ring of integers of F . Suppose
that we have a flat oF -scheme X and a line bundle L on X extending X and L,
respectively. The set of integral points X (oF ) ⊂ X(F ) is an open compact set in
the analytic topology. Let U ⊂ X be a Zariski open subset trivializing L by a
section s ∈ H0(U ,L). For any integral point σ ∈ U(oF ) extending P ∈ X(F ), σ∗L
11
defines a oF -lattice in a F -vector space LP (F ). We define a metric on LP (F ) by
for any l ∈ LP (F ), ‖l‖ ≤ 1 if and only if l ∈ σ∗L(oF ).
It follows that ‖σ∗s‖ = 1 so that this metric is smooth. Since analytic open sets of
the form U(oF ) cover X (oF ), we induced a smooth metric on X (oF ). In particular,
when X is a projective model, this construction provides a smooth metric on X(F ).
Example 3. Let X = PnF = Proj(F [x0, · · · , xn]) be a projective space over F .
Consider a standard integral model X = Proj(oF [x0, · · · , xn]) → Spec(oF ). Let
L = OX (1) be the dual of the tautological bundle, which is extending L = OX(1).
The induced metric from this integral model is given by
‖l‖ =|l(x)|
max|x0|, · · · , |xn|,
where [x] = [x0 : · · · : xn] ∈ X(F ) and l ∈ L[x](F ).
1.1.2 Adelic metrics and heights
Let F be a number field and Val(F ) the set of normalized norms of F .
Definition 4 (Adelic metrics). Let X be a smooth projective variety defined over
F and L a line bundle on X. An adelic metric on L is a collection of v-adic
smooth metrics ‖ · ‖v on the associated analytic line bundles L(Fv) on the Fv-
analytic varieties X(Fv), for all v ∈ Val(F ) such that there exist a Zariski open
subset U ⊂ Spec(oF ), a projective flat morphism X → U , and a line bundle L on
X , extending X and L respectively, and satisfying that for any non-archimedean
place v ∈ U , ‖ · ‖v is the induced metric from (X ,L).
12
Remark 5. It is well-known that two projective flat models are isomorphic at
almost all places, so they define the same metrics at these places.
Definition 6 (Heights). Let X be a smooth projective variety defined over F
and L = (L, ‖ · ‖) an adelically metrized line bundle on X. Let P ∈ X(F ) and
l ∈ LP (F ). For almost all v ∈ Val(F ), we have ‖l‖v = 1 so that
∏v∈Val(F )
‖l‖v,
absolutely converges. We define a height function HL : X(F )→ R>0 by
HL(P ) =∏
v∈Val(F )
‖l‖−1v .
This function is well-defined because of the product formula (1.2).
Example 7. Let F = Q, and we consider Example 3. We define a smooth metric
at the archimedean place ∞ by
‖l‖∞ =|l(x)|∞√
x20 + · · ·+ x2
n
,
where [x] = [x0 : · · · : xn] ∈ X(Q) and l ∈ L[x](Q). It follows from the product
formula that
HL([x]) =∏
p:prime
max|x0|p, · · · , |xn|p ·√x2
0 + · · ·+ x2n.
This coincides with a height introduced in Introduction. See the equation (1).
Let X be a smooth projective variety defined over F and L an adelically
13
metrized line bundle on X. A logarithmic function
log HL : X(F )→ R,
agrees with Weil’s classical height machine up to O(1). See [HS00, Theorem B.3.2
and Theorem B.10.7]. In particular, they share some important properties:
Theorem 8. Let X be a smooth projective variety defined over F and L = (L, ‖·‖)
an adelically metrized line bundle on X. Then we have
1. let B be the base locus of the linear system |L|. Then
log HL(P ) ≥ O(1),
for any P ∈ (X \B)(F );
2. when L is ample, the set
P ∈ X(F ) | HL(P ) ≤ B
is finite where B > 0 is any positive real number.
For a proof, see [HS00, Theorem B.3.2].
1.1.3 Adelic heights
Let F be a number field. The ring of adeles AF of F is the restricted product
of all local fields Fv for all v ∈ Val(F ), with respect to ov for non-archimedean
places v. It is a locally compact topological ring. Also it is known that F ⊂ AF is
discrete and AF/F is compact. See [Tat67] for more details.
14
Let X be a smooth variety defined over F . A locally compact topological space
X(AF ), which is called the space of adelic points on X, is defined as follows: Choose
an integral model X of X over a Zariski open subset U ⊂ Spec(oF ). Then, the set
X(AF ) consists of all (Pv) ∈∏
v∈Val(F ) X(Fv) such that Pv ∈ X (ov) for almost all
v ∈ U . In other words, X(AF ) is the restricted product of X(Fv) with respect to
X (ov) for all v ∈ U . Since such integral models are isomorphic at almost all places,
the set X(AF ) does not depend on the choice of integral models. Moreover, since
X (ov) is open and compact for all v ∈ U , we can endow the restricted product
topology on X(AF ). In particular, when X is projective, then X(AF ) =∏
vX(Fv)
and the topology is just given by the product topology. See, for example, [Sal98,
Section 4] for more details.
Definition 9 (Adelic heights). Let X be a smooth projective variety defined over
F and L = (L, ‖ · ‖) an adelically metrized line bundle on X. Choose a non-zero
global section s ∈ H0(X,L), and consider a Zariski open subset X := X \ div(s).
We define an adelic height HL, s : X(AF )→ R>0 on an adelic space X(AF ) by
HL, s(P ) :=∏
v∈Val(F )
HL, s,v(Pv), for P = (Pv) ∈ X(AF ).
where HL, s,v(Pv) := ‖s(Pv)‖−1v , which is called the local height associated to L
and s ∈ H0(X,L). For any P = (Pv) ∈ X(AF ), ‖s(Pv)‖v = 1 for almost all
non-archimedean places v so that this adelic height is well-defined. Note that the
definition does depend on the choice of sections s ∈ H0(X,L).
Adelic heights satisfy some important properties which are analogous to prop-
erties in Theorem 8:
Lemma 10. Let X = X \ div(s). Then,
15
1. A function HL, s : X(AF )→ R>0 is continuous;
2. A logarithmic function is bounded from below, i.e.,
log HL, s(P ) ≥ O(1),
for any P ∈ X(AF );
3. when L is ample, the set
P ∈ X(AF ) | HL, s(P ) ≤ B ,
is compact in the adelic topology.
For proofs, see [CLT10, Lemma 2.3.2].
1.2 Measures
Let F be a number field. For v ∈ Val(F ), Fv is a locally compact topological
field, and we have an additive Haar measure µv on Fv. We fix µv so that µv(ov) = 1
for almost all non-archimedean places v ∈ Val(F ).
1.2.1 Measures on local fields
Let F be a local field and X a smooth variety defined over F . Let n =
dimX and ω ∈ H0(X,KX) a n-form. Consider a F -analytic manifold X(F ).
Let x1, · · · , xn be F -analytic coordinates on an analytic open subset U ⊂ X(F ).
16
Then ω has the following form:
ω = f(x)dx1 ∧ · · · ∧ dxn,
where f(x) is an analytic function. The product measure µn determines a measure
on U , and we denote this measure by |dx1| · · · |dxn|. We define a measure |ω| by
|ω| = |f(x)||dx1| · · · |dxn|.
This measure does not depend on the choice of analytic coordinates. Thus, by
using a partition of unity, we can extend a measure |ω| to X(F ). Slightly different
construction of |ω| can be found in [Sal98, Example 1.13].
Definition 11 (Measures induced from metrics). Let X be a smooth variety de-
fined over F . Choose a metrization ‖ · ‖ on the canonical line bundle KX . For
any local non-vanishing top form ω, we define a measure by |ω|/‖ω‖. Since this
construction does not depend on the choice of ω, we can patch these measures
and define a global measure on X(F ). We denote this measure by τX . For more
details, see [CLT10, Section 2] or [Sal98, Section 1].
The following formula is due to Weil [Wei82]:
Theorem 12. Let F be a p-adic field. Let X be a smooth variety defined over F
and X a smooth oF -model of X. We consider a smooth metric on KX which is
induced from X . Suppose that µ(oF ) = 1. Then, we have
∫X (oF )
τX = q−dimX |X (k)|,
where k is the residue field of oF and q is the size of k.
17
For a proof, see [Wei82] or [Sal98, Corollary 2.15].
1.2.2 Tamagawa measures on adelic spaces
Let F be a number field. For any non-archimedean place v, we denote the
residue field of ov by kv and the size of kv by qv. Let X be a smooth projective
variety defined over F . In this subsection, we always assume that
H1(X,OX) = H2(X,OX) = 0.
This holds when X is a smooth rationally connected variety. Also note that the
dimensions of these cohomology spaces are birationally invariant (see [CLT10, Re-
mark 2.4.8]). The assumption guarantees that Pic(X)Q = NS(X)Q. We consider
an adelically metrized canonical line bundle KX = (KX , ‖ · ‖). One might hope
to define a finite measure on a compact topological space X(AF ) =∏
vX(Fv) by
considering ∏v∈Val(F )
τX,v,
where τX,v is a measure on X(Fv), induced by ‖ · ‖v. However, Theorem 12 claims
that for almost all non-archimedean places v ∈ Val(F ),
τX,v(X(Fv)) =|X (kv)|qdimXv
,
where X is a flat projective oF -model of X, and the infinite product
∏v∈Val(F )
τX,v(X(Fv)),
18
never converges absolutely. Indeed, by Deligne’s theory of the Weil conjecture, one
can conclude that for almost all non-archimedean places v,
|X (kv)|qdimXv
= 1 +1
qvTr(Frv | Pic(Xkv)Q) +O
(1
q3/2v
), (1.3)
where Frv is the geometric Frobenius acting on Pic(Xkv)Q. See [Pey95, p. 117].
Thus, we need to introduce a regularization to obtain convergence:
Definition 13 (the Artin L-functions). Let X be a smooth projective model of
X over a Zariski open subset U ⊂ Spec(oF ) such that every fiber is geometrically
connected. For any v ∈ U , we define the local L-function at v ∈ U by
Lv(s,Pic(Xkv)Q) := det(1− q−sv Frv | Pic(Xkv)Q)−1,
and we define the global L-function by
LU(s,Pic(XF )Q) :=∏v∈U
Lv(s,Pic(Xkv)Q).
This infinite product converges absolutely and uniformly for <(s) > 1. Moreover,
LU(s,Pic(XF )Q) has a meromorphic continuation to the entire plane with a pole
of order r = Pic(X) at s = 1. See [Pey95, Lemma 2.2.5].
[CLT10, Lemma 2.4.5] claims that Lv(σ,Pic(Xkv)Q) is a positive real number
for any σ > 0. We define convergence factors λv by
λv =
1/Lv(1,Pic(Xkv)Q) if v ∈ U ,
1 if v /∈ U .
19
Then we have
Theorem 14. Assume that H1(X,OX) = H2(X,OX) = 0. The infinite product
∏v∈Val(F )
λvτX,v(X(Fv)),
converges absolutely.
Proof. Since we have,
det(1− q−1v Frv | Pic(Xkv)Q) = 1− q−1
v Tr(Frv | Pic(Xkv)Q) +O(q−3/2v ),
it follows from 1.3 that
λvτX,v(X(Fv)) = 1 +O(q−3/2v ).
Thus, our assertion follows from this. For more details, see [CLT10, Theorem
2.4.7].
Let
LU∗ (1,Pic(XF )Q) := lims→1
(s− 1)rLU(s,Pic(XF )Q) > 0.
Definition 15 (Tamagawa measure). Let X be a smooth projective variety defined
over F such that
H1(X,OX) = H2(X,OX) = 0.
The Tamagawa measure on the adelic space X(AF ) is defined as follows:
τX := LU∗ (1,Pic(XF )Q)∏
v∈Val(F )
λvτX,v.
20
Note that this definition does not depend on the choice of the model X . For more
information regarding this measure, see [Sal98, Definition 6.12]. [CLT10] provides
generalizations of Tamagawa measures for an open variety U with a smooth com-
pactification X satisfying the above assumptions. See [CLT10, Definition 2.4.11].
21
Chapter 2
Manin’s conjecture and Height
zeta functions
Manin’s conjecture was proposed in [FMT89] and [BM90], and more precise re-
finement concerning the leading constant of the asymptotic formula was introduced
in [Pey95] and [BT98b]. The conjecture has been confirmed to be true for large
classes of varieties, including flag varieties, complete intersections of small degree,
toric varieties, certain del Pezzo surfaces, and other equivariant compactifications
of algebraic groups, and these results are based on several different methods. One
of them is the Height zeta functions method which was introduced in [FMT89].
In this chapter, we describe the general picture of Manin’s conjecture, and we
introduce Height zeta functions. Then we will discuss the strategy to obtain mero-
morphic continuations of Height zeta functions of equivariant compactifications of
algebraic groups by using spectral theory on adelic groups. The main references
of this chapter are [Tsc09] and [CL10].
22
2.1 Manin’s conjecture
2.1.1 Geometric invariants
In this subsection, we introduce global geometric invariants which play central
roles in the context of Manin’s conjecture. Here we assume that the ground field
is an algebraically closed field of characteristic zero.
Let X be a smooth projective variety. We denote its Picard group by Pic(X).
Let Pic0(X) be the subgroup consisting of isomorphism classes of line bundles
algebraically equivalent to the trivial bundle OX . The Neron-Severi group NS(X)
is defined by
NS(X) = Pic(X)/Pic0(X).
It is well-known that NS(X) is a finitely generated abelian group, and its rank
is called Picard rank, denoted by ρ(X). The Neron-Severi group is unchanged by
algebraically closed base extension. See [MP09, Proposition 3.1]. We denote the
real Neron-Severi group NS(X)⊗ R by NS(X)R. We use
Λeff(X) ⊂ NS(X)R,
to denote the cone of pseudo-effective divisors, i.e., the closure of the cone of
effective Q-divisors on X in the real Neron-Severi group NS(X)R. This cone plays
a central role in the Minimal Model Program. See, for example, [HK00].
Suppose that X is a smooth Fano variety, i.e., −KX is ample. Kodaira vanish-
ing theorem implies that H1(OX) = 0, so we have Pic(X)Q = NS(X)Q. The recent
development of the Minimal Model Program [BCHM10, Corollary 1.3.1] concludes
that X is a Mori dream space (see [HK00] for a definition) so that in particular, the
23
cone of pseudo-effective divisors Λeff(X) is finitely generated by effective divisors.
Let L be a big line bundle, i.e., the numerical class [L] is in the interior of
Λeff(X) (see [Laz04a, Section 2.2] for a definition and other characterizations of
big line bundles). We define invariants a(L) and b(L) as follows:
a(L) := mina ∈ R | a[L] + [KX ] ∈ Λeff(X),
b(L) := the codimension of the minimal face of Λeff(X) containing a(L)L+KX .
Since Λeff(X) is finitely generated by effective divisors, a(L) is a rational number.
We will explore some birational geometric aspects of these invariants in Part II.
2.1.2 Manin’s conjecture
Let F be a number field. Let X be a smooth projective variety defined over
F and L = (L, ‖ · ‖) an adelically metrized big line bundle on X. Every big line
bundle is linearly equivalent to a sum of an ample Q-divisor A and a effective
Q-divisor E (see [Laz04a, Corollary 2.2.7]). Let B be the stable base locus of E
(see [Laz04a, Definition 2.1.20]). Note that we can choose A and E to be defined
over F so that B is also defined over F . Theorem 8 ensures that for any Zariski
open subset X ⊂ X \B, the following counting function of the number of rational
points of bounded height is finite:
N(X,L,B) := #P ∈ X(F ) | HL(P ) ≤ B <∞,
where B > 0 is any positive real number. Note that in general, it is possible that
N(X,L,B) = ∞ unless L is ample, so it is important to consider a sufficiently
24
small Zariski open subset X.
By the general principle of Diophantine geometry, one hopes to relate the
asymptotic formula for the counting function N(X,L,B) as B → ∞, for a suit-
able Zariski open subset X, to global geometric invariants of X when −KX is
sufficiently positive, e.g., −KX is ample. Manin and others proposed the following
conjecture in [FMT89] and [BM90]:
Conjecture 16. Let X be a smooth Fano variety defined over F and L an adel-
ically metrized big line bundle on X. Then, after passing to a finite extension,
there exists a Zariski open subset X ⊂ X and a positive constant c(L) > 0 such
that
N(X,L,B) = c(L)Ba(L) log(B)b(L)−1(1 + o(1)),
as B → +∞ where a(L) and b(L) were introduced in Subsecton 2.1.1. An adelic
interpretation of the constant c(L) will be discussed in the next subsection 2.1.3.
Example 17. Let F be a number field and X = PnF . We consider the following
height function associated to O(1)X :
H([x]) :=∏
v∈Val(F )
max |x0|v, · · · , |xn|v ,
where [x] = [x0 : · · · : xn] ∈ X(F ). Note that this height function is slightly
different from the height defined in Example 7. The following formula is due to
Schanuel [Sch79]:
N(X,H,B) = c(F, n)Bn+1(1 + o(1)).
25
The constant c(F, n) is given by
c(F, n) =hR/ω
ζF (n+ 1)
(2r1(2π)r2√
DF
)n+1
(n+ 1)r1+r2−1,
where each invariant is given by
h : class number of F ,
R : regulator of F ,
ω : number of roots of unity in F ,
ζF : zeta function of F ,
r1 : number of real embeddings of F ,
r2 : number of complex embeddings of F ,
DF : absolute value of the discriminant of F/Q.
See [Sch79] or [HS00, Section B.6].
Remark 18. It is essential to take a sufficiently small Zariski open subset X ⊂ X
even when L is ample. For example, let X be a smooth cubic surface in P3. The
anticanonical line bundle L = −KX gives us an embedding into P3. Manin’s con-
jecture predicts that, at least after taking a finite extension, the counting function
N(X,L,B) behaves like B log(B)6 for a suitable Zariski open subset X. On the
other hand, there are 27 lines on X, and the number of rational points on these
lines behaves like B2. Thus rational points will be accumulating on these lines,
and we need to remove all lines. Conjecturely, we believe that removing all lines
will suffice. One of the concrete examples, for which Manin’s conjecture has been
26
confirmed, is the singular cubic surface X ⊂ P3 defined by
xyz = w3.
This is a singular cubic surface with three isolated singularities of type A2, which
contains three lines x = w = 0, y = w = 0, z = w = 0. Moreover,
X is a toric variety, so we can apply results of [BT98a] and [BT96a]. Let X
be the complement of three lines. [BT98a] and [BT96a] conclude that for the
anticanonical line bundle L = −KX , the counting function behaves like
N(X,L,B) ∼ cB log(B)6,
which coincides with the prediction of Manin’s conjecture. However, the num-
ber of rational points on three lines behaves like B2, so rational points are more
accumulating on these lines than the complement.
Example 19. Conjecture 16 admits counter examples found by Batyrev and
Tschinkel in [BT96b]. Let X be a hypersurface in P3x × P3
y defined by
3∑i=0
xiy3i = 0.
X is a smooth Fano 5-fold of Picard rank 2, and the first projection π1 to P3x
provides us a diagonal cubic surfaces-fibration on X. Manin’s conjecture asserts
that for L = −KX , the counting function N(X,L,B) behaves like B log(B) for a
suitable Zariski open subset X ⊂ X. On the other hand, a general fiber Xx of
π1 is a diagonal cubic surface. Suppose that F contains Q(√−3). Then the rank
of Pic(Xx) is 7 whenever xi’s are cubic in F , and a more precise conjecture for
27
del Pezzo surfaces predicts that the number of F -rational points on these fibers
behaves like B log(B)6. Thus Conjecture 16 is not consistent. Actually Batyrev
and Tschinkel proved that X cannot satisfy Conjecture 16. This failure is also
related to failure of the balance property, which we will discuss in Part II.
2.1.3 Tamagawa number
Let F be a number field and X a smooth Fano variety defined over F . Consider
an adelically metrized canonical line bundle KX = (KX , ‖ · ‖). In this subsection,
we discuss an adelic interpretation of the constant c(−KX) in Conjecture 16, which
was proposed by Peyre in [Pey95].
For any v ∈ Val(F ), we normalize a Haar measure µv in the following way:
µv =
ordinal Lebesgue measure dx if Fv = R,
2dudv where x = u+ iv if Fv = C,
a measure normalized by µv(ov) = (Ndv)− 1
2 if v is non-archimedean,
where dv is the local absolute different. See [Tat67].
Let τX be a Tamagawa measure introduced in Subsection 1.2.2.
Definition 20. Let X(F ) ⊂ X(AF ) be the closure in the adelic topology. We
define the Tamagawa number τ(−KX) by
τ(−KX) :=
∫X(F )
τX .
Let (A,Λ) be a pair consisting of a lattice and a strictly convex closed cone Λ ⊂
AR, i.e., Λ ∩ −Λ = 0. Let (A∗,Λ∗) be the dual of (A,Λ), i.e., A∗ = Hom(A,Z)
28
and
Λ∗ := a∗ ∈ A∗R | (a∗, a) ≥ 0 for any a ∈ Λ.
We normalize a Haar measure da∗ on A∗R by vol(A∗R/A∗) = 1.
Definition 21. The X -function of Λ is defined as the integral:
XΛ(s) =
∫Λ∗e−(s,a∗)da∗.
This integral converges absolutely and uniformly when <(s) is contained in any
compact subset in Λ where Λ is the interior of Λ. See [BT98a, Section 5] for
more details.
Definition 22. We define α(X) as follows:
α(X) = XΛeff(X)(−KX).
Conjecture 23. The leading constant c(−KX) in Conjecture 16 is given by
c(−KX) =α(X)β(X)τ(−KX)
(rk Pic(X)− 1)!,
where β(X) = #Br(X)/Br(F ). Here Br(X) and Br(F ) denote Brauer groups of
X and F respectively. See [Man86] for more details.
Example 24. [Pey95, Proposition 6.1.1] Let X = PnF and consider the height
29
introduced in Example 17. Then, α(X) = 1/(n+ 1) and β(X) = 1. We have
τX,v(X(Fv)) =
2n(n+ 1) if Fv = R,
(2π)n(n+ 1) if Fv = C,
(Ndv)−n/2q−nv
qn+1v −1qv−1
if v is non-archimedean.
The Artin L-function is given by
Lv(s,Pic(Xkv)) = (1− q−sv )−1, L(s,Pic(XF )) = ζF (s).
Since X(F ) = X(AF ), we conclude that
c(−KX) =hR/ω
ζF (n+ 1)
(2r1(2π)r2√
DF
)n+1
(n+ 1)r1+r2−1,
which coincides with the constant in Example 17.
Generalization of the constant c(L) for arbitrary big line bundles is obtained
by Batyrev and Tschinkel. See [BT98b].
2.2 Height zeta functions
2.2.1 Height zeta functions
Let F be a number field and X a smooth projective variety defined over F .
Let L = (L, ‖ · ‖) be an adelically metrized big line bundle on X. Inspired by
Conjecture 16, we consider the following Dirichlet series, which is called the Height
zeta function:
Z(X,L, s) =∑
P∈X(F )
HL(P )−s,
30
where X is a Zariski open subset and s ∈ C.
Definition 25 (Abscissa of convergence). The abscissa of convergence αX(L) is
αX(L) := inf
σ ∈ R
∣∣∣∣∣∣∑
P∈X(F )
HL(P )−s absolutely converges for <(s) > σ.
.
When the above set is empty, we let αX(L) = +∞. Since HL(P ) is a positive real
number, αX(L) is also equal to the infimum of σ such that Z(X,L, s) converges
for <(s) > σ. It could be that 0 ≤ αX(L) ≤ +∞ or αX(L) = −∞. The latter
happens exactly when X(F ) is finite. Also note that αX(L) does not depend on
the choice of metrizations. Z(X,L, s) is a holomorphic function in the right half
plane <(s) > αX(L), and Z(X,L, s) cannot extend holomorphically to any larger
right half plane <(s) > αX(L)− ε for any ε > 0. See [HR64] for more details.
Lemma 26. We have
1. if L is ample, then αX(L) < +∞ for any Zariski open set X;
2. if L is big, then αX(L) < +∞ for a sufficiently small Zariski open set X.
Proof. When L is ample, Example 17 implies that the counting function N(X,L,B)
grows at most polynomially. [HR64, Theorem 8] concludes that αX(L) < +∞.
Suppose that L is big. A big line bundle L can be expressed as a sum of an ample
Q-divisor A and a effective Q-divisor E. Let B be the stable base locus of E and
X ⊂ X \ B a Zariski open subset. We fix adelic metrizations A = (A, ‖ · ‖) and
E = (E, ‖ · ‖). It follows from Theorem 8 that
HL(P ) HA(P ),
31
for any P ∈ X(F ). Our assertion follows from this.
The connection between Manin’s conjecture and Height zeta functions is given
by a Tauberian theorem:
Theorem 27 (Tauberian theorem). Let hn be an increasing sequence of positive
real numbers such that limn→∞ hn = +∞. Let an be another sequence of positive
real numbers and consider
Z(s) :=+∞∑n=1
anhsn.
Assume that this series converges absolutely to an analytic function for <(s) > a
where a is some positive real number. Furthermore suppose that Z(s) has the
following meromorphic continuation:
Z(s) =f(s)
(s− a)b,
where f(s) is an analytic function in the right half plane <(s) > a− ε, for some
ε > 0, b ∈ N, and f(a) = c > 0. Then,
N(B) :=∑hn≤B
an =c
a(b− 1)!Ba log(B)b−1(1 + o(1)),
as B→∞.
See [Ten95, Section II.7, Theorem 15]. Thus our goal here is to obtain the
meromorphic continuation of a Height zeta function. This task is extremely hard
in general, and we will discuss how one can attack this problem when X is an
equivariant compactification of a linear algebraic group.
The following theorem is also important in the analysis of Height zeta functions.
32
For U ⊂ Rn, we define a tube domain:
TU := s ∈ Cn | <(s) ∈ U.
Then we have
Theorem 28 (Convexity principle). Let U ⊂ Rn be a connected open domain.
Suppose that f is a holomorphic function on TU . Let U be the convex hull of U .
Then f has a holomorphic extension to U .
See [FG02, Exercises in Section II-7].
2.2.2 Height zeta functions of equivariant compactifica-
tions of algebraic groups
In this subsection, we explain the general strategy, which is described in [Tsc09,
Subsection 6.3], to obtain a meromorphic continuation of a Height zeta function
of an equivariant compactification of a connected linear algebraic group by using
spectral theory on an adelic group.
Let F be a number field and G a connected linear algebraic group defined
over F . Suppose that X is a smooth projective equivariant compactification of G
defined over F , i.e, X is a smooth projective variety with a group action by G such
that the group action admits the open dense orbit which is isomorphic to G. Since
X is a rational variety, we have Pic(X)Q = NS(X)Q. We assume that G is acting
on X from the right.
Choose a basis of Pic(X)Q consisting of ample line bundles L1, · · · , Lr and fix
33
adelic metrizations Li = (Li, ‖ · ‖). For each Li, we have a height function
HLi : X(F )→ R>0,
and we define a height system H : Pic(X)C×X(F )→ C by extending linearly, i.e.,
H
(∑i
siLi, P
)=∏i
HLi(P )si .
We are interested in a meromorphic continuation of a Height zeta function
Z(s) =∑
γ∈G(F )
H(s, γ)−1.
This infinite series converges absolutely and uniformly to a holomorphic function
when <(si) 0 for any i.
Step 1 : One defines an adelic height pairing
H =∏v
Hv : Pic(X)C ×G(AF )→ C,
whose restriction to Pic(X)× G(F ) → R>0 descends to the above height system.
One of important properties of the height pairing is K-invariance. The fact that
the right action extends to X implies that the local height
Hv : Pic(X)C ×G(Fv)→ C,
is invariant under the right action of a compact subgroup Kv ⊂ G(Fv) for any
non-archimedean place v. Moreover, Kv = G(ov) for almost all non-archimedean
places v. This property plays a significant role in the analysis of a Height zeta
34
function. Let K :=∏
v finite Kv.
Step 2 : We consider a Height zeta function on an adelic space:
Z(s, g) =∑
γ∈G(F )
H(s, γg)−1,
where s ∈ Pic(X)C and g ∈ G(AF ). One can prove that this infinite series con-
verges to a function which is continuous in g and is holomorphic in s when <(s) is
sufficiently large. Moreover, one can prove that
Z(s, g) ∈ L2(G(F )\G(AF ))K,
for a sufficiently large <(s). Thus, we can apply spectral theory, i.e., the decom-
position of L2-space into unitary irreducible representations, and we obtain
Z(s, g) =∑π
Zπ(s, g), (2.1)
where π runs over all unitary irreducible representations occurring in L2(G(F )\G(AF ))K.
Step 3 : Analysis of each term in (2.1) can be done by using geometric integral
techniques developed in [CLT10]. For example, when π is trivial, then the trivial
part is given by
Z0(s, g) =
∫G(AF )
H(s, g)−1dg =∏v
∫G(Fv)
Hv(s, gv)−1dgv,
where dgv is the right invariant Haar measure on G(Fv). The study of this type
of Height integrals has been carried out in complete generality in [CLT10], and
in many examples, this trivial part coincides with the main term of a Height zeta
35
function. Other terms also can be studied by using techniques in [CLT10], and
one hopes to obtain a meromorphic continuation from this spectral decomposition
with geometric integration techniques.
This strategy has been carried out and lead to a proof of Manin’s conjecture
for the following varieties:
• toric varieties [BT95], [BT98b], [BT96a];
• equivariant compactifications of additive groups Gna [CLT02];
• equivariant compactifications of unipotent groups [ST04], [ST];
• wonderful compactifications of semi-simple groups of adjoint type [STBT07].
Moreover, applications of Langland’s theory of Eisenstein series allowed proofs of
Manin’s conjecture for flag varieties [FMT89], their twisted products [Str01], and
horospherical varieties [ST99], [CLT01].
In the next chapter, we apply this strategy to a relatively simple group, e.g.,
the ax+ b-group. However, our results contain new ingredients. In previous work,
the proofs of a suitable meromorphic continuation of such spectral decomposition
relied on the fact that both the left and right action of G on itself extended to
actions on X. This ensured that the sum in (2.1) was actually over a discrete
set. In the noncommutative situation, this does not hold, generally. In [TT12],
we conducted analysis that relied on tight control of local integrals with rapidly
oscillating phase. Applying analytic techniques from [CLT10], we succeeded in ob-
taining a meromorphic continuation of a Height zeta function for one-sided actions
under some geometric conditions.
36
Chapter 3
Height zeta functions of
equivariant compactifications of
semi-direct products of algebraic
groups
In this chapter, we discuss results of [TT12]. First, we review general geometric
facts concerning equivariant compactifications of solvable algebraic groups. Then
we discuss the main terms of Height zeta functions of equivariant compactifica-
tions of solvable algebraic groups and confirm that the main terms coincide with
the prediction of Manin’s conjecture. In the last section, we discuss Height zeta
functions of equivariant compactifications of the ax + b-group and prove Manin’s
conjecture under some geometric conditions. We closely follow the exposition of
[TT12].
37
3.1 Geometry of equivariant compactifications
In this section, we review geometric facts concerning equivariant compactifi-
cations of connected solvable linear algebraic groups. Here we assume that the
ground field is an algebraically closed field of characteristic zero. This section is
extracted from [TT12, Section 1].
3.1.1 Equivariant compactifications
Let G be a connected linear algebraic group. In dimension 1, the only examples
are the additive group Ga := Spec(k[x]) and the multiplicative group Gm :=
Spec(k[x, x−1]). Let X∗(G) := Hom(G,Gm), the group of algebraic characters of
G. For any connected linear algebraic group G, this is a torsion-free Z-module of
finite rank.
Let X be a projective equivariant compactification of G. When X is normal,
then it follows from Hartog’s theorem (see [Har77, Proposition 6.3A]) that the
boundary
D := X \G,
is a Weil divisor. Moreover, after applying equivariant resolution of singularities,
if necessary, we may assume that X is smooth and that the boundary
D = ∪ιDι,
is a divisor with normal crossings. Here Dι are irreducible components of D. Let
PicG(X) be the group of equivalence classes of G-linearized line bundles on X.
(See [MFK94, Chapter 1, Section 3] for a definition.)
38
Generally, we will identify divisors, associated line bundles, and their classes in
Pic(X), resp. PicG(X).
The following proposition is taken from [TT12, Proposition 1.1]:
Proposition 29. Let X be a smooth and proper equivariant compactification of a
connected solvable linear algebraic group G. Then,
1. we have an exact sequence
0→ X∗(G)→ PicG(X)→ Pic(X)→ 0,
2. PicG(X) = ⊕ι∈IZDι, and
3. the closed cone of pseudo-effective divisors of X is spanned by the boundary
components:
Λeff(X) =∑ι∈I
R≥0Dι.
Proof. The first claim follows from the proof of [MFK94, Proposition 1.5]. The
crucial point is to show that the Picard group ofG is trivial. As an algebraic variety,
a connected solvable group is a product of an algebraic torus and an affine space.
The second and third assertions hold since every finite-dimensional representation
of a solvable group has a fixed vector.
The following proposition is taken from [TT12, Proposition 1.2]:
Proposition 30. Let X be a smooth and proper equivariant compactification for
the left action of a connected linear algebraic group. Then the right invariant top
39
degree differential form ω on X := G ⊂ X satisfies
−div(ω) =∑ι∈I
dιDι,
where dι > 0. The same result holds for the right action and the left invariant
form.
Proof. This fact was proved in [HT99, Theorem 2.7] or [CLT02, Lemma 2.4]. Sup-
pose that X has the left action. Let g be the Lie algebra of G. For any ∂ ∈ g, the
global vector field ∂X on X is defined by
∂X(f)(x) = ∂gf(g · x)|g=1,
where f ∈ OX(U) and U is a Zariski open subset of X. Note that this is a right
invariant vector field on X = G. Let ∂1, · · · , ∂n be a basis for g. Consider a global
section of det TX ,
δ := ∂X1 ∧ · · · ∧ ∂Xn ,
which is the dual of ω on X. The proof of [CLT02, Lemma 2.4] implies that δ
vanishes along the boundary. Thus our assertion follows.
The following proposition is taken from [TT12, Proposition 1.3]:
Proposition 31. Let X be a smooth and proper equivariant compactification of a
connected linear algebraic group. Let f : X → Y be a birational morphism to a
normal projective variety Y . Then Y is an equivariant compactification of G such
that the contraction map f is a G-morphism.
Proof. This fact was proved in [HT99, Corollary 2.4]. Choose an embedding Y →
40
PN , and let L be the pullback of O(1) on X. Since Y is normal, Zariski’s main
theorem implies that the image of the complete linear series |L| is isomorphic to
Y . According to [MFK94, Corollary 1.6], after replacing L by a multiple of L, if
necessary, we may assume that L carries G-linearizations. Fix one G-linearization
of L. This defines the action of G on H0(X,L) and so on P(H0(X,L)∗). Now note
that the morphism
Φ|L| : X → P(H0(X,L)∗),
is a G-morphism with respect to this action. Thus our assertion follows.
3.1.2 Semi-direct products of Ga and Gm
The simplest solvable groups are Ga and Gm, as well as their products. New
examples arise as semi-direct products. For example, let
ϕd : Gm → Gm = GL1,
a 7→ ad
and put
Gd := Ga oϕd Gm,
where the group law is given by
(x, a) · (y, b) = (x+ ϕd(a)y, ab).
Note that Gd ' G−d. Now we collect several results illustrating specific phenomena
connected with noncommutativity of Gd and with the necessity to distinguish
actions on the left, on the right, or on both sides. These play a role in the analysis
41
of height zeta functions in following sections. The following lemma is taken from
[TT12, Lemma 1.4]:
Lemma 32. Let X be a biequivariant compactification of a semi-direct product
GoH of linear algebraic groups. Then X is a one-sided (left- or right-) equivariant
compactification of G×H.
Proof. Fix one section s : H → GoH. Define a left action by
(g, h) · x = g · x · s(h)−1,
for any g ∈ G, h ∈ H, and x ∈ X.
In particular, there is no need to invoke noncommutative harmonic analysis in
the treatment of height zeta functions of biequivariant compactifications of general
solvable groups since such groups are semi-direct products of tori with unipotent
groups and the lemma reduces the problem to a one-sided action of the direct
product. Height zeta functions of direct products of additive groups and tori can
be treated by combining the methods of [BT98a] and [BT96a] with [CLT02], see
Theorem 48. However, Manin’s conjectures are still open for one-sided actions of
unipotent groups, even for the Heisenberg group.
The next observation is that the projective plane P2 is an equivariant compact-
ification of Gd, for any d. Indeed, the embedding
(x, a) 7→ (x : a : 1) ∈ P2
defines a left-sided equivariant compactification, with boundary a union of two
42
lines. The left action is given by
(x, a) · (x0 : x1 : x2) 7→ (adx0 + xx2 : ax1 : x2).
The following proposition is taken from [TT12, Proposition 1.5]:
Proposition 33. If d 6= 1, 0, or −1, then P2 is not a biequivariant compactification
of Gd.
Proof. Assume otherwise. Proposition 29 implies that the boundary must consist
of two irreducible components. Let D1 and D2 be the two irreducible boundary
components. Since O(KP2) ∼= O(−3), it follows from Proposition 30 that either
both components D1 and D2 are lines or one of them is a line and the other a
conic. Let ω be a right invariant top degree differential form. Then ω/ϕd(a) is a
left invariant differential form. If one of D1 and D2 is a conic, then the divisor of
ω takes the form
−div(ω) = −div(ω/ϕd(a)) = D1 +D2,
but this is a contradiction. If D1 and D2 are lines, then without loss of generality,
we can assume that
−div(ω) = 2D1 +D2 and − div(ω/ϕd(a)) = D1 + 2D2.
However, div(a) is a multiple of D1 −D2, which is also a contradiction.
Combining this result with Proposition 31, we conclude that a del Pezzo surface
is not a biequivariant compactification of Gd, for d 6= 1, 0, or,−1.
Another sample result in this direction is the following proposition which is
taken from [TT12, Proposition 1.6]:
43
Proposition 34. Let S be the singular quartic del Pezzo surface of type A3 + A1
defined by
x20 + x0x3 + x2x4 = x1x3 − x2
2 = 0
Then S is a one-sided equivariant compactification of G1, but not a biequivariant
compactification of Gd if d 6= 0.
Proof. For the first assertion, see [DL10, Section 5]. Assume that S is a biequivari-
ant compactification of Gd. Let π : S → S be its minimal desingularization. Then
S is also a biequivariant compactification of Gd because the action of Gd must fix
the singular locus of S. See [DL10, Lemma 4]. It has three (−1)-curves L1, L2,
and L3, which are the strict transforms of
x0 = x1 = x2 = 0, x0 + x3 = x1 = x2 = 0, and x0 = x2 = x3 = 0,
respectively, and has four (−2)-curves R1, R2, R3, and R4. The nonzero intersec-
tion numbers are given by:
L1.R1 = L2.R1 = R1.R2 = R2.R3 = R3.L3 = L3.R4 = 1.
Since the cone of curves is generated by the components of the boundary, these
negative curves must be in the boundary because each generates an extremal ray.
Since the Picard group of S has rank six, it follows from Proposition 29 that the
number of boundary components is seven. Thus, the boundary is equal to the
union of these negative curves.
Let f : S → P2 be the birational morphism which contracts L1, L2, L3, R2, and
R3. According to Proposition 31, this induces a biequivariant compactification on
44
P2. The birational map f π−1 : S 99K P2 is given by
S 3 (x0 : x1 : x2 : x3 : x4) 7→ (x2 : x0 : x3) ∈ P2.
The images of R1 and R4 are y0 = 0 and y2 = 0 and we denote them by D0
and D2, respectively. The images of L1 and L2 are (0 : 0 : 1) and (0 : 1 : −1),
respectively; so that the induced group action on P2 must fix (0 : 0 : 1), (0 : 1 : −1),
and D0 ∩D2 = (0 : 1 : 0). Thus, the group action must fix the line D0, and this
fact implies that all left and right invariant vector fields vanish along D0. It follows
that
−div(ω) = −div(ω/ϕd(a)) = 2D0 +D2,
which contradicts d 6= 0.
The following examples are taken from [TT12, Examples 1.7 and 1.8]:
Example 35. Let l ≥ d ≥ 0. The Hirzebruch surface Fl = PP1((O ⊕O(l))∗) is a
biequivariant compactification of Gd. Indeed, we may take the embedding
Gd → Fl
(x, a) 7→ ((a : 1), [1⊕ xσl1]),
where σ1 is a section of the line bundle O(1) on P1 such that
div(σ1) = (1 : 0).
Let π : Fl → P1 be the P1-fibration. The right action is given by
((x0 : x1), [y0 ⊕ y1σl1]) 7→ ((ax0 : x1), [y0 ⊕ (y1 + (x0/x1)dxy0)σl1]),
45
on π−1(U0 = P1 \ (1 : 0)) and
((x0 : x1), [y0 ⊕ y1σl0]) 7→ ((ax0 : x1), [aly0 ⊕ (y1 + (x1/x0)l−dxy0)σl0]),
on U1 = π−1(P1 \ (0 : 1)). Similarly, one defines the left action. The boundary
consists of three components: two fibers f0 = π−1((0 : 1)), f1 = π−1((0 : 1)) and
the special section D characterized by D2 = −l.
Example 36. Consider the right actions in Examples 35. When l > d > 0, these
actions fix the fiber f0 and act multiplicatively, i.e., with two fixed points, on the
fiber f1. Let X be the blowup of two points (or more) on f0 and of one fixed point
P on f1 \D. Then X is an equivariant compactification of Gd which is neither a
toric variety nor a G2a-variety. Indeed, there are no equivariant compactifications
of G2m on Fl fixing f0, so X cannot be toric. Also, if X were a G2
a-variety, we
would obtain an induced G2a-action on Fl fixing f0 and P . However, the boundary
consists of two irreducible components and must contain f0, D, and P because D
is a negative curve. This is a contradiction.
In Section 3.3, we prove Manin’s conjecture for X with l ≥ 3.
3.2 The main terms of Height zeta functions of
equivariant compactifications of solvable al-
gebraic groups
In this section, we discuss the main term of a Height zeta function of an equiv-
ariant compactification of a solvable group. This subsection is based on results in
46
[TT12, Section 2].
Let F be a number field and G a connected solvable linear algebraic group
defined over F . (See [Bor85] for a definition.) We may fix a flat group scheme G
over Spec(oF ) whose the generic fiber is isomorphic to G. For example, choose an
embedding G → GLn and take the Zariski closure of G in GLn over Spec(oF ). We
frequently denote the set of integral points G(ov) by G(ov) for a non-archimedean
place v. For connected solvable linear algebraic groups, the maximal tori are all
conjugate to each other. Let T be a maximal torus. For simplicity, we assume
that T is split, i.e., T ∼= Grm over F . Let U be the set of unipotent elements in G.
It is known that U is a closed, connected, nilpotent, normal subgroup of G, and
G/U is isomorphic to T so that G is a semi-direct product of T and U . For more
information, see [Spr81, Subsection 6.3]. From now on, we always assume that U
is unipotent and hence U is a closed subgroup of the group of upper triangular
matrices whose diagonal entries are 1.
Let X be a smooth projective equivariant compactification of G defined over F .
We assume that G is acting on X from the right. Consider the boundary divisor
D = X \G and its irreducible decomposition
D = ∪ι∈IDι.
We assume that the irreducible components are geometrically irreducible and a
boundary divisor D =∑
ιDι is a divisor with strict normal crossings, i.e., compo-
nents are geometrically meeting transversely.
47
3.2.1 Height pairings and Height zeta functions
Let Lι = O(Dι) and fι a F -section corresponding to an effective divisor Dι.
Fix adelic metrizations Lι = (Lι, ‖ · ‖). We consider the local height functions
defined in Subsection 1.1.3
HLι,fι,v : G(Fv)→ R>0,
for v ∈ Val(F ). According to Proposition 29, there is a canonical isomorphism
PicG(X) ∼= ⊕ι∈IZDι.
With this identification, we define the local height pairing
Hv : PicG(X)C ×G(Fv)→ C
and the adelic height pairing
H : PicG(X)C ×G(AF )→ C
as follows:
Hv(s, gv) =∏ι
HLι,fι,v(gv)sι , H(s, g) =
∏v∈Val(F )
Hv(s, gv),
where s =∑
ι sιDι and g = (gv)v∈Val(F ) ∈ G(AF ). By adjusting adelic metrizations,
we may assume that the following property holds: For any f ∈ X∗(G), let div(f) =
48
∑ι dιDι. Then
Hv(s, gv)|f(gv)|v = Hv(s− d, gv).
Due to the product formula, the restriction of H to PicG(X)Q × G(F ) descends
to the height system on Pic(X)Q×G(F ), which is introduced in Subsection 2.2.2.
The following lemma plays a central role in the study of Height zeta functions:
Lemma 37. For any non-archimedean place v ∈ Val(F ), there exists a compact
open subgroup Kv ⊂ G(ov) such that Hv(s, gv) is invariant under the right action
of Kv. Moreover, Kv = G(ov) for almost all non-archimedean places v ∈ Val(F ).
For a proof, see [CLT02, Lemma 3.2 and Proposition 3.3]. We define
K =∏v:finite
Kv.
We consider the Height zeta function on an adelic space:
Z(s, g) =∑
γ∈G(F )
H(s, γg),
where s ∈ PicG(X)C and g ∈ G(AF ).
Lemma 38. Z(s, g) converges absolutely and uniformly to a function which is
holomorphic in s and is continuous in g when <(s) is sufficiently large.
Proof. Consider an ample divisor L =∑
ι tιDι where tι ∈ Z>0. We may assume
that the abscissa of convergence αG(L) < 1 so that
∑γ∈G(F )
H(L, γ)−1,
49
converges absolutely. Let g = (gv)v ∈ G(AF ). For any archimedean place v,
let Kv ⊂ G(Fv) be a sufficiently small open neighborhood containing e and let
U = K ×∏
v|∞Kv. Then gU ⊂ G(AF ) is an open neighborhood of g. Since the
right action extends, r∗hfι is also a section corresponding to Dι where rh is the right
multiplication map by h ∈ G. This implies that
log ‖fι(γgvkv)‖v = log ‖fι(γ)‖v +O(1),
where γ ∈ G(F ) and kv ∈ Kv. Moreover, for almost all non-archimedean places v,
we have
‖fι(γgvkv)‖v = ‖fι(γ)‖v.
Hence we can conclude that
∑γ∈G(F )
H(L, γg′)−1 ∑
γ∈G(F )
H(L, γ)−1 <∞,
where g′ ∈ gU.
In general, let <(sι) > tι. According to Lemma 10, there exists a positive
constant Cι > 0 such that
HLι,fι(g) ≥ Cι,
for any g ∈ G(AF ). Thus we conclude that
∑γ∈G(F )
H(<(s), gu)−1 ≤∑
γ∈G(F )
H(L, γgu)−1∏ι
C−(<(sι)−tι)ι ,
where u ∈ U. Our assertion follows from this.
Let duv be a Haar measure on U(Fv). Since every unipotent group has no
50
algebraic characters, duv is both left and right invariant. We normalize them by
∫U(ov)
duv = 1,
for almost all non-archimedean places v. Let du =∏
v duv. This is a Haar measure
on U(AF ). Since U(F )\U(AF ) is compact, we may assume that
∫U(F )\U(AF )
du = 1.
Let da×v be a multiplicative Haar measure on Gm(Fv) which is normalized by
da×v =
dxv/|xv|v if v is archimedean,
NpvNpv−1
dxv/|xv|v if v is non-archimedean,
where dxv is a normalized additive Haar measure which is introduced in Subsec-
tion 2.1.3. Let dtv be a Haar measure on T (Fv) which is a product of da×v , and
let dt =∏
v dtv which is a Haar measure on T (AF ). For a semi-direct product
G = U o T , we consider the following Haar measures:
dgv = duvdtv, dg =∏v
dgv.
They are right invariant Haar measures, but not left invariant.
Proposition 39. If <(s) is sufficiently large, then
Z(s, g) ∈ L1(G(F )\G(AF ), dg) ∩ L2(G(F )\G(AF ), dg)K.
Proof. A proof here is a generalization of the proof in [TT12, Lemma 5.2].
51
Integrability follows from [CLT10, Proposition 4.3.4] since
∫G(F )\G(AF )
|Z(s, g)| dg ≤∫G(F )\G(AF )
∑γ∈G(F )
H(<(s), γg)−1 dg
=
∫G(AF )
H(<(s), g)−1 dg.
To prove that Z(s, g) is square-integrable, we prove that Z(s, g) ∈ L∞(G(F )\G(AF )).
Consider a toric variety T ⊂ Y = (P1)r and an equivariant projective compactifi-
cation Z of U . Let π : X → X be a resolution of indeterminacy such that rational
maps X 99K Y mapping ut 7→ t and X 99K Z mapping ut 7→ u are honest mor-
phisms where u ∈ U and t ∈ T . Note that the right action of G does not extend to
X since a rational map X 99K Z mapping ut 7→ u is not a G-rational map unless
G is a product group of U and T . Let H1 and H2 be pull backs of ample boundary
divisors on Y and Z respectively. Fix positive real numbers c, d. It follows from
Lemma 10 that
H(<(s), g) H(cH1, g) H(dH2, g),
assuming that <(s) is sufficiently large and is contained in a compact subset of
PicG(X)R. Write g = ut ∈ G(AF ) where u ∈ U(AF ) and t ∈ T (AF ).
∑γ∈G(F )
H(<(s), γg)−1 ∑
γ∈G(F )
H(cH1, γg)−1 H(dH2, γg)−1
=∑
α∈T (F ), β∈U(F )
H(cH1, αt)−1 H(dH2, βϕαu)−1
=∑
α∈T (F )
H(cH1, αt)−1
∑β∈U(F )
H(dH2, βϕαu)−1,
where ϕ : T → Aut(U) is a homomorphism given by conjugation. Note that the
52
following Dirichlet series
Z2(d, u) =∑
β∈U(F )
H(dH2, βu)−1,
is a Height zeta function of an equivariant compactification Z, and Lemma 38
implies that it is continuous in u ∈ U(AF ) when d is sufficiently large. Moreover,
U(F )\U(AF ) is compact, so it follows that Z2(d, u) is bounded. Hence we need to
bound
Z1(c, t) =∑
α∈T (F )
H(cH1, αt)−1,
but this is a Height zeta function of Y . For our purpose, we only need to bound it
for a specific height, so we may assume that
H(H1, t) =r∏i=1
∏v
max|ti,v|v, |ti,v|−1v ,
where t = (t1, · · · , tr) ∈ Grm(AF ). Then Z1(c, t) is a product of a Height zeta
function of P1, so we may assume that r = 1. Note that
Z1(c, t) ≤∑
α∈T (F )
Hfin(cH1, αt)−1,
where Hfin(H1, t) =∏
v:finite max|tv|v, |tv|−1v . The later series converges absolutely
when c > 1. Since the later series is T (F )-periodic, finiteness of the class group of
F implies that it takes only finitely many values. Thus our assertion follows.
53
3.2.2 Poisson formula and the main term of a Height zeta
function
Poisson formula was used in the analysis of Height zeta functions of additive
groups and toric varieties (see [BT96a], [BT98a], and [CLT02]):
Theorem 40. [DE09, Theorem 3.6.3] Let G be a locally compact abelian group
with a Haar measure dg, H ⊂ G a closed subgroup with a Haar measure dh. Let
f ∈ L1(G) and suppose that its Fourier transform f is also L1-function on H⊥
where H⊥ is the group of topological characters χ : G → S1 which are trivial on
H. Then ∫H
f(h) dh =
∫H⊥
f(χ) dχ,
where dχ is the Plancherel Haar measure on H⊥.
It is expected that for the anticanonical bundle −KX , the main term of a Height
zeta function Z(s, g) is given by
Z0(s, g) =
∫U(F )\U(AF )
Z(s, ug) du.
Note that Z0(s, ut) = Z0(s, t) where u ∈ U(AF ) and t ∈ T (AF ), and so we have
Z0(s, t) ∈ L1(T (F )\T (AF ), dt) ∩ L2(T (F )\T (AF ), dt)
54
which follows from Jensen’s inequality. We apply Poisson formula 40 and obtain
Z0(s, id) =
∫(T (F )\T (AF ))∗
∫G(F )\G(AF )
Z(s, g) χ(g) dg dχ
=
∫(T (F )\T (AF ))∗
∫G(AF )
H(s, g)−1 χ(g) dg dχ
=
∫(T (F )\T (AF ))∗
H(s, χ) dχ,
where (T (F )\T (AF ))∗ is the set of automorphic characters and
H(s, χ) =
∫G(AF )
H(s, g)−1 χ(g) dg.
For any non-archimedean place v, let KT,v be the image of Kv by a homomor-
phism G → T . KT,v ⊂ T (ov) is a compact open subgroup, and KT,v = T (ov)
for almost all non-archimedean places v. For simplicity, we may assume that the
equality holds for all non-archimedean places. This is possible after replacing met-
rics by the average of metrics over compact subgroups T (ov). For any archimedean
place v, let KT,v = ±1r if Fv = R and KT,v = (S1)r if Fv = C. Again, we may
assume that the height pairing is right KT,v-invariant. Let KT =∏
v∈Val(F ) KT,v.
Since the local height pairing H(s, g) is right K-invariant (Lemma 37), we have
H(s, χ) = 0,
if χ /∈ (T (F )KT\T (AF ))∗. It follows that
Z0(s, id) =
∫(T (F )KT \T (AF ))∗
H(s, χ) dχ. (3.1)
We now describe the set of characters on T (F )KT\T (AF ). LetM = Hom(T,Gm) ∼=
55
Zr and N = M∗ = Hom(M,Z). We have a natural surjective map:
T (AF )→ NR,
mapping T (AF ) 3 (tv) 7→ (M 3 m 7→∑
v log |m(tv)|v ∈ R) ∈ NR. We denote its
kernel by T 1(AF ).
Proposition 41. We have the following properties:
1. T 1(AF )\T (AF ) ∼= NR;
2. T (F )\T 1(AF ) is compact;
3. T (F )KT\T 1(AF ) is isomorphic to a product of a finite abelian group and a
compact abelian group.
4. w(T ) = KT ∩ T (F ) is a finite abelian group of torsion elements.
See [BT96a, Proposition 3.2]. We look at the third statement in more detail.
First we have a natural surjective homomorphism to the class group of T :
T (F )KT\T 1(AF )→ Cl(T )→ 0,
and its kernel is given by T (oF )KT\(∏
v-∞ T (ov)× T ′) where
T ′ = (tv) ∈∏v|∞
T (Fv) |∑v|∞
log |m(tv)|v = 0, for any m ∈M.
We have T (oF )KT\(∏
v-∞ T (ov) × T ′) = (T (oF )∏
v|∞KT,v)\T ′ and a natural iso-
morphism
⊕v|∞T (Fv)/KT,v∼= ⊕v|∞NR =: NR,∞.
56
Let N1R,∞, ET ⊂ NR,∞ be the images of T ′ and T (oF ) under this isomorphism
respectively. Dirichlet’s unit theorem implies that N1R,∞/ET is compact and ET ∼=
T (oF )/w(T ). We have the following isomorphism:
(T (oF )∏v|∞
KT,v)\T ′ ∼= N1R,∞/ET .
Thus non-canonically T (F )KT\T 1(AF ) is isomorphic to the product of N1R,∞/ET
and Cl(T ). Let UG = (T (F )KT\T 1(AF ))∗. UG is a finitely generated abelian
group, and
(T (F )KT\T (AF ))∗
is non-canonically isomorphic toMR×UG. The natural mapMR → (T (F )KT\T (AF ))∗
is given by
M 3 m 7→ ((tv)v 7→∏v
|m(tv)|iv) ∈ (T (F )KT\T (AF ))∗.
Let M1R,∞ be the quotient of the diagonal map:
MR 3 m 7→ ⊕v|∞m ∈MR,∞ := ⊕v|∞MR,
which is the dual of N1R,∞. We have a natural map
Φ : (T (F )KT\T (AF ))∗ →M1R,∞
whose image is the kernel of M1R,∞ → E∗T and is a maximal rank lattice. The kernel
of Φ is non-canonically isomorphic to the product of MR and Cl(T )∗.
57
Next we discuss the local height integrals and its Euler product:
Hv(s, χv) =
∫G(Fv)
Hv(s, g)−1χv(g) dgv
H(s, χ) =∏v
Hv(s, χv).
Let
div(ω) = −∑ι
κιDι,
where ω is the top degree right invariant form. When χ = 1, then Hv(s, χv) has
the following analytic properties:
Proposition 42. For any v ∈ Val(F ),
Hv(s, 1) =
∫G(Fv)
Hv(s, g)−1 dgv
is holomorphic when <(sι) > κι − 1, and
∏ι∈I
ζFv(sι − κι + 1)−1Hv(s, 1)
is holomorphic when <(sι) − κι + 1 > −1/2. Here ζFv(s) = (1 − q−sv )−1 is the
Dedekind zeta function of Fv.
See [CLT10, Lemma 4.1.1 and Proposition 4.1.2]. Moreover, for almost all non-
archimedean places v, we have a formula of J. Denef ([Den87, Theorem 3.1]). Let
S be a finite set of non-archimedean places such that for any v /∈ S, we have a
smooth integral model X at v, the metric at v is induced from X , the reduction of
boundary components are smooth geometrically irreducible divisors with normal
58
crossings, and µv(ov) = 1. For any I ⊂ I, we define
XI := ∩ι∈IDι, XI := XI \ ∪I′)IXI′ ,
which is the stratification of the boundary; by convention X∅ = G. Then we have
Proposition 43. [CLT10, Proposition 4.1.6] For any finite place v /∈ S,
Hv(s, 1) =
∫G(Fv)
Hv(s, gv)−1 dgv = τv(G)−1
(∑I⊂I
#X I (kv)
qdim(X)v
∏ι∈I
qv − 1
qsι−κι+1v − 1
),
where τv(G) is the local Tamagawa number of G,
τv(G) =#G(kv)
qdim(G).
Convergence of Euler products follows from Proposition 43:
Proposition 44.
H(s, 1) =∏v
Hv(s, 1),
is holomorphic when <(sι) > κι, and
∏ι
ζF (sι − κι + 1)−1H(s, 1),
has a holomorphic continuation to the domain defined by <(sι) > κι − 1/2.
See [CLT10, Proposition 4.3.4].
Let χ = χaχu where χa ∈ MR and χu ∈ UT . Let m(χa) be the coordinate of
59
the image of a character χa by X∗(G)R → PicG(X)R = ⊕ιRDι. Then we have
H(s, χ) = H(s− im(χa), χu).
Let χu ∈ T (F )KT\T (AF ) be a character of finite order. Choose a basis χ1, · · · , χr
for X∗(T ) = M . Let χ1, · · · , χr be the dual basis for X∗(T ) = Hom(Gm, T ). For
any ι ∈ I, we define a finite order character from Gm as follows:
χu,ι =r∏i=1
(χu χi)mι(χi).
This definition does not depend on the choice of a basis. Analytic properties of
H(s, χ) also can be obtained by using techniques in [CLT10].
Proposition 45. Write χ = χaχuχ∞ where χa ∈ MR, χu : a finite order charac-
ter, and χ∞ : a character at archimedean places. Then
Φ(s, χ) :=∏ι∈I
LF (sι − κι + 1− imι(χa), χu,ι)−1H(s, χ),
has a holomorphic continuation to the domain defined by <(sι) > κι − 1/2 where
LF (s, χu,ι) is a Hecke L-function of χu,ι. Fix a compact set K in the domain defined
by <(sι) > κι − 1/2. Then we have
|Φ(s, χ)| = O(1),
for any <(s) ∈ K and χ ∈ (T (F )KT\T (AF ))∗.
For a proof, one can apply techniques in [CLT10]. Also see [BT96a, Proposition
4.11] for example.
60
So far we have not justified the identity (3.1) obtained from the Poisson formula.
To do this, one can combine integration by parts with respect to left invariant
vector fields ti∂/∂ti at archimedean places, where (t1, · · · , tr) ∈ T , with techniques
in [CLT02, Proposition 2.2 and Proposition 8.4]. Fix a norm ‖ · ‖ on MR,∞ =
⊕v|∞MR. We consider the natural map
m∞ : (T (F )KT\T (AF ))∗ →MR,∞.
The image of this homomorphism is non-canonically isomorphic to the direct sum
of MR and the lattice in M1R,∞.
Proposition 46. For any N ∈ N and a compact set K in a domain defined by
<(sι)− κι > −1, there exists a positive constant C > 0 such that
∣∣∣∣∣∣∏v|∞
Hv(s, χ)
∣∣∣∣∣∣ ≤ C
(1 + ‖m∞(χ)‖)N,
for any <(s) ∈ K.
For a proof, see [CLT02, Proposition 2.2 and Proposition 8.4]. Upshot is:
• A function
F(s) =∏ι∈I
(sι − κι)∑χu∈UG
H(s, χu),
is holomorphic when <(sι)− κι > −1/2.
• The main term of the Height zeta function
Z0(s, id) =
∫χa∈MR
∏ι∈I
(sι − κι − imι(χa))−1F(s− im(χa)) dχa, (3.2)
61
is holomorphic in the interior of the shifted cone −KX + Λeff(X).
Assertions follow from Propositions 45 and 46. Note that for any χ ∈ X∗(G) and
t ∈ R,
Z0(s + itm(χ), id) = Z0(s, id).
Proposition 29 implies that Z0(s, id) descends to a function on Pic(X)C, and it is
holomorphic in −KX + Λeff(X) because the image of a simplicial cone is exactly
the cone of effective divisors.
[CLT01, Section 3] discussed this type of integration (3.2), and the main
theorem is that the analytic properties of such integrals match those of the X -
function of Λeff(X), which is introduced in Subsection 2.1.3. Analytic properties
of X -functions are explained in [BT98a, Section 5], and for any big line bundle
L ∈ Λeff(X), the pole and its order of the X -function
XΛeff(X)(sL+KX),
coincides with geometric invariants a(L) and b(L) introduced in Subsection 2.1.1.
See [BT98a, Proposition 5.3 and 5.4].
At last, we discuss the leading constant c(−KX). We have not determined the
normalization of the Plancherel measure dχ. Let dn be the Lebesgue measure on
NR ∼= T (AF )/T 1(AF ) normalized by vol(NR/N) = 1. Let dt = ω1dn where ω1 is a
measure on T 1(AF ). We define b(T ) by
b(T ) =
∫T 1(AF )/T (F )
ω1.
62
Lemma 47. We have
Z0(s, id) =1
(2π)rb(T )
∫χa∈MR
( ∑χu∈UG
H(s− im(χa), χu)
)dχa,
where r = rankM and dχa is the Lebesgue measure on MR normalized by
vol(MR/M) = 1.
See [BT98a] or [BT96a]. Let U ′G be the set of characters χu ∈ UG such that the
restriction of χu to T (AF,fin) is trivial. Then Proposition 45 implies that
lims→κ
F(s) = lims→κ
∏ι∈I
(sι − κι)∑χu∈U ′G
H(s, χu).
It follows from the Poisson formula ([BT96a, Lemma 5.7]) that
∑χu∈U ′G
H(s, χu) =
∫G(AF )
H(s, g)−1 dg.
Hence it follows from [CLT10, Proposition 4.3.4] that
lims→κ
F(s) = lims→κ
∏ι∈I
(sι − κι)∫G(AF )
H(s, g)−1 dg
= (ζ∗F (1))r∫X(AF )
dτX
= (ζ∗F (1))r τ(−KX),
where ζ∗F (1) is the residue of Dedekind zeta function at s = 1. Note that since T is
split and U is isomorphic to an affine space over F , the weak approximation holds
63
for X. According to [BT98a, Definition 2.4 and Theorem 2.9], b(T ) = (ζ∗F (1))r.
Now [BT98a, Theorem 5.5] concludes that the residue of Z0(−sKX , id) at s = 1 is
given by
α(X)τ(−KX).
Note that since X is rational over F , β(X) = 1.
Our conclusion of this subsection is:
Theorem 48. Let G be an extension of a splitting algebraic torus T by a unipo-
tent group U over a number field F . Let X be a smooth projective equivariant
compactification of G over F and
Z(s, g) =∑
γ∈G(F )
H(s, γg)−1,
the height zeta function with respect to an adelic height pairing. Let
Z0(s, g) =
∫Zχ(s, g) dχ,
be the integral over automorphic characters. Then Z0(s, id) satisfies Manin’s con-
jecture.
3.3 Height zeta functions of equivariant compact-
ifications of the ax + b-group
In this section, we study Height zeta functions of equivariant compactifications
of the ax + b-group. First, we discuss harmonic analysis on semi-direct products
of Ga and Gm.
64
3.3.1 Harmonic analysis
In this subsection we study the local and adelic representation theory of
G := Ga oϕ Gm,
an extension of T := Gm by N := Ga via a homomorphism ϕ : Gm → GL1. The
group law given by
(x, a) · (y, b) = (x+ ϕ(a)y, ab).
This subsection is extracted from [TT12, Section 3]. We fix the standard Haar
measures which are introduced in Subsection 2.1.3 and 3.2.1:
dx =∏v
dxv and da× =∏v
da×v ,
on N(AF ) and T (AF ). Note that G(AF ) is not unimodular, unless ϕ is trivial. The
product measure dg := dxda× is a right invariant measure on G(AF ) and dg/ϕ(a)
is a left invariant measure.
Let % be the right regular unitary representation of G(AF ) on the Hilbert space:
H := L2(G(F )\G(AF ), dg).
We now discuss the decomposition of H into irreducible representations. Let
ψ =∏v
ψv : AF → S1,
be the standard automorphic character which is introduced in [Tat67, Subsection
65
2.2]. Let ψn be the character defined by
x→ ψ(nx),
for n ∈ F×. Let
W := ker(ϕ : F× → F×),
which is a finite cyclic group, and
πn := IndG(AF )N(AF )×W (ψn),
for n ∈ F×. More precisely, the underlying Hilbert space of πn is L2(W\T (AF )),
and the group action is given by
(x, a) · f(b) = ψn(ϕ(b)x)f(ab),
where f is a square-integrable function on T (AF ).
Proposition 49. Irreducible automorphic representations, i.e., irreducible unitary
representations occurring in H = L2(G(F )\G(AF )), are parametrized as follows:
H = L2(T (F )\T (AF ))⊕⊕
n∈(F×/ϕ(F×))πn,
Remark 50. Up to unitary equivalence, the representation πn does not depend
on the choice of a representative n ∈ F×/ϕ(F×).
Proof. Define
H0 := φ ∈ H |φ((x, 1)g) = φ(g) ,
66
and let H1 be the orthogonal complement of H0. It is straightforward to prove
that
H0∼= L2(T (F )\T (AF )).
Lemma 52 concludes our assertion.
Lemma 51. For any φ ∈ L1(G(F )\G(AF )) ∩ H, the projection of φ onto H0 is
given by
φ0(g) :=
∫N(F )\N(AF )
φ((x, 1)g) dx
Proof. It is easy to check that φ0 ∈ H0. Also, for any φ′ ∈ H0, we have
∫G(F )\G(AF )
(φ− φ0)φ′dg = 0.
Lemma 52. We have
H1∼=⊕
n∈F×/ϕ(F×)πn.
Proof. For φ ∈ C∞c (G(F )\G(AF )) ∩H1, define
fn,φ(a) :=
∫N(F )\N(AF )
φ(x, a)ψn(x) dx.
67
Then,
‖ φ ‖2L2 =
∫T (F )\T (AF )
∫N(F )\N(AF )
|φ(x, a))|2 dxda×
=
∫T (F )\T (AF )
∑α∈F
∣∣∣∣∫N(F )\N(AF )
φ(x, a)ψ(αx)dx
∣∣∣∣2 da×
=
∫T (F )\T (AF )
∑α∈F×
∣∣∣∣∫N(F )\N(AF )
φ(x, a)ψ(αx)dx
∣∣∣∣2 da×
=
∫T (F )\T (AF )
∑α∈F×
∑n∈F×/ϕ(F×)
1
#W
∣∣∣∣∫N(F )\N(AF )
φ(x, a)ψ(nϕ(α)x)dx
∣∣∣∣2 da×
=
∫T (F )\T (AF )
∑α∈F×
∑n∈F×/ϕ(F×)
1
#W
∣∣∣∣∫N(F )\N(AF )
φ(x, αa)ψ(nx)dx
∣∣∣∣2 da×
=∑
n∈F×/ϕ(F×)
1
#W
∫T (AF )
∣∣∣∣∫N(F )\N(AF )
φ(x, a)ψn(x)dx
∣∣∣∣2 da×
=∑
n∈F×/ϕ(F×)
‖ fn,φ ‖2L2 .
The second equality is the Plancherel theorem for N(F )\N(AF ). Third equality
follows from Lemma 51. The fifth equality follows from the left G(F )-invariance
of φ. Thus, we obtain an unitary operator:
I : H1 →⊕
n∈F×/ϕ(F×)πn.
Compatibility with the group action is straightforward, so I is actually a morphism
of unitary representations. We construct the inverse map of I explicitly. For
f ∈ C∞c (W\T (AF )), define
φn,f (x, a) :=1
#W
∑α∈F×
ψn(ϕ(α)x)f(αa).
68
The orthogonality of characters implies that
∫N(F )\N(AF )
φn,f (x, a) · φn,f (x, a) dx
= (#W )2∫N(F )\N(AF )
(∑
α∈F× ψn(ϕ(α)x)f(αa)) · (∑
α∈F× ψn(ϕ(α)x)f(αa)) dx
= 1#W
∑α∈F× |f(αa)|2.
Substituting, we obtain
‖ φn,f ‖2 =
∫T (F )\T (AF )
∫N(F )\N(AF )
|φn,f (x, a)|2 dxda×
=1
#W
∫T (F )\T (AF )
∑α∈F×
|f(αa)|2da× =‖ f ‖2n .
Lemma 51 implies that φf ∈ H1 and we obtain a morphism
Θ :⊕
n∈F×/ϕ(F×)πn → H1.
Now we only need to check that ΘI = id and IΘ = id. The first follows from the
Poisson formula: For any φ ∈ C∞c (G(F )\G(AF )) ∩H1,
ΘIφ =∑
n∈F×/ϕ(F×)
1
#W
∑α∈F×
ψn(ϕ(α)x)
∫N(F )\N(AF )
φ(y, αa)ψn(y)dy
=∑
n∈F×/ϕ(F×)
1
#W
∑α∈F×
∫N(F )\N(AF )
φ(ϕ(α)y, αa)ψn(ϕ(α)(y − x))dy
=∑
n∈F×/ϕ(F×)
1
#W
∑α∈F×
∫N(F )\N(AF )
φ((y + x, a))ψn(ϕ(α)y)dy
=∑α∈F
∫N(F )\N(AF )
φ((y, 1)(x, a))ψ(αy)dy = φ(x, a),
69
where we apply Poisson formula for the last equality. The other identity, IΘ = id
is checked by a similar computation.
To simplify notation, we now restrict to F = Q. For our applications in Sec-
tions 3.3.2, we need to know an explicit orthonormal basis for the unique infinite-
dimensional representation π = L2(A×Q) of G = G1. For any n ≥ 1, define compact
subgroups of G(Zp)
G(pnZp) := (x, a) |x ∈ pnZp, a ∈ 1 + pnZp.
Let vp : Qp → Z be the discrete valuation on Qp.
Lemma 53. Let Kp = G(pnZp).
• When n = 0, an orthonormal basis for L2(Q×p )Kp is given by
1pjZ×p | j ≥ 0.
• When n ≥ 1, an orthonormal basis for L2(Q×p )Kp is given by
λp(·/pj)1pjZ×p | j ≥ −n, λp ∈ Mp,
where Mp is the set of characters on Z×p /(1 + pnZp).
Moreover, let Kfin =∏
p Kp, where the local compact subgroups are given by Kp =
G(pnpZp), with np = 0 for almost all p. Let S be the set of primes with np 6= 0 and
N =∏
p pnp. Then an orthonormal basis for L2(A×Q,fin)Kfin is given by
⊗p∈Sλp(ap · p−vp(ap))1mN
Z×p (ap)⊗′p/∈S 1mZ×p (ap) |m ∈ N, λp ∈ Mp.
70
Proof. For the first assertion, let f ∈ L2(Q×p )Kp where Kp = G(Zp). Since it is
Kp-invariant, we have
f(bp · ap) = f(ap),
for any b ∈ Z×p . Hence f takes the form of
f =∞∑
j=−∞
cj1pjZ×p
where cj = f(pj) and∑∞
j=−∞ |cj|2 < +∞. On the other hand we have
ψp(ap · xp)f(ap) = f(ap)
for any xp ∈ Zp. This implies that f(pj) = 0 for any j < 0. Thus the first assertion
follows. The second assertion is treated similarly. The last assertion follows from
the first and the second assertions.
We denote these vectors by vm,λ where m ∈ N and λ ∈ M :=∏
p∈S Mp. Note
that M is a finite set. Also we define
θm,λ,t(g) := Θ(vm,λ ⊗ | · |it∞)(g)
=∑α∈Q×
ψ(αx)vm,λ(αafin) |αa∞|it∞.
The following proposition is a combination of Lemma 53 and the standard
Fourier analysis on the real line:
Proposition 54. Let f ∈ HK1 . Suppose that
71
1. I(f) is integrable, i.e.,
I(f) ∈ L2(A×Q)K ∩ L1(A×Q),
2. the Fourier transform of f is also integrable, i.e.,
∫ +∞
−∞|(f, θm,λ,t)| dt < +∞,
for any m ∈ N and λ ∈M.
Then we have
f(g) =∑λ∈M
∞∑m=1
1
4π
∫ +∞
−∞(f, θm,λ,t)θm,λ,t(g) dt
where
(f, θm,λ,t) =
∫G(Q)\G(AQ)
f(g)θm,λ,t(g) dg.
Proof. For simplicity, we assume that np = 0 for all primes p. Let I(f) = h ∈
L2(A×Q)K ∩ L1(A×Q). It follows from the proof of Lemma 52 that
f(g) = Θ(h)(g) =∑α∈Q×
ψ(αx)h(αa).
Note that this infinite sum exists in both L1 and L2 sense. It is easy to check that
∫A×Qh(a)vm(afin)|a∞|−it∞ da× = (f, θm,t).
Write
h =∑m
vm ⊗ hm,
72
where hm ∈ L2(R>0, da×∞). The first and the second assumptions imply that hm
and the Fourier transform of hm both are integrable. Hence the inverse formula of
Fourier transformation on the real line implies that
h(a) =∑m
1
4π
∫ +∞
−∞(f, θm,t)vm(afin)|a∞|it∞ dt.
Apply Θ to both sides, and our assertion follows.
We recall some results regarding Igusa integrals with rapidly oscillating phase,
studied in [CLT09]:
Proposition 55. Let p be a finite place of Q and d, e ∈ Z. Let
Φ : Q2p × C2 → C,
be a function such that for each (x, y) ∈ Q2p, Φ((x, y), s) is holomorphic in s =
(s1, s2) ∈ C2. Assume that the function (x, y) 7→ Φ(x, y, s) belongs to a bounded
subset of the space of smooth compactly supported functions when <(s) belongs to a
fixed compact subset of R2. Let Λ be the interior of a closed convex cone generated
by
(1, 0), (0, 1), (d, e).
Then, for any α ∈ Q×p ,
ηα(s) =
∫Q2p
|x|s1p |y|s2p ψp(αxdye)Φ(x, y, s)dx×p dy×p ,
is holomorphic on TΛ. The same argument holds for the infinite place when Φ is
a smooth function with compact supports.
73
Proof. For the infinite place, use integration by parts and apply the convexity
principle. For finite places, assume that d, e are both negative. Let δ(x, y) = 1 if
|x|p = |y|p = 1 and 0 else. Then we have
ηα(s) =∑n,m∈Z
∫Q2p
|x|s1p |y|s2p ψp(αxdye)Φ(x, y, s)δ(p−nx, p−my) dx×p dy×p
=∑n,m∈Z
p−(ns1+ms2) · ηα,n,m(s),
where
ηα,n,m(s) =
∫|x|p=|y|p=1
ψp(αpnd+mexdye)Φ(pnx, pmy, s) dx×p dy×p .
Fix a compact subset of C2 and assume that <(s) is in that compact set. The
assumptions in our proposition mean that the support of Φ(·, s) is contained in a
fixed compact set in Q2p, so there exists an integer N0 such that ηα,n,m(s) = 0 if
n < N0 or m < N0. Moreover our assumptions imply that there exists a positive
real number δ such that Φ(·, s) is constant on any ball of radius δ in Q2p. This
implies that if 1/pn < δ, then for any u ∈ Z×p ,
ηα,n,m(s) =
∫ψp(αp
nd+mexdyeud)Φ(pnxu, pmy, s) dx×p dy×p
=
∫ψp(αp
nd+mexdyeud)Φ(pnx, pmy, s) dx×p dy×p
=
∫ ∫Z×pψp(αp
nd+mexdyeud) du×Φ(pnx, pmy, s) dx×p dy×p ,
and the last integral is zero if n is sufficiently large because of [CLT09, Lemma
2.3.5]. Thus we conclude that there exists an integer N1 such that ηα,n,m(s) = 0 if
74
n > N1 or m > N1. Hence we obtained that
ηα(s) =∑
N0≤n,m≤N1
p−(ns1+ms2) · ηα,n,m(s),
and this is holomorphic everywhere.
The case of d < 0 and e = 0 is treated similarly.
Next assume that d < 0 and e > 0. Then again we have a constant c such that
ηα,n,m(s) = 0 if 1/pn < δ and n|d| −me > c. We may assume that c is sufficiently
large so that the first condition is unnecessary. Then we have
|ηα(s)| ≤∑N0≤n
∑m
p−ne
(e<(s1)+|d|<(s2)) · p(n|d|−me)
e<(s2) · |ηα,n,m(s)|
≤∑N0≤n
p−ne
(e<(s1)+|d|<(s2)) · pce<(s2)
1− p−<(s2)e
Thus ηα(s) is holomorphic on TΛ.
From the proof of Proposition 55, we can claim more for finite places:
Proposition 56. Let ε > 0 be any small positive real number. Fix a compact
subset K of Λ, and assume that <(s) is in K. Define:
κ(K) :=
max
0,−<(s1)
|d| ,−<(s2)|e|
if d < 0 and e < 0,
max
0,−<(s1)|d|
if d < 0 and e ≥ 0,.
Then we have
|ηα(s)| 1/|α|κ(K)+εp
75
as |α|p → 0.
Proof. Let |α|p = p−k, and assume that both d, e are negative. By changing
variables, if necessary, we may assume that N0 in the proof of Proposition 55 is
zero. If k is sufficiently large, then one can prove that there exists a constant c
such that ηα,n,m(s) = 0 if n|d|+m|e| > k + c. Also it is easy to see that
|p−(ns1+ms2)| ≤ p(n|d|+m|e|)κ(K).
Hence we can conclude that
|ηα(s)| k21/|α|κ(K)p 1/|α|κ(K)+ε
p .
The case of d < 0 and e = 0 is treated similarly.
Assume that d < 0 and e > 0. Then we have a constant c such that ηα,n,m(s) = 0
if n|d| −me > k + c. Thus we can conclude that
|ηα(s)| ≤∑m≥0
∑n≥0
p−(n<(s1)+m<(s2))|ηα,n,m(s)|
∑m≥0
p−m<(s2)(me+ k)p(me+k)κ(K,s2)
k1/|α|κ(K,s2)p
∑m≥0
(m+ 1)p−m(<(s2)−eκ(K,s2))
1/|α|κ(K,s2)+εp .
where
κ(K, s2) = max
0,−<(s1)
|d|: (<(s1),<(s2)) ∈ K
.
76
Thus we can conclude that
|ηα(s)| 1/|α|κ(K,s2)+εp 1/|α|κ(K)+ε
p .
3.3.2 the ax+ b-group
In this subsection, we study Height zeta functions of equivariant compactifica-
tions of the ax+b-group by using harmonic analysis developed in Subsection 3.3.1.
This subsection is extracted from [TT12, Section 5]. Our main theorem is:
Theorem 57. Let X be a smooth projective equivariant compactification of G = G1
over Q, under the right action. Assume that the boundary divisor has strict normal
crossings. Let a, x ∈ Q(X) be rational functions, where (x, a) are the standard
coordinates on G ⊂ X. Let E be the Zariski closure of x = 0 ⊂ G. Assume
that:
• the union of the boundary and E is a divisor with strict normal crossings,
• div(a) is a reduced divisor, and
• for any pole Dι of a, one has −ordDι(x) > 1.
Then Manin’s conjecture holds for X.
The remainder of this subsection is devoted to a proof of this fact. Blowing up
the zero-dimensional subscheme
Supp(div0(a)) ∩ Supp(div∞(a)),
77
if necessary, we may assume that
Supp(div0(a)) ∩ Supp(div∞(a)) = ∅.
Here div0 and div∞ stand for the divisors of zeroes, respectively poles, of the
rational functions a on X. The local height functions are invariant under the
right action of some compact subgroup Kp ⊂ G(Zp). Moreover, we can assume
that Kp = G(pnpZp), for some np ∈ Z>0. Let S be the set of bad places for
X; a priori, this set depends on a choice of an integral model for X and for
the action of G. Specifically, we insist that for p /∈ S, the reduction of X at
p is smooth, the reduction of the boundary is a union of smooth geometrically
irreducible divisors with normal crossings, and the action of G lifts to the integral
models. In particular, we insist that np = 0 for all p /∈ S. The proof works with
X being any sufficiently large finite set. For simplicity, we assume that the height
function at the infinite place is invariant under the action of K∞ = (0,±1).
Lemma 58. We have
Z(s, g) ∈ L2(G(Q)\G(AQ))K ∩ L1(G(Q)\G(AQ)).
See Proposition 39. By Proposition 49, the height zeta function decomposes as
Z(s, id) = Z0(s, id) + Z1(s, id).
Analytic properties of Z0(s, id) were established in Subsection 3.2.2. It remains to
show that Z1(s, id) is holomorphic on a tube domain over an open neighborhood
of the shifted effective cone −KX + Λeff(X). To conclude this, we use the spectral
78
decomposition of Z1:
Lemma 59. We have
Z1(s, id) =∑λ∈M
∞∑m=1
1
4π
∫ +∞
−∞(Z(s, g), θm,λ,t)θm,λ,t(id) dt,
for sufficiently large <(s).
Proof. To apply Proposition 54, we need to check that Z1 satisfies the assumptions
of Proposition 54. The proof of Lemma 52 implies that
I(Z1) =
∫N(Q)\N(A)
Z(s, g)ψ(x) dx.
Thus we have
∫T (A)
|I(Z1)| da× =
∫T (A)
∣∣∣∣∫N(Q)\N(A)
Z(s, g)ψ(x) dx
∣∣∣∣ da×
≤∑α∈Q×
∫T (A)
∣∣∣∣∫N(A)
H(s, g)−1ψ(αx) dx
∣∣∣∣ da×
=∑α∈Q×
∏p
∫T (Qp)
∣∣∣∣∣∫N(Qp)
Hp(s, gp)−1ψp(αxp) dxp
∣∣∣∣∣ da×p×∫T (R)
∣∣∣∣∫N(R)
H∞(s, g∞)−1ψ∞(αx) dx∞
∣∣∣∣ da×∞.
Assume that p /∈ S. Since the height function is right Kp-invariant, we obtain that
79
for any yp ∈ Zp,
∫N(Qp)
Hp(s, gp)−1ψp(αxp) dxp =
∫N(Qp)
Hp(s, (xp + apyp, ap))−1ψp(αxp) dxp
=
∫N(Qp)
Hp(s, gp)−1ψp(αxp)
∫Zpψp(αapyp) dyp dxp
= 0 if |αap|p > 1.
Hence we can conclude that
∫T (Qp)
∣∣∣∣∣∫N(Qp)
Hp(s, gp)−1ψp(αxp) dxp
∣∣∣∣∣ da×p ≤∫G(Qp)
Hp(<(s), gp)−11Zp(αap) dgp.
Similarly, for p ∈ S, we can conclude that
∫T (Qp)
∣∣∣∣∣∫N(Qp)
Hp(s, gp)−1ψp(αxp) dxp
∣∣∣∣∣ da×p ≤∫G(Qp)
Hp(<(s), gp)−11 1
NZp(αap) dgp,
where N is an integer introduced in Lemma 53. Then the convergence of the
following sum
∑α∈Q×
∏p
∫G(Qp)
H−1p 1 1
NZp(αap) dgp ·
∫T (R)
∣∣∣∣∫N(R)
H−1∞ ψ∞(αx) dx∞
∣∣∣∣ da×∞,
can be verified from the detailed study of the local integrals which we will conduct
later. See proofs of Lemmas 62, 66, and 67.
Next we need to check that
∫ +∞
−∞|(Z(s, g), θm,λ,t)| dt < +∞.
80
Note that
(Z(s, g), θm,λ,t) =
∫G(Q)\G(AQ)
Z(s, g)θm,λ,t dg
=
∫G(AQ)
H(s, g)−1θm,λ,t dg
=∑α∈Q×
∫G(AQ)
H(s, g)−1ψ(αx)vm,λ(αafin)|αa∞|−it∞ dg
=∑α∈Q×
∏p
H′p(s,m, λ, α) · H′∞(s, t, α),
where H′p(s,m, λ, α) is given by
=
∫G(Qp)
Hp(s, gp)−1ψp(αxp)1mZ×p (αap) dgp, p /∈ S
=
∫G(Qp)
Hp(s, gp)−1ψp(αxp)λp(αap/p
vp(αap))1mN
Z×p (αap) dgp, p ∈ S
and
H′∞(s, t, α) =
∫G(R)
H∞(s, g∞)−1ψ∞(αx∞)|αa∞|−it∞ dg∞.
The integrability follows from the proof of Lemma 66. Thus we can apply Proposi-
tion 54, and the identity in our statement follows from the continuity of Z(s, g).
We obtained that
Z1(s, id) =∑λ∈M
∞∑m=1
1
4π
∫ +∞
−∞(Z(s, g), θm,λ,t)θm,λ,t(id) dt
=∑
λ∈M, λ(−1)=1
∞∑m=1
1
2π
∫ +∞
−∞(Z(s, g), θm,λ,t)
∏p∈S
λp
(mN· p−vp(m/N)
) ∣∣∣mN
∣∣∣it∞
dt.
81
We will use the following notation:
λS(αap) :=∏q∈S
λq(pvp(αap)), p /∈ S
λS,p(αap) := λp
(αap
pvp(αap)
)∏q∈S\p
λq(pvp(αap)), p ∈ S.
Proposition 60. If <(s) is sufficiently large, then
Z1(s, id) =∑
λ∈M, λ(−1)=1
∑α∈Q×
1
2π
∫ +∞
−∞
∏p
Hp(s, λ, t, α) · H∞(s, t, α) dt,
where Hp(s, λ, t, α) is given by
∫G(Qp)
Hp(s, gp)−1ψp(αxp)λS(αap)1Zp(αap)|ap|−itp dgp, p /∈ S∫
G(Qp)
Hp(s, gp)−1ψp(αxp)λS,p(αap)1 1
NZp(αap)|ap|
−itp dgp, p ∈ S
and
H∞(s, t, α) =
∫G(R)
H∞(s, g∞)−1ψ∞(αx∞)|a∞|−it∞ dg∞
Proof. For simplicity, we assume that S = ∅. We have seen that
Z1(s, id) =∞∑m=1
∑α∈Q×
1
2π
∫ +∞
−∞
∏p
H′p(s,m, α) · H′∞(s, t, α)|m|it∞ dt.
On the other hand, we have
Hp(s, t, α) =∞∑j=0
∫G(Qp)
Hp(s, gp)−1ψp(αxp)1pjZ×p (αap)
∣∣∣∣pjα∣∣∣∣−itp
dgp.
82
Hence we have the formal identity:
∏p
Hp(s, t, α) · H∞(s, t, α) =∞∑m=1
∏p
H′p(s,m, α) · H′∞(s, t, α)|m|it∞,
and our assertion follows from this. To justify the above identity, we need to ad-
dress convergence issues; this will be discussed below. (See the proof of Lemma 62.)
Thus we need to study the local integrals in Proposition 60. We introduce some
notation:
I1 = ι ∈ I |Dι ⊂ Supp(div0(a))
I2 = ι ∈ I |Dι ⊂ Supp(div∞(a))
I3 = ι ∈ I |Dι 6⊂ Supp(div(a)).
Note that I = I1 t I2 t I3 and I1 6= ∅. Also Dι ⊂ Supp(div∞(x)) for any ι ∈ I3
because
D = ∪ι∈IDι = Supp(div(a)) ∪ Supp(div∞(x)).
Let
−div(ω) =∑ι∈I
dιDι,
where ω = dxda/a is the top degree right invariant form on G. Note that ω defines
a measure |ω| on an analytic manifold G(Qv), and for any finite place p,
|ω| =(
1− 1
p
)dgp,
83
where dgp is the standard Haar measure defined in Section 3.3.1.
Lemma 61. Consider an open convex cone Ω in PicG(X)R, defined by the following
relations: sι − dι + 1 > 0 if ι ∈ I1
sι − dι + 1 + eι > 0 if ι ∈ I2
sι − dι + 1 > 0 if ι ∈ I3
where eι = |ordDι(x)|. Then Hp(s, λ, t, α) and H∞(s, t, α) are holomorphic on TΩ.
Proof. First we prove our assertion for H∞. We can assume that
Hv(s, t) = Hv(s− itm(a), 0),
where m(a) ∈ X∗(G) ⊂ PicG(X) is the character associated to the rational function
a (by choosing appropriate adelic metrizations). It suffices to discuss the case when
t = 0. Choose a finite covering Uη of X(R) by open subsets and local coordinates
yη, zη on Uη such that the union of the boundary divisor D and E is locally defined
by yη = 0 or yη · zη = 0. Choose a partition of unity θη; the local integral takes
the form
H∞(s, α) =∑η
∫G(R)
H∞(s, g∞)−1ψ∞(αx∞)θη dg∞.
Each integral is an oscillatory integral in the variables yη, zη. For example, assume
that Uη meets Dι, Dι′ , where ι, ι′ ∈ I2. Then
∫G(R)
H∞(s, g∞)−1ψ∞(αx∞)θηdg∞
=
∫R2
|yη|sι−dι|zη|sι′−dι′ψ∞
(αf
yeιη zeι′η
)Φ(s, yη, zη)dyη dzη,
84
where Φ is a smooth function with compact support and f is a nonvanishing ana-
lytic function. Shrinking Uη and changing variables, if necessary, we may assume
that f is a constant. Proposition 55 implies that this integral is holomorphic
everywhere. The other integrals can be studied similarly.
Next we consider finite places. Let p be a prime of good reduction. Since
Supp(div0(a)) ∩ Supp(div∞(a)) = ∅,
the smooth function 1Zp(αap) extends to a smooth function h on X(Qp). Let
U = h = 1.
Then
Hp(s, λ, α) =
∫U
Hp(s, gp)−1ψp(αxp)λS(αap)dgp.
Now the proof of [CLT10, Lemma 4.4.1] implies that this is holomorphic on TΩ
because U ∩ (∪ι∈I2Dι(Qp)) = ∅. Places of bad reduction are treated similarly.
Lemma 62. Let |α|p = pk > 1. Then, for any compact set in Ω and for any δ > 0,
there exists a constant C > 0 such that
|Hp(s, λ, t, α)| < C|α|−minι∈I1<(sι)−dι+1−δp ,
for <(s) in that compact set.
Proof. First assume that p is a good reduction place. Let ρ : X (Zp) → X (Fp)
be the reduction map modulo p where X is a smooth integral model of X over
85
Spec(Zp). Note that
ρ(|a|p < 1) ⊂ ∪ι∈I1Dι(Fp),
where Dι is the Zariski closure of Dι in X . Thus Hp(s, λ, α) is given by
Hp(s, λ, α) =∑
x∈∪ι∈I1Dι(Fp)
∫ρ−1(x)
Hp(s, gp)−1ψp(αxp)λS(αap)1Zp(αap)dgp.
Let x ∈ Dι(Fp) for some ι ∈ I1, but x /∈ Dι′(Fp) for any ι′ ∈ I \ ι. Since p is a
good reduction place, we can find analytic coordinates y, z such that
∣∣∣∣∫ρ−1(x)
∣∣∣∣ ≤ ∫ρ−1(x)
Hp(<(s), gp)−11Zp(αap)dgp
=
(1− 1
p
)−1 ∫ρ−1(x)
Hp(<(s)− d, gp)−11Zp(αap)dτX,p
=
(1− 1
p
)−1 ∫m2p
|y|<(sι)−dιp 1Zp(αy)dypdzp
=1
p· p−k(<(sι)−dι+1)
1− p−(<(sι)−dι+1),
where dτX,p is the local Tamagawa measure. For the construction of such local
analytic coordinates, see [Wei82], [Den87], or [Sal98]. If x ∈ Dι(Fp) ∩ Dι′(Fp) for
ι ∈ I1, ι′ ∈ I3, then we can find local analytic coordinates y, z such that
∣∣∣∣∫ρ−1(x)
∣∣∣∣ ≤ (1− 1
p
)∫m2p
|y|<(sι)−dι+1p |z|<(sι′ )−dι′+1
p 1Zp(αy)dy×p dz×p
=
(1− 1
p
)p−k(<(sι)−dι+1)
1− p−(<(sι)−dι+1)
p−(<(sι′ )−dι′+1)
1− p−(<(sι′ )−dι′+1).
If x ∈ Dι(Fp) ∩ Dι′(Fp) for ι, ι′ ∈ I1, ι 6= ι′, then we can find analytic coordinates
86
x, y such that
∣∣∣∣∫ρ−1(x)
∣∣∣∣ ≤ (1− 1
p
)∫m2p
|y|<(sι)−dι+1p |z|<(sι′ )−dι′+1
p 1Zp(αyz) dy×p dz×p
≤(
1− 1
p
)∫m2p
|yz|min<(sι)−dι+1,<(sι′ )−dι′+1p 1Zp(αyz) dy×p dz×p
=
(1− 1
p
)((k − 1)
p−kr
1− p−r+
p−(k+1)r
(1− p−r)2
),
where
r = min<(sι)− dι + 1, <(sι′)− dι′ + 1.
It follows from these inequalities and Lemma 9.4 in [CLT02] that there exists a
constant C > 0, independent of p, satisfying the inequality in the statement.
Next assume that p is a bad reduction place. Choose an open covering Uη of
∪ι∈I1Dι(Qp) such that
(∪ηUη) ∩ (∪ι∈I2Dι(Qp)) = ∅,
and each Uη has analytic coordinates yη, zη. Moreover, we can assume that the
boundary divisor is defined by yη = 0 or yη ·zη = 0 on Uη. Let V be the complement
of ∪ι∈I1Dι(Qp), and consider the partition of unity for Uη, V which we denote
by θη, θV . If k is sufficiently large, then
1 1N
Zp(αa) = 1 ∩ Supp(θV ) = ∅.
87
Hence if k is sufficiently large, then
|Hp(s, λ, α)| ≤∑η
∫Uη
Hp(<(s), gp)−11 1
NZp(αap) · θη dgp.
When Uη meets only one component Dι(Qp) for ι ∈ I1, then
∫Uη
≤∫
Q2p
|yη|<(sι)−dιp 1cZp(αyη)Φ(s, yη, zη) dyη,pdzη,p p−k(<(sι)−dι+1),
as k → ∞, where c is some rational number and Φ is a smooth function with
compact support. Other integrals are treated similarly.
We record the following useful lemmas (see, e.g., [CLT09, Lemma 2.3.1]):
Lemma 63.
∫Z×pψp(bx)db× =
1 if |x|p ≤ 1,
− 1p−1
if |x|p = p,
0 otherwise.
Lemma 64. Let d be a positive integer and a ∈ Qp. If |a|p > p and p - d, then
∫Z×pψp(ax
d) dx×p = 0.
Moreover, if |a|p = p and d = 2, then
∫Z×pψp(ax
d)dx×p =
√p−1
p−1or
i√p−1
p−1if pa is a quadratic residue,
−√p−1
p−1or−i√p−1
p−1if pa is a quadratic non-residue.
Lemma 65. Let |α|p = p−k < 1. Consider an open convex cone Ωε in Pic(X)R,
88
defined by the following relations:
sι − dι + 1 > 0 if ι ∈ I1
sι − dι + 2 + ε > 0 if ι ∈ I2
sι − dι + 1 > 0 if ι ∈ I3
where 0 < ε < 1/3. Then, for any compact set in Ωε, there exists a constant C > 0
such that
|Hp(s, λ, t, α)| < C|α|−23
(1+2ε)p ,
for <(s) in that compact set.
Proof. First assume that p is a good reduction place and that p - eι, for any ι ∈ I2.
We have
Hp(s, λ, α) =∑
x∈X (Fp)
∫ρ−1(x)
Hp(s, gp)−1ψp(αxp)λS(αap)1Zp(αap) dgp.
A formula of J. Denef (see [Den87, Theorem 3.1] or [CLT10, Proposition 4.1.7])
and Lemma 9.4 in [CLT02] give us an uniform bound:
|∑
x/∈∪ι∈I2Dι(Fp)
| ≤∑
x/∈∪ι∈I2Dι(Fp)
∫ρ−1(x)
Hp(<(s), gp)−1 dgp.
Hence we need to study
∑x∈∪ι∈I2Dι(Fp)
∫ρ−1(x)
Hp(s, gp)−1ψp(αxp)λS(αap)1Zp(αap) dgp.
Let x ∈ Dι(Fp) for some ι ∈ I2, but x /∈ Dι′(Fp)∪E(Fp) for any ι′ ∈ I \ ι, where
89
E is the Zariski closure of E in X . Then we can find local analytic coordinates y, z
such that
∫ρ−1(x)
=
(1− 1
p
)−1 ∫m2p
|y|sι−dιp ψp(αf/yeι)λS(αy−1)1Zp(αy
−1) dypdzp,
where f ∈ Zp[[y, z]] such that f(0) ∈ Z×p . Since p does not divide eι, there exists
g ∈ Zp[[y, z]] such that f = f(0)geι . After a change of variables, we can assume
that f = u ∈ Z×p . Lemma 64 implies that
∫ρ−1(x)
=1
p
∫mp
|y|sι−dι+1p λS(αy−1)
∫Z×pψp(αub
eι/yeι)db×p 1Zp(αy−1)dy×p
=1
p
∫p−(k+1)≤|yeι |p
|y|sι−dι+1p λS(αy−1)
∫Z×pψp(αub
eι/yeι) db×p dy×p
Thus it follows from the second assertion of Lemma 64 that
∣∣∣∣∫ρ−1(x)
∣∣∣∣ ≤ 1
p
∫p−(k+1)≤|yeι |
|y|<(sι)−dι+1p
∣∣∣∣∣∫
Z×pψp(αub
eι/yeι)db×p
∣∣∣∣∣ dy×p≤ 1
pkp
keι
(1+ε) +1
ppk+1eι
(1+ε) ×
1 if eι > 2
1√p−1
if eι = 2
1
pkp
23k(1+ε).
90
If x ∈ Dι(Fp) ∩ E(Fp), for some ι ∈ I2, then we have
∫ρ−1(x)
=
(1− 1
p
)−1 ∫m2p
|y|sι−dιp ψp(αz/yeι)λS(αy−1)1Zp(αy
−1) dypdzp
=
∫mp
|y|sι−dι+1p λS(αy−1)1Zp(αy
−1)
∫mp
ψp(αz/yeι)dzpdy
×p
=1
p
∫p−(k+1)≤|y|eιp <1
|y|sι−dι+1p λS(αy−1) dy×p .
Hence we obtain that
∣∣∣∣∫ρ−1(x)
∣∣∣∣ ≤ 1
p
∫p−(k+1)≤|y|eιp <1
|y|<(sι)−dι+1p dy×p ≤ kp
keι
(1+ε) < kp23k(1+ε).
If x ∈ Dι(Fp)∩Dι′(Fp) for some ι ∈ I2 and ι′ ∈ I3, then it follows from Lemma 64
∫ρ−1(x)
=
(1− 1
p
)−1 ∫m2p
|y|sι−dιp |z|sι′−dι′p ψp
(αu
yeιzeι′
)λS(αy−1)1Zp(αy
−1) dypdzp
=
(1− 1
p
)−1 ∫|y|sι−dιp |z|sι′−dι′p λS(αy−1)
∫Z×pψp
(αubeι
yeιzeι′
)db×p dypdzp,
where the last integral is over the domain
(y, z) ∈ m2p | p−(k+1) ≤ |yeιzeι′ |p.
We conclude that
∣∣∣∣∫ρ−1(x)
∣∣∣∣ ≤ (1− 1
p
)−1 ∫p−(k+1)≤|yeιzeι′ |p
|y|<(sι)−dιp |z|<(sι′ )−dι′
p dypdzp
≤(
1− 1
p
)−1 ∫p−k≤|yeι |p<1
|y|<(sι)−dιp dyp
∫mp
|z|<(sι′ )−dι′p dzp
≤ kpkeι
(1+ε) p−(<(sι′ )−dι′+1)
1− p−(<(sι′ )−dι′+1).
91
If x ∈ Dι(Fp) ∩ Dι′(Fp) for some ι, ι′ ∈ I2, then the local integral on ρ−1(x) is:
(1− 1
p
)−1 ∫m2p
|y|sι−dιp |z|sι′−dι′p ψp
(αu
yeιzeι′
)λS(αy−1z−1)1Zp(αy
−1z−1) dypdzp
=
(1− 1
p
)∫m2p
|y|sι−dιp |z|sι′−dι′p λS(αy−1z−1)
∫Z×pψp
(αubeι
yeιzeι′
)db×p dy×p dz×p .
We can assume that eι ≤ eι′ . Then we can conclude that
∣∣∣∣∫ρ−1(x)
∣∣∣∣ ≤ ∫p−k≤|yeιzeι′ |p
|yeιzeι′ |− 1eι
(1+ε)p dy×p dz×p
+
∫p−(k+1)=|yeιzeι′ |p
|yeιzeι′ |− 1eι
(1+ε)p
∣∣∣∣∣∫
Z×pψp
(αubeι
yeιzeι′
)db×
∣∣∣∣∣ dy×p dz×p
≤ k2pkeι
(1+ε) + kpk+1eι
(1+ε) ×
1 if eι > 2
1√p−1
if eι = 2
k2p23k(1+ε).
Thus our assertion follows from these estimates and Lemma 9.4 in [CLT02].
Next assume that p is a place of bad reduction or that p divides eι, for some
ι ∈ I2. Fix a compact subset of Ωε and assume that <(s) is in that compact set.
Choose a finite open covering Uη of ∪ι∈I2Dι(Qp) with analytic coordinates yη, zη
such that the union of the boundary D(Qp) and E(Qp) is defined by yη = 0 or
yη · zη = 0. Let V be the complement of ∪ι∈I2Dι(Qp), and consider a partition of
unity θη, θV for Uη, V . Then it is clear that
∫V
Hp(s, gp)−1ψp(αxp)λS,p(αap)1 1
NZp(αap)θV dgp,
92
is bounded, so we need to study
∫Uη
Hp(s, gp)−1ψp(αxp)λS,p(αap)1 1
NZp(αap)θUηdgp.
Assume that Uη meets only one Dι(Qp) for some ι ∈ I2. Then, the above integral
looks like
∫Uη
=
∫Q2p
|yη|sι−dιp ψp(αf/yeιη ))λS,p(αg/yη)1 1
NZp(αg/yη)Φ(s, yη, zη)dyη,pdzη,p,
where f and g are nonvanishing analytic functions, and Φ is a smooth function
with compact support. By shrinking Uη and changing variables, if necessary, we
can assume that f and g are constant. The proof of Proposition 56 implies our
assertion for this integral. Other integrals are treated similarly.
Lemma 66. For any compact set in an open convex cone Ω′, defined by
sι − dι − 1 > 0 if ι ∈ I1
sι − dι + 3 > 0 if ι ∈ I2
sι − dι + 1 > 0 if ι ∈ I3
there exists a constant C > 0 such that
|H∞(s, t, α)| < C
|α|2(1 + t2),
for <(s) in that compact set.
Proof. Consider the left invariant differential operators ∂a = a∂/∂a and ∂x =
93
a∂/∂x. Assume that <(s) 0. Integrating by parts, we have
H∞(s, t, α) = − 1
t2
∫G(R)
∂2aH∞(s, g∞)−1ψ∞(αx∞)|a∞|−it∞ dg∞
=1
(2π)2|α|2t2
∫G(R)
∂2
∂x2(∂2aH∞(s, g∞)−1)ψ∞(αx∞)|a∞|−it∞ dg∞.
According to Proposition 2.2. in [CLT02],
∂2
∂x2(∂2aH∞(s, g∞)−1) = |a|−2∂2
x∂2aH∞(s, g∞)−1
= H∞(s− 2m(a), g∞)−1 × (a bounded smooth function).
Moreover, Lemma 4.4.1. of [CLT10] tells us that
∫G(R)
H∞(s− 2m(a), g∞)−1dg∞,
is holomorphic on TΩ′ . Thus we can conclude our lemma.
Lemma 67. The Euler product
∏p
Hp(s, λ, t, α) · H∞(s, t, α)
is holomorphic on TΩ′.
Proof. First we prove that the Euler product is holomorphic on TΩ′ . To conclude
this, we only need to discuss:
∏p/∈S∪S3, |α|p=1,
Hp(s, λ, t, α),
94
where S3 = p | p | eι for some ι ∈ I3. Let p be a prime such that p /∈ S ∪ S3
and |α|p = 1. Fix a compact subset of Ω′, and assume that <(s) is sitting in that
compact set. From the definition of Ω′, there exists ε > 0 such that
sι − dι + 1 > 2 + ε for any ι ∈ I1
sι − dι + 1 > ε for any ι ∈ I3.
Since we have
|a|p ≤ 1 = X(Qp) \ ρ−1(∪ι∈I2Dι(Fp)),
we can conclude that
Hp(s, λ, α) =∑
x/∈∪ι∈I2Dι(Fp)
∫ρ−1(x)
Hp(s, gp)−1ψp(αxp)λS(ap)dgp.
Note that ∑x/∈∪ι∈IDι(Fp)
∫ρ−1(x)
=
∫G(Zp)
1 dgp = 1.
Also it follows from a formula of J. Denef (see [Den87, Theorem 3.1] or [CLT10,
Proposition 4.1.7]) and Lemma 9.4 in [CLT02] that there exists an uniform bound
C > 0 such that for any x ∈ ∪ι∈I1Dι(Fp),
∣∣∣∣∫ρ−1(x)
∣∣∣∣ < ∫ρ−1(x)
Hp(<(s), gp)−1dgp <
C
p2+ε.
Hence we need to obtain uniform bounds of∫ρ−1(x)
for
x ∈ ∪ι∈I3Dι(Fp) \ ∪ι∈I1∪I2Dι(Fp).
95
Let x ∈ Dι(Fp) for some ι ∈ I3, but x /∈ ∪ι∈I1∪I2Dι(Fp) ∪ E(Fp). Then it follows
from Lemmas 63 and 64 that
∫ρ−1(x)
=
(1− 1
p
)−1 ∫m2p
|y|sι−dιp ψp(u/yeι) dypdzp
=1
p− 1
∫mp
|y|sι−dιp
∫Z×pψp(ub
eι/yeι) db×p dyp
=
0 if eι > 1
−p−(sι−dι+2)
p−1if eι = 1.
If x ∈ Dι(Fp) ∩ E(Fp) for some ι ∈ I3, then we have
∫ρ−1(x)
=
(1− 1
p
)−1 ∫m2p
|y|sι−dιp ψp(z/yeι)dypdzp
=
(1− 1
p
)−1 ∫mp
|y|sι−dιp
∫mp
ψp(z/yeι) dzpdyp
=
0 if eι > 1
p−(sι−dι+2) if eι = 1.
If x ∈ Dι(Fp) ∩ Dι′(Fp) for some ι, ι′ ∈ I3, then it follows from Lemma 64 that
∫ρ−1(x)
=
(1− 1
p
)−1 ∫m2p
|y|sι−dιp |z|sι′−dι′p ψp
(u
yeιzeι′
)dypdzp
=
(1− 1
p
)−1 ∫m2p
|y|sι−dιp |z|sι′−dι′p
∫Z×pψp
(ubeι
yeιzeι′
)db×p dypdzp
= 0.
Thus we can conclude from these estimates and Lemma 9.4 in [CLT02] that there
96
exists an uniform bound C ′ > 0 such that
∣∣∣Hp(s, λ, t, α)− 1∣∣∣ < C ′
p1+ε
Our assertion follows from this.
Lemma 68. Let Ω′ε be an open convex cone, defined by
sι − dι − 2− ε > 0 if ι ∈ I1
sι − dι + 2 + 2ε > 0 if ι ∈ I2
sι − dι + 1 > 0 if ι ∈ I3
where ε > 0 is sufficiently small. Fix a compact subset of Ω′ε and ε δ > 0. Then
there exists a constant C > 0 such that
|∏p
Hp(s, λ, t, α) · H∞(s, α, t)| < C
(1 + t2)|β| 43− 83ε−δ|γ|1+ε−δ
,
for <(s) in that compact set, where α = βγ
with gcd(β, γ) = 1.
Proof. This lemma follows from Lemmas 62, 65, and 66, and from the proof of
Lemma 67.
Theorem 69. The zeta function Z1(s, id) is holomorphic on the tube domain over
an open neighborhood of the shifted effective cone −KX + Λeff(X).
Proof. Let 1 ε δ > 0. Lemma 68 implies that
Z1(s, id) =∑
λ∈M, λ(−1)=1
∑α∈Q×
1
2π
∫ +∞
−∞
∏p
Hp(s, λ, t, α) · H∞(s, t, α) dt,
97
is absolutely and uniformly convergent on Ω′ε, so Z1(s, id) is holomorphic on TΩ′ε .
Now note that the image of Ω′ε by PicG(X)→ Pic(X) contains an open neighbor-
hood of −KX + Λeff(X). Thus our theorem is concluded.
Remark 70. Theorem 69 claims that only Z0(s, id) contributes to the main term
of a Height zeta function for any big line bundle. However, this cannot be true in
general. A reason why we achieved this result is that assumptions in Theorem 69
ensure that for any torus T ∼= Gm, a G-rational map X 99K P1 mapping G→ T\G
never be a honest morphism. In terms of balanced line bundles which we will
introduce in next part, we can phrase this phenomenon in the following way: Let
Y be the Zariski closure of T . Then any big line bundle is balanced with respect
to Y . See Part II and [BT98b].
98
Part II
Balanced line bundles
99
Chapter 4
Balanced line bundles
From now on, we discuss the notion of balanced line bundles. Balanced line
bundles are introduced in the paper [HTT12], which is joint work with Brendan
Hassett and Yuri Tschinkel, and the motivation of this research is coming from the
study of ergodic method, mixing in [GTBT11]. We found that this concept can
be explained in terms of birational geometry, e.g., the Minimal Model Program
and Iitaka fibration. Such connections between Manin’s program and birational
geometry were already observed by Victor Batyrev and Yuri Tschinkel in [BT98b]
and [Tsc03], and we further develop the circle of these ideas in [HTT12]. In this
chapter, first we discuss general properties of geometric invariants showing up in
the context of Manin’s conjecture, and then introduce the notion of balanced line
bundles. In this part, unless it is stated, we always assume that the ground field
is an algebraically closed field of characteristic zero.
4.1 Generalities
This section is extracted from [HTT12, Section 2].
100
Let X be a Q-factorial projective variety, i.e., X is a normal projective variety
such that for any Weil divisor D, some multiple of D is a Cartier divisor. A rigid
effective divisor is an effective Weil divisor D ⊂ X such that
H0(X,OX(nD)) = 1,
for sufficiently divisible n ∈ N. If D is rigid with irreducible components D1, . . . , Dr
then
H0(OX(n1D1 + . . .+ nrDr)) = 1,
for sufficiently divisible n1, · · · , nr ∈ N and
span(D1, . . . , Dr) ∩ Λeff(X) = ∅.
4.1.1 The invariant a(L)
First we generalize the invariant a(L) in Conjecture 16 for a Q-factorial pro-
jective varieties with only canonical singularities. See [KM98, Definition 2.34] for
definitions of various singularities.
Definition 71. Let X be a Q-factorial projective varieties with only canonical
singularities. Assume that KX is not pseudo-effective and L is a big line bundle
on X. The Fujita invariant is defined by
a(X,L) = infa ∈ R : aL+KX effective
= mina ∈ R : a[L] + [KX ] ∈ Λeff(X).
Remark 72. A smooth projective variety X is uniruled if and only if KX is not
101
pseudo-effective [BDPP04], [Laz04b, Cor. 11.4.20].
For technical purposes, we need a slightly more general definition: Let (X,∆)
denote a Kawamata log terminal pair, i.e., a normal projective variety X and
an effective Q-divisor ∆ on X, such that KX + ∆ is Q-Cartier and (X,∆) has
Kawamata log terminal singularities. The notions of pseudo-effective and big Q-
Cartier divisors make sense in this context [Laz04a, §2.2.B]. When KX + ∆ is
not pseudo-effective and L is a big line bundle, then the natural extension of
Definition 71 may still be used:
a((X,∆), L) = infa ∈ R : aL+KX + ∆ effective .
The following result was conjectured by Fujita and proved by Batyrev for threefolds
and [BCHM10, Cor. 1.1.7] in general.
Theorem 73. Let (X,∆) be a projective Kawamata log terminal pair and L an
ample line bundle on X. Then, a((X,∆), L) is rational.
However, this property can fail when L is big, but not ample:
Example 74. [Leh08, Example 4.9] Let Y be an abelian surface with Picard
rank at least 3. The cone of nef divisors and the cone of pseudo-effective divisors
coincide, and its boundary is a quadratic cone. Let N be a line bundle on Y such
that −N is ample. We consider X := P(O⊕O(N)), and we denote the projection
morphism to Y by π. Let S ⊂ X be the section corresponding to the quotient map
O ⊕ O(N) → O(N). Every line bundle on X is linearly equivalent to a divisor
tS+π∗D where D is a divisor on Y , and in particular the canonical line bundle KX
is linearly equivalent to −2S + π∗N . The cone of pseudo-effective divisors Λeff(X)
is generated by S and π∗Λeff(Y ).
102
Consider a big Q-divisor L = tS + π∗D where t > 0 and D is a big Q-divisor
on Y . If t is sufficiently large, then a(X,L) is defined by a[D] + [N ] ∈ ∂Λeff(Y ).
However, the boundary of Λeff(Y ) is a quadratic cone, and this is not solvable in
Q in general.
From a point of view of Manin’s conjecture, global geometric invariants involv-
ing Manin’s conjecture should be birationally invariant, and this observation is
true for Fujita invariant.
Proposition 75. Let β : X → X be a birational morphism of projective varieties,
where X is smooth and X has only canonical singularities. Assume KX is not
pseudo-effective and L is big. Setting L = β∗L, we have
a(X,L) = a(X, L).
Proof. Since X has canonical singularities, we have
KX = β∗KX +∑i
diEi,
where the Ei are the irreducible exceptional divisors and the di are nonnegative
rational numbers. It follows that for integers m,n > 0 we have
Γ(OX(mKX + nL)) = Γ(OX(mKX −∑i
bmdicEi) + nL)
= Γ(OX(mKX + nL)),
where the second equality reflects the fact that allowing poles in the exceptional
locus does not increase the number of global sections. In particular, effective
103
divisors supported in the exceptional locus of β are rigid. Note that it follows from
the assumption that any multiple of KX is not effective, and at least this implies
that a(X, L) ≥ 0. So definition 71 gives a(X, L) = a(X,L) > 0.
Remark 76. This proposition enables us to define Fujita invariants for singular
varieties via passage to a smooth model.
4.1.2 The invariant of b(L)
Next, we discuss the second invariant showing up in the context of Manin’s
conjecture. To fix our terminology, we recall some basic definitions:
Definition 77. Let N be a finite dimensional vector space over R. A closed convex
cone Λ ⊂ N is a closed subset which is closed under linear combinations with non-
negative real coefficients. An extremal face F ⊂ Λ is a closed convex subcone of Λ
such that u, v ∈ Λ and u+ v ∈ F imply that u, v ∈ F . It is well-known that every
extremal face F is supported by a supporting function σ, i.e., there exists a linear
functional σ : N → R such that σ ≥ 0 on Λ and
F = σ = 0 ∩ Λ.
Definition 78. Let X be a projective variety with only Q-factorial terminal sin-
gularities such that KX is not pseudo-effective. Let L be a big line bundle on X.
Then we define
b(X,L) = the codimension of the minimal extremal face of
Λeff(X) containing a R-divisor a(X,L)L+KX .
The birational invariance of b(X,L) is also true in general:
104
Proposition 79. Let X be a Q-factorial terminal projective variety such that KX
is not pseudo-effective and L a big line bundle on X. Let β : X → X be a smooth
resolution. Setting L = β∗L we have
b(X,L) = b(X, L).
Proof. Let F be the minimal extremal face of Λeff(X) containing a R-divisor
a(X,L)L+KX and F be the minimal extremal face of Λeff(X) containing a(X, L)L+
KX . We denote vector spaces generated by F and F by MF and MF respectively.
There exists a nef cycle ξ ∈ NM1(X) such that
F = ξ = 0 ∩ Λeff(X).
Here NM1(X) is the cone of nef curves which is the dual of the cone of pseudo-
effective divisors. Let E1, · · · , En be irreducible components of the exceptional
locus of β. Note that the negativity lemma ([BCHM10, Lemma 3.6.2]) implies
that NS(X) is a direct sum of β∗NS(X) and [Ei]’s. Since X has only terminal
singularities, it follows from Proposition 75 that
a(X, L)L+KX = β∗(a(X,L)L+KX) +∑i
diEi,
where di’s are positive rational numbers. This implies that F contains β∗(a(X,L)L+
KX) and Ei’s. Thus a(X,L)L+KX is contained in an extremal face supported by
a supporting function β∗ξ, i.e.,
β∗ξ = 0 ∩ Λeff(X),
105
so this extremal face also contains F . Thus we can define a well-defined injective
map
Φ : MF →MF/(∑i
REi).
On the other hand, let η ∈ NM1(X) be a nef cycle supporting F . Consider a
linear functional η : NS(X)→ R defined by
η ≡ η on β∗NS(X) and η · Ei = 0 for any i.
The projection from NS(X) to β∗NS(X) maps pseudo-effective divisors to pseudo-
effective divisors so that η ∈ NM1(X). Moreover,
η · (a(X, L)L+KX) = η · (β∗(a(X,L)L+KX) +∑i
diEi) = 0,
so η = 0 ∩ Λeff(X) contains F . This implies that Φ is actually bijective. Our
assertion follows from this.
Remark 80. This proposition enables us to define b(X,L) for singular projective
varieties via passage to a smooth model.
Example 81. Let X be a Q-factorial terminal projective variety with the big
anticanonical line bundle. Let β : X → X be a smooth resolution, and E1, · · · , En
be irreducible components of the exceptional locus. Since X has only terminal
singularities, we have
KX = β∗KX +∑i
diEi,
where di’s are positive rational numbers. Hence the minimal extremal face con-
taining −β∗KX + KX contains a simplicial cone F := ⊕iR≥0[Ei]. We claim that
106
F is an extremal face.
Take u, v ∈ Λeff(X) such that u + v ∈ F . The projection from NS(X) to
β∗NS(X) maps pseudo-effective divisors to pseudo-effective divisors so that u, v
are linear combinations of [Ei]’s. Thus we need to prove that if E =∑
i eiEi =
E+ − E−, where E+ ≥ 0 and E− > 0 are effective R-divisors with no common
component, then E is not pseudo-effective. By cutting by hyperplanes, we reduce
to the case when X is a surface. Suppose that E is pseudo-effective. By per-
turbing by Ei’s, we may assume that E is a Q-divisor. We consider the Zariski
decomposition of E = P + N where P is a nef Q-divisor and N is an effective
Q-divisor. Then, P and N are numerically equivalent to linear combinations of
Ei’s. However (Ei ·Ej) is negative definite so that P ≡ 0. We have E+ ≡ E−+N .
We may assume that E+ and N have no common component. Then negativity
implies that 0 ≥ (E+)2 = E+ · (E− + N) ≥ 0, so we conclude that E+ = 0. This
is a contradiction since −E− cannot be pseudo-effective. Thus F is an extremal
face, and we conclude that
b(X,−β∗KX) = rk NS(X,R)− n = rk NS(X,R) = b(X,−KX).
However, the geometric meaning of the invariant b(X,L) is unclear in general,
and we introduce some situations where it is more clear:
Definition 82. Let X be a Q-factorial terminal and projective variety such that
KX is not pseudo-effective. Let L be a big line bundle on X. We say a(X,L)L+KX
is locally rational polyhedral if there exist finitely many linear functionals
λi : NS(X,Q)→ Q
107
such that λi(a(X,L)L+KX) > 0 and
Λeff(X) ∩ v : λi(v) ≥ 0 for any i,
is finite rational polyhedral and generated by effective Q-divisors.
There are a number of situations where Λeff(X) is generated by a finite collec-
tion of effective divisors:
• A projective variety X is log Fano if there exists an effective Q-divisor ∆ on X
such that (X,∆) is divisorially log terminal and −(KX +∆) is ample. If X is
log Fano then the Cox ring of X is finitely generated [BCHM10, Cor. 1.3.2], so
in particular, Λeff(X) is finite rational polyhedral and generated by effective
divisors.
• Let X be a smooth projective variety that is toric or an equivariant compact-
ification of the additive group Gna . Then Λeff(X) is generated by boundary
divisors, i.e., irreducible components of the complement of the open orbit,
[HT99, Thm. 2.5], [BT95, Prop.1.2.11].
Let X be a smooth projective variety with Λeff(X) generated by a finite number
of effective divisors and Pic(X)Q = NS(X,Q). Since each irreducible rigid effective
divisor on X is a generator of Λeff(X), we have
Z := ∪rigid effectiveD
is a Zariski closed proper subset of X.
Theorem 83. Let X be a Q-factorial terminal projective variety such that KX is
not pseudo-effective. Let L be a big line bundle on X. Suppose that a(X,L)L+KX
108
has the form of c(A+KX +∆) where A is an ample R-divisor, (X,∆) a kawamata
log terminal pair, and c > 0 a positive number. Then a(X,L)L + KX is locally
rational polyhedral so that a(X,L) is rational.
Proof. The local finiteness of the pseudo-effective boundary is proved in [Leh08,
Proposition 3.3] by using the finiteness of the ample models [BCHM10, Corollary
1.1.5], and moreover Lehmann proved that the pseudo-effective boundary is locally
defined by movable curves. The generation by effective Q-divisors follows from the
non-vanishing theorem [BCHM10, Theorem D].
In particular, a(X,L)L + KX is locally rational polyhedral when L is ample.
As we observed in Example 74, the local finiteness is no longer true if we only
assume that L is big. However, there are certain cases where the local finiteness
of a(X,L)L+KX still holds for any big line bundle L:
Example 84 (Surfaces). Let X be a smooth projective surface such that KX is not
pseudo-effective. Let L be a big line bundle on X. Suppose that D = a(X,L)L+
KX is a non-zero pseudo-effective divisor. Then D is locally rational polyhedral.
We consider the Zariski decomposition of D = P + N where P is a nef R-divisor
and N is a negative part of D. [Bou04] proved that the boundary of Λeff(X) is
locally rational polyhedral away from the nef cone. (See [Bou04, Theorem 3.19]
and [Bou04, Theorem 4.1].) Thus if N is non-zero, then our assertion follows.
Suppose that N is zero. Since a(X,L)L · P + KX · P = D · P = 0 and L · P > 0,
we have KX · P < 0. Thus our assertion follows from Mori’s cone theorem. In
particular, a(X,L) is a rational number.
Example 85 (Equivariant compactifications of the additive groups). Let X be
a smooth projective equivariant compactification of the additive group Gna . Then
109
Λeff(X) is a simplicial cone generated by boundary components. (See [HT99, Theo-
rem 2.5].) However this phenomena cannot be explained from Theorem 83. Indeed,
consider the standard embedding of G3a into P3:
G3a 3 (x, y, z) 7→ (x : y : z : 1) ∈ P3.
This is an equivariant compactification, and the group action fixes any points on
the boundary divisor D which is a hyperplane section. Let X be an equivariant
blow up of 12 generic points on a smooth cubic curve in the hyperplane D. Write
H for the pullback of the hyperplane class and E1, · · · , E12 for the exceptional
divisors. Consider
L = 4H − E1 − · · · − E12.
Then L is a big and nef divisor, but not semi-ample. (See [Laz04a, Section 2.3.A]
for more details.) In particular, the section ring of L is not finitely generated. On
the other hand, we consider
Λadj(X) = Γ ∈ Λeff(X) | Γ = c(A+KX + ∆)
where A is an ample R-divisor, (X,∆) a Kawamata log terminal pair, and c a
positive number. Then Λadj(X) forms a covex cone. The existence of non finitely
generated divisors and [BCHM10, Corollary 1.1.9] imply that Λadj(X) $ Λeff(X).
It is natural to expect that the invariant b(X,L) is related to the canonical
fibration associated to a(X,L)L+KX . A sample result in this direction is:
Proposition 86. Let X be a smooth projective variety such that KX is not pseudo-
effective. Let L be a big line bundle and assume that D = a(X,L)L+KX is locally
110
rational polyhedral and semi-ample. Let π : X → Y be the semi-ample fibration of
D. Then
b(X,L) = rk NS(X)− rk NSπ(X),
where NSπ(X) is the lattice generated by π-vertical divisors, i.e., divisors M ⊂ X
such that π(M) ( Y
Proof. Let F be the minimal extremal face of Λeff(X) containing D = a(X,L)L+
KX and MF the vector space generated by F . We claim that MF = NSπ(X). Let
H be an ample Q-divisor on Y such that π∗H = D. Let M be a π-vertical divisor
on X. Then for sufficiently large m, there exists an effective Cartier divisor H ′
such that mH ∼ H ′ and the support of H ′ contains π(M). Thus mD = mπ∗H ∼
π∗H ′ ∈ F and the support of π∗H ′ contains M . We conclude that M ∈ F , and
this proves that NSπ(X) ⊂ MF . Next, let Xy be a general fiber of π and C ⊂ Xy
be a movable curve on X such that [C] is in the interior of NM1(Xy). Then the
following set
FC = [C] = 0 ∩ Λeff(X),
is an extremal face containing D. The minimality implies F ⊂ FC . On the
other hand, the local rational finiteness of D implies that there exist effective
Q-divisors D1, · · · , Dn ∈ F such that D1, · · · , Dn form a basis of MF . Since
D1 · C = · · · = Dn · C = 0, the supports of Di’s are π-vertical. Hence it follows
that MF ⊂ NSπ(X).
Remark 87. When L is ample, it follows from [KMM87, Lemma 3.2.5] that the
codimension of the minimal extremal face of Nef cone containing D is equal to the
relative Picard rank ρ(X/Y ).
We explored this idea more in the case of toric varieties. See Section 6.3.
111
4.2 Balanced line bundles
This section is extracted from [HTT12, Section 3].
Definition 88. A big line bundle L on X is weakly balanced with respect to an
irreducible subvariety Y ⊂ X if
• L|Y is big;
• a(Y, L|Y ) 6 a(X,L);
• if a(Y, L|Y ) = a(X,L) then b(Y, L|Y ) 6 b(X,L).
A big line bundle is called balanced with respect to Y if it is weakly balanced and
one of the two inequalities is strict.
It is weakly balanced on X if there exists a Zariski closed subset Z ⊂ X such that
L is weakly balanced with respect to every irreducible subvariety Y not contained
in Z. The subset Z will be called exceptional.
A line bundle is called balanced on X if it is weakly balanced on X and if it is
balanced with respect to every irreducible subvariety not contained in Z.
Example 89. Every ample line bundle on P2 is balanced. Consider X = P1× P1.
Direct arguments, or Proposition 96, show that an ample L is balanced on X if
and only if L is proportional to −KX .
One hopes to use the invariant a(X,L) to identify the exceptional locus X,
however, this is highly non-trivial even in the following simple situations:
Conjecture 90 (Debarre-de Jong conjecture). Let X ⊂ Pn be a Fano hypersurface
of degree d ≤ n. Then the dimension of the variety of lines is 2n− d− 3.
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In particular, when d = n, for any line C, we have
a(C,−KX |C) = 2 > 1 = a(X,−KX).
The conjecture predicts that the dimension of the variety of lines is n− 3 so that
lines will not sweep out X.
Proposition 91. Let X be a smooth Fano variety of Picard rank one and Y ⊂ X
an irreducible smooth effective divisor. Then −KX is balanced with respect to Y .
Proof. For smooth divisors Y ⊂ X the claim follows from adjunction formula,
−KX |Y +KY = Y |Y ∈ Λeff(Y ),
because Y is an ample divisor on X. Thus we obtain
a(Y,−KX |Y ) < a(X,−KX) = 1.
However this statement is no longer true when Y is singular:
Example 92 (Mukai-Umemura 3-folds, [MU83]). Consider the standard action of
SL2 on V = Cx⊕Cy. Let R12 = Sym12(V ) be a space of homogeneous polynomials
of degree 12 in two variables and f ∈ R a general form. Let X be a Zariski closure
of SL2-orbit SL2 · [f ] ⊂ P(R12). Then X is a smooth Fano 3-fold of index 1 with
Pic(X) = Z for general f . The complement of the open orbit SL2 · [f ] is an
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irreducible divisor
D = SL2 · [x11y] = SL2 · [x11y] ∪ SL2 · [x12].
D is a hyperplane section on P(R12) and generates Pic(X). Furthermore D is the
image of P1 × P1 by a linear series of bidegree (11, 1), which is injective, open
immersion outside of the diagonal, but not along the diagonal. In particular, D is
singular along the diagonal.
Let β : D → D be the normalization of D which is isomorphic to P1×P1. Then
−β∗KX |D is a line bundle of bidegree (11, 1) so that
a(D,−KX |D) = a(D,−β∗KX |D) = 2 > 1 = a(X,−KX).
Thus Proposition 91 does not hold for D.
Remark 93. We still do not know whether Proposition 91 holds for singular
surfaces Y in P3. This question is related to an arithmetic question whether linear
growth conjecture holds for big line bundles on very singular rational surfaces of
general type in P3. See [Tsc09, Conjecture 4.10.1 and Conjecture 4.10.3].
Example 94. Let X ⊂ P4 be a smooth cubic threefold. The Picard group of X is
spanned by the hyperplane class. By Proposition 91, −KX is balanced with respect
to every smooth divisor on X. Let Y ⊂ X be a line. Note that O(2) on P4 restricts
to the anticanonical line bundle on X and on Y , and b(Y, L|Y ) = b(X,L) = 1. Thus
−KX is weakly balanced, but not balanced, with respect to Y . Since the family
of lines dominates X, −KX is not balanced on X.
114
Chapter 5
Fano varieties
In this chapter, we discuss balanced line bundles on del Pezzo surfaces and
smooth Fano 3-folds of Picard rank 1.
5.1 Del Pezzo surfaces
This section is extracted from [HTT12, Section 4].
Let X be a smooth projective surface with ample −KX , i.e., a Del Pezzo
surface. These are classified by the degree of the canonical class d := (KX , KX).
Basic examples are P2 and P1 × P1, more examples are obtained by blowing up
9− d general points on P2. We have
• rk NS(X) = 10− d;
• for 2 6 d 6 8 the cone Λeff(X) is generated by classes of exceptional curves,
i.e., smooth rational curves of self-intersection −1.
Let L be a big line bundle on X. When is it balanced? The only subvarieties
of X on which we need to test the values of a and b are rational curves C ⊂ X,
115
and b(C,L|C) = 1.
It is easy to characterize curves breaking the balanced condition for the Fujita
invariant:
Lemma 95. Let X be a del Pezzo surface of degree d. Let C be an irreducible
rational curve with (C,C) 6= −1 and L a big line bundle on X. Then
a(C,L|C) 6 a(X,L), (5.1)
i.e., L is weakly balanced on X, where the exceptional set Z is the union of excep-
tional curves.
Proof. First observe that if C is an irreducible rational curve with (−KX , C) = 1
then C is exceptional. Indeed, if (C,C) < 0, then (C,C) = −1, by adjunction,
and C is exceptional. On the other hand, the Hodge index theorem implies that
d(C,C) − 1 < 0, i.e., (C,C) = −1 or 0. The second case is not possible since
(KX , C) + (C,C) must be even.
Let C ⊂ X be a rational curve which is not exceptional. After rescaling, we
may assume that a(X,L) = 1, in particular, we do not assume that L is an integral
divisor. Writing L + KX = D, where D is an effective Q-divisor, and computing
the intersection with C we obtain
(L,C) = (−KX , C) + (D,C) > (−KX , C)
Since C is not exceptional, (−KX , C) > 2, i.e., (L,C) > 2. It follows that
(L,C) + deg(KC) = (L,C)− 2 > 0,
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where C is the normalization of C, i.e., a(C,L|C) 6 1, as claimed.
We proceed with a characterization of b(X,L). Consider the Zariski decompo-
sition
a(X,L)L+KX = P + E,
where P is a nef Q-divisor and E =∑n
i=1 eiEi, ei ∈ Q>0, (Ei, Ej) < 0. We have
(P,E) = 0. By basepoint free theorem (see [KM98, Theorem 3.3]), the divisor P
is semi-ample and defines a semi-ample fibration
π : X → B.
We have two cases:
Case 1. B is a point. Then
a(X,L)L+KX =n∑i=1
eiEi
is rigid, which implies that the classes Ei are linearly independent in NS(X). In
particular, ⊕iR>0Ei is an extremal face of Λeff(X), and in fact the minimal extremal
face containing a(X,L)L+KX . It follows that
b(X,L) = rk NS(X)− n.
Case 2. B is a smooth rational curve. Then the minimal extremal face con-
117
taining
a(X,L)L+KX = P +n∑i=1
eiEi
is given by
NSπ(X) ∩ Λeff(X) = P = 0 ∩ Λeff(X),
where NSπ(X) ⊂ NS(X) is the subspace generated by vertical divisors, i.e., divisors
D ⊂ X not dominating B. It follows that
b(X,L) = rk NS(X)− rk NSπ(X) = 1.
Proposition 96. Let X be a del Pezzo surface and L a big line bundle on X.
Then L is balanced if and only if a(X,L)L+KX = D where D is a rigid effective
divisor.
Proof. Assume that a(X,L) = 1. In Case 1, we must have
L+KX = D =n∑i=1
eiEi, ei > 0,
with Ei disjoint exceptional curves. Assume that L is not balanced so that
b(X,L) = 1. Let π : X → P2 be the blowdown of E1, . . . , En and h a hyper-
plane class on P2. Then
L = −KX +D = 3π∗h+n∑i=1
(ei − 1)Ei.
Let C be an irreducible rational curve which is not exceptional. If C does not meet
any of the Ei then
(L,C) = (3π∗h,C) > 3 > 2.
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If C meets at least one of the Ei then
(L,C) = (−KX , C) + (D,C) > 2
since the first summand is > 2. It follows that a(C,LC) < 1, i.e., L is balanced,
contradicting our assumption.
In Case 2, we have
L+KX = D = P +n∑i=1
eiEi, ei > 0,
where P is nef and Ei are disjoint exceptional divisors. Let π : X → P1 be the
fibration induced by the semi-ample line bundle P . The general fiber F of π is a
conic and
rk NS(X)− rk NSπ(X) = 1.
We have (F, F ) = 0, (−KX , F ) = 2, and the class of F is proportional to P . Hence,
for any such F ,
a(F,L|F ) = a(X,L), b(F,L|F ) = b(X,L) = 1.
Thus L is not balanced.
Example 97. Let X = Blp1,p2(P2) be the blowup of P2 in two points and ` the
line passing through p1, p2. Let ˜⊂ X be the strict transform of ` and E1, E2 ⊂ X
the exceptional divisors. Then
Λeff(X) = R>0˜⊕ R>0E1 ⊕ R>0E2.
119
We have
−KX = 3˜+ 2E1 + 2E2, (˜, ˜) = (Ei, Ei) = −1.
Every L such that L + KX is on the face spanned by E1 and E2 is balanced. On
the other hand, the Zariski decomposition of
L+KX = λ`+ εEi, λ > 0, ε > 0,
is given by λ(˜+ Ei) + (ε− λ)Ei, when λ 6 ε,
ε(˜+ Ei) + (λ− ε)˜ otherwise.
In particular, such L are balanced if and only if either λ or ε vanish.
5.2 Del Pezzo surface fibrations
This section is extracted from [HTT12, Section 5].
As we observed in Example 19, the balance property can fail in general:
Example 98. [BT96b] Let f, g be general cubic forms on P3 and
X := sf + tg = 0 ⊂ P1 × P3
the Fano threefold obtained by blowing up the base locus of the pencil. The
projection onto the first factor exhibits a cubic surface fibration
π : X → P1,
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so that −KX restricts to −KY , for every smooth fiber Y of π. Thus
a(Y,−KY ) = a(X,−KX) = 1.
Furthermore, the Neron-Severi rank of a smooth fiber of π is 7. On the other hand,
by Lefschetz theorem, we have rk NS(X) = 2. It follows that
7 = b(Y,−KY ) > b(X,−KX) = 2,
i.e., −KX is not balanced on X.
Let Z be the union of singular fibers of π, −KY -lines in general smooth fibers
Y , and the exceptional locus of the blow up to P3. −KX is balanced with respect
to every rational curve on X which is not contained in Z.
Failure of the balance property for Fano fibrations is related to the monodromy
action on the Neron-Severi space. A sample result in this direction is:
Proposition 99. Let π : X → Y be a Fano fibration from a smooth Fano variety
to a smooth projective curve. Take a general smooth fiber Xy, and assume that the
monodromy action on NS(Xy) is trivial. Then
rk NS(Xy) < rk NS(X).
Proof. See [dFH11, Proposition 6.5]. Also see [KM92, Section 12].
The following proposition follows from the rigidity of Fano varieties:
Proposition 100. Let π : X → B be a flat family of Fano varieties with terminal
Q-factorial singularities over a connected smooth curve B and L a π-big line bundle
121
on X . Then
a(Xb, Lb) = a(Xb′ , Lb′), b(Xb, Lb) = b(Xb′ , Lb′),
for all b, b′ ∈ B.
Proof. By [dFH09, Corollary 5.1], the cone of pseudo-effective divisors for Fano
varieties is locally constant in families.
5.3 Fano threefolds
In this section we investigate the geometry of smooth Fano threefolds of Picard
rank one. There are finitely many families of smooth Fano threefolds; a list of
these can be found in [MM82], [Sha99, Chapter 12], and also in [MM04], which
includes one case that was missing previously. Here we follow the classification
scheme in [Man93].
The basic invariants of Fano threefolds are:
• the rank of the Picard group, i.e., b(X,−KX);
• the index r = r(X), which is the maximal integer such that KX is divisible
by r in Pic(X);
• the degree d(X) := (−KX)3;
• the Mori invariant m(X) which is the smallest integer m such that through
every point of X passes a rational curve C with (−KX , C) 6 m.
The following proposition from [Man93, Section 2] describes smooth Fano three-
folds:
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Proposition 101. Every smooth Fano threefold over an algebraically closed field
of characteristic zero is isomorphic to one of the following:
(1) a generalized flag variety P\G;
(2) a variety X with m(X) = 2;
(3) a blowup of varieties of type (1) or (2);
(4) a direct product of P1 and a del Pezzo surface.
We recall the finer classification from [Man93]. When rk Pic(X) = 1, we have
the following possibilities for X:
• P3, or a quadric, or
• r(X) = 2 and d(X) ∈ 8, 16, 24, 32, 40, or
• r(X) = 1 and d(X) ∈ 2, 4, 6, 8, 10, 12, 14, 16, 18, 22.
The first two cases are covered by Proposition 103. By Proposition 91, −KX is
balanced with respect to smooth divisors on X. Thus we need to focus on curves.
Varieties in the second class are dominated by −KX-conics and there are no curves
of smaller degree. Varieties in the third class are also dominated by −KX-conics
but are not dominated by −KX-lines. Thus, −KX is weakly balanced but not
balanced on X.
Remark 102. Let X be a Fano variety of the second or third class, defined over
a number field. The families of −KX-conics dominating X are surfaces of general
type, which embedded into their Albanese varieties. By Falting’s theorem, conics
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defined over a fixed number field lie on a proper subvariety and cannot dominate
X.
124
Chapter 6
Equivariant geometry
In this chapter, we discuss the balance property for projective varieties with
group actions, e.g., flag varieties, toric varieties, and equivariant compactifications
of homogeneous spaces. Manin’s conjecture has been proved for many classes of
these varieties, but, nevertheless, proving balanceness for these varieties is quite
nontrivial. Results in this chapter have arithmetic applications in [CLT02] and
[GTBT11].
6.1 Generalized flag varieties
Let G be a connected semi-simple group, and P ⊂ G a parabolic subgroup.
Let X = P\G be a generalized flag variety. Manin’s conjecture for X was proved
in [FMT89]. The following proposition is extracted from [HTT12, Section 3].
Proposition 103. Let G be a connected semi-simple algebraic group, P ⊂ G a
parabolic subgroup and X = P\G the associated generalized flag variety. Let L be
a big line bundle on X. Then
125
• if L is not proportional to −KX , then L is not balanced;
• if L is proportional to −KX , then L is balanced with respect to any smooth
subvariety Y ⊂ X.
Proof. Recall that for generalized flag varieties, the nef cone and the pseudo-
effective cone coincide so that L is ample. Moreover, it is well-known that the
nef cone of a flag variety is finitely generated by semi-ample line bundles. Also
note that since every rationally connected smooth proper variety is simply con-
nected, all parabolic subgroups are connected.
Assume that L is not proportional to the anticanonical bundle, i.e., D =
a(X,L)L + KX is a non-zero effective Q-divisor. Let π : X → Y be the semi-
ample fibration of D. Then Y is also a G-variety so that there exists a parabolic
subgroup P ′ ⊃ P such that Y = P ′\G and π is the natural projection map. We
have the following exact sequence:
0→ Pic(P ′\G)Q → Pic(P\G)Q → Pic(P\P ′)Q → 0.
Indeed, the surjectivity follows from [KKV89, Proposition 3.2(i)]. Then the exact-
ness of other parts follows from [KMM87, Lemma 3.2.5]. Let W be a fiber of π.
Then we have a(X,L) = a(W,L|W ) since KX |W = KW . The exact sequence and
Remark 87 imply that
b(X,L) = ρ(X/Y ) = rk Pic(W ) = b(W,L|W ).
Thus L is not balanced with respect to any fiber of π.
Let L = −KX and Y ⊂ X a smooth subvariety. Let g be the Lie algebra of G.
126
For any ∂ ∈ g, we can construct a global vector field ∂X on X such that for any
open set U ⊂ X and any f ∈ OX(U),
∂X(f)(x) = ∂gf(x · g)|g=1.
It follows that NY/X is globally generated where NY/X is the normal bundle of Y
in X. Thus we conclude that
−KX |Y +KY = det(NY/X) ∈ Λeff(Y ),
so that a(Y,−KX |Y ) ≤ a(X,−KX) = 1.
Suppose that a(Y,−KX |Y ) = a(X,−KX) = 1. Our goal is to prove that
b(Y,−KX |Y ) < b(X,−KX) = rk NS(X).
First we assume that det(NY/X) is trivial so that NY/X is the trivial vector
bundle of rank r = codim(Y,X). The above construction of vector fields defines a
surjective map:
ϕ : g→ H0(Y,NY/X).
We may assume that e = P ∈ Y so that the Lie algebra p of P is contained in the
kernel of ϕ. We consider the Hilbert scheme Hilb(X). Note that H0(Y,NY/X) is
naturally isomorphic to the Zariski tangent space of Hilb(X) at [Y ]. Consider the
following morphism:
π : G 3 g 7→ [Y · g] ∈ Hilb(X).
Since Y is Fano-type, H1(Y,NY/X) = 0. This implies that Y is unobstructed, and
127
it follows that Hilb(X) is smooth at [Y ] and
dim[Y ] Hilb(X) = r.
Moreover, since ϕ is surjective, we conclude that π is a smooth morphism and
π(G) is a smooth open subscheme in Hilb(X). Let H be the connected component
of Hilb(X) containing [Y ]. Let P ′ = Stab(Y ). Since the kernel of ϕ contains p, we
have P ⊂ P ′. This implies that π(G) = P ′\G is open and closed so that
H = π(G) = P ′\G.
In particular, dimG − dimP ′ = r, so the kernel of ϕ is exactly equal to the Lie
algebra p′ of P ′. Consider the universal family U ⊂ X ×H on H. It follows that
G acts on U transitively, and we conclude that U = P\G and Y = P\P ′. Our
assertion follows from the exact sequence which we discussed before.
In general case, we consider the semi-ample fibration associated to det(NY/X).
Its general smooth fibers are smooth subvarieties with trivial normal bundles so
that they can be realized as fibers of a fibration ρ : X → B. Then Y is a pullback
of a subvariety H ⊂ B. Theorem 83 and Proposition 86 imply that
b(Y,−KX |Y ) = rk NS(Y )− rk NSρ(Y ).
The restriction map:
Φ : NS(Y )/NSρ(Y )→ NS(Yh),
is injective where Yh is a general fiber of ρ. (For example, this follows from
128
[KMM87, Lemma 3.2.5].) Hence we have
b(Y,−KX |Y ) ≤ b(Yh,−KX |Yh) < b(X,−KX).
6.2 Equivariant compactifications of homogeneous
spaces
This section is extracted from [HTT12, Section 7].
Let G be a connected linear algebraic group, H ⊂ G a closed subgroup, and
X a projective equivariant compactification of X := H\G. Using equivariant
resolution of singularities we may assume that X is smooth and that the boundary
∪α∈ADα = X \X
is a divisor with normal crossings. If H is a parabolic subgroup of a semi-simple
group G, then there is no boundary, i.e., A is empty, and H\G is a generalized
flag variety which was discussed in Section 6.1. Throughout, we will assume that
A is not empty.
Let X(G)∗ be the group of algebraic characters of G and
X(G,H)∗ = χ : G→ Gm |χ(hg) = χ(g), ∀h ∈ H
the subgroup of characters whose restrictions to H are trivial. Let PicG(X) be
the group of equivalence classes of G-linearized line bundles on X and Pic(X) the
129
Picard group of X. For L ∈ PicG(X), the subgroup H ⊂ G acts linearly on the
fiber Lx at x = H ∈ H\G. This defines a homomorphism
PicG(X)→ X(H)∗
to characters of H. Let Pic(G,H)(X) be the kernel of this map. We will identify
line bundles and divisors with their classes in Pic(X).
Proposition 104. Let G be a connected linear algebraic group and H a closed
subgroup of G. Let X be a smooth projective equivariant compactification of X :=
H\G with a boundary divisor ∪α∈ADα. Then
1. we have an exact sequence
0→ X(G,H)∗ → ⊕α∈AZDα → Pic(X)→ Pic(X)→ 0;
2. we have an exact sequence
0→ X(G,H)∗Q → Pic(G,H)(X)Q → Pic(X)Q;
and the last homomorphism is surjective when
C(G,H) := Coker(X(G)∗ → X(H)∗)
is finite. (Equivalently, Pic(X) is finite.)
3. and we have a canonical injective homomorphism
Ψ : ⊕α∈AQDα → Pic(G,H)(X)Q;
130
and Ψ is an isomorphism when C(G,H) is finite.
Proof. The first exact sequence easily follows. The second assertion follows from
[MFK94, Corollary 1.6] and [KKV89, Proposition 3.2(i)].
For the last assertion: Corollary 1.6 of [MFK94] implies that some multiple of
Dα is G-linearizable. We may assume that G acts on the finite-dimensional vector
space H0(X,OX(Dα)), via this G-linearization. Let sα be the section corresponding
to Dα. Then sα ∈ H0(X,OX(Dα)) is an eigenvector of the action by G. After
multiplying by a character of G, if necessary, we may assume that sα is fixed by
the action of G. We let Ψ(Dα) be this G-linearization.
Suppose that Φ(∑
α dαDα) = OX , with trivial G-linearization, where dα ∈ Z.
Then there exists a rational function f such that
div(f) =∑α
dαDα.
We may assume that f is a character of G whose restriction to H is trivial. By the
definition of Ψ, the function f must be fixed by the G-linearization. This implies
that f ≡ 1. When C(G,H) is finite, the surjectivity of Ψ follows from (1) and
(2).
Proposition 105. Let G be a connected linear algebraic group and H a closed
subgroup of G. Let X be a smooth projective equivariant compactification of X :=
H\G with a boundary divisor ∪α∈ADα. If C(G,H) is finite, then the anticanonical
divisor −KX is big.
Proof. Let g, resp. h, be the Lie algebra of G, resp. H. For any ∂ ∈ g, there is
a unique global vector field ∂X on X such that for any open set U ⊂ X and any
131
f ∈ OX(U),
∂X(f)(x) = ∂gf(x · g)|g=1.
Let ∂1, . . . , ∂n ∈ g be a lift of a basis for g/h. Consider the following global section
of the anticanonical bundle det(TX):
δ = ∂X1 ∧ · · · ∧ ∂Xn .
Note that this section is nonzero at x = H ∈ H\G = X. The proof of [CLT02,
Lemma 2.4] implies that this section vanishes along the boundary. Hence we can
conclude that
div(δ) =∑α
dαDα + (an effective divisor in X),
where nα > 0. When C(G,H) is finite, then Pic(X)Q is generated by boundary
components so that every ample divisor can be expressed as a linear combination
of Dα’s. This implies that∑
α dαDα is big so div(δ) is also big.
From now on we consider the following situation: LetH ⊂M ⊂ G be connected
linear algebraic groups. Typical examples arise when G is a unipotent group or
a product of absolutely simple groups. Let X be a smooth projective equivariant
compactification of H\G, and Y the induced compactification of H\M .
Lemma 106. Let π : X → X ′ be an equivariant morphism onto a projective
equivariant compactification of M\G. Assume that the projection G → M\G
admits a rational section. Then
• π(Dα) = X ′ if and only if Dα ∩ Y 6= ∅;
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• if Dα ∩ Y 6= ∅ then Dα ∩ Y is irreducible;
• if Dα ∩ Y 6= ∅ and Dα′ ∩ Y 6= ∅, for α 6= α′ then Dα ∩ Y 6= Dα′ ∩ Y .
Proof. We have the diagram
H\G
⊂ X
π
⊃ Y
π
M\G ⊂ X ′ ⊃ M · e = point
The first claim is evident. To prove the second assertion, choose a rational section
σ : M\G 99K G of the projection G → M\G. We may assume that a rational
section is well-defined at a point M ∈M\G. Consider the diagram
Dα
π
⊃ Dα
π
X ′ ⊃ M\G
where Dα = Dα ∩ π−1(M\G). We define a rational map
Ψ : Dα 99K Dα ∩ Y
x 7→ x · (σ π(x))−1.
This map is dominant which implies that Dα ∩ Y is irreducible.
Since the G-orbit of Dα ∩ Y is Dα, the third claim follows.
Remark 107. When M is a connected solvable group, then G is birationally
isomorphic to M×(M\G) so that the projection G→M\G has a rational section.
See [Bor91, Corollary 15.8].
133
Theorem 108. Let
H ⊂M ⊂ G
be connected linear algebraic groups. Let X be a smooth projective equivariant
compactification of H\G and Y ⊂ X the induced compactification of H\M . Let L
be a big line bundle on X. Assume that
• the projection G→M\G admits a rational section;
• and D := a(X,L)L+KX is a rigid effective Q-divisor.
Furthermore, we assume that either
1. Λeff(X) is finitely generated by effective divisors;
2. or there exists a birational contraction map f : X 99K Z contracting D where
Z is a normal projective variety.
Then L is balanced with respect to Y .
Proof. Let X ′ be any smooth projective equivariant compactification of M\G. We
consider a G-rational map π : X 99K X ′ mapping
π : G 3 g 7→Mg ∈M\G.
After applying a G-equivariant resolution of the indeterminacy of the projection
π if necessary, we may assume that π is a surjective morphism and X is a smooth
equivariant compactification of H\G with a boundary divisor ∪αDα. Note that
Y is a general fiber of π so that Y is smooth. Write a rigid effective Q-divisor
134
D = a(X,L)L+KX by
D = a(X,L)L+KX =n∑i=1
eiEi,
where Ei’s are irreducible components of a(X,L)L + KX and ei ∈ Q>0. Our goal
is to show that
(a(Y, L|Y ), b(Y, L|Y )) < (a(X,L), b(X,L)).
Since Ei’s are rigid effective divisors, they are parts of boundary components. This
implies that
a(X,L)L|Y +KY = (a(X,L)L+KX)|Y =n∑i=1
eiEi|Y ∈ Λeff(Y ).
It follows that
a(Y, L|Y ) ≤ a(X,L).
Assume that a(Y, L|Y ) = a(X,L) =: a. Let F be the minimal extremal face of
Λeff(X) containing D = aL + KX =∑eiEi and VF a vector subspace generated
by F . Either condition (1) or (2) guarantees that F is generated by Ei’s so that
b(X,L) = rk NS(X)− n.
(See Proposition 79 and Example 81.) Let F ′ be the minimal extremal face of
Λeff(Y ) containing
D|Y = aL|Y +KY =∑
eiEi|Y .
Let V ′ be a vector subspace generated by all components of Ei ∩ Y . Since F ′
contains all components of Ei ∩ Y , we have b(Y, L|Y ) ≤ codimV ′. Consider the
135
restriction map:
Φ : NS(X)/VF → NS(Y )/V ′.
It follows from [KKV89, Proposition 3.2(i)], Lemma 106, and the exact sequence (1)
in Proposition 104 that Φ is surjective. On the other hand, π∗NS(X ′) is contained
in the kernel of Φ, so Φ has the nontrivial kernel. We conclude that
b(Y, L|Y ) ≤ codimV ′ < b(X,L).
Corollary 109. Let H ⊂ M ⊂ G be connected linear algebraic groups and X a
smooth projective equivariant compactification of H\G with the big anticanonical
bundle. Let Y ⊂ X be the induced compactification of H\M . Assume that the
projection G → M\G admits a rational section. Then −KX is balanced with
respect to Y .
Example 110. Let G = PGL2, M = B, a Borel subgroup of G and H = 1. Let
X = P3 be the standard equivariant compactification of G given by
PGL2 3
a b
c d
7→ [a : b : c : d] ∈ P3,
with boundary D := ad − bc = 0 = P1 × P1. Then Y = P2; with boundary
DY = Y \ B = `1 ∪ `2, a union of two intersecting lines. We have X ′ = P1. The
projection
π : X 99K X ′
136
has indeterminacy along one of lines, say `1. Blowing up `1, we obtain a fibration
π : X → P1.
We have
a(X,−KX) = a(Y ,−KX |Y ) = a(Y ,−KY ) = 1.
Every boundary component of X dominates the base P1, since the G-action is
transitive on the base. Lemma 106 shows that the number of boundary components
of Y is equal to the number of boundary components of X, which equals the rank
of NS(X) = 2. However, X(B)∗ = Z, and in particular, the rank of the Picard
group of Y is one less than the number of boundary components, i.e.,
b(Y ,−KX |Y ) = 1 < 2 = b(X,−KX).
6.3 Toric varieties
In this section, we illustrate how to use the Minimal Model Program to deter-
mine balanced line bundles. This section is extracted from [HTT12, Section 8].
We refer to [FS04] for details concerning toric Mori theory.
Proposition 111. Let X be a Q-factorial projective toric variety. Let D =∑
iDi
be the complement of the big torus regarded as a reduced divisor and its irreducible
decomposition. Then
1. KX +D ∼ 0;
2. (X,∑
i aiDi) is klt (resp. log-canonical) where 0 ≤ ai < 1 (resp, 0 ≤ ai ≤ 1);
137
3. the cone of curves and the cone of pseudo-effective divisors are convex ratio-
nal polyhedral cones and generated by finitely many effective cycles;
4. X is a log Fano variety.
Proof. See Remark 2.6, Proposition 2.10, and Theorem 4.1 in [FS04]. Finite gener-
ation of the cone of pseudo-effective divisors was addressed in [BT95, Proposition
1.2.11]. The last statement follows from (1), (2), and (3).
Next the following two propositions are shared with all Mori dream spaces (see
[HK00, Section 1]):
Proposition 112 (D-Minimal Model Program). Let X be a Q-factorial projective
toric variety and D be a Q-divisor. Then the minimal model program with respect
to D runs i.e.
1. for any extremal ray R of NE1(X), there exists the contraction morphism
ϕR;
2. for any small contraction ϕR of a D-negative extremal ray R, the D-flip
ψ : X 99K X+ exists;
3. any sequence of D-flips terminates in finite steps;
4. and every nef line bundle is semi-ample.
Proof. See Theorem 4.5, Theorem 4.8, Theorem 4.9, and Proposition 4.6 in [FS04].
Proposition 113 (Zariski decomposition). Let X be a Q-factorial projective toric
variety and D a Q-effective divisor. We apply D-MMP and obtain a birational
138
contraction map f : X 99K X ′ and the proper transform D′ of D, which is nef.
Consider a common resolution:
Xµ
~~
ν
X
f//___ X ′
Then we have
1. µ∗D = ν∗D′ + E where E is a ν-exceptional effective Q-divisor;
2. the support of E contains all divisors contracted by f ;
3. let g : X → Y be the semi-ample fibration associated to ν∗D′, then for any
ν-exceptional effective Cartier divisor E ′, the natural map
OY → g∗O(E ′),
is an isomorphism.
Proof. The assertions (1) and (2) follow from the Negativity lemma (see [FS04,
Lemma 4.10]). Also see [FS04, Theorem 5.4].
The invariant b(X,L) can be characterized in terms of Zariski decomposition
of a(X,L)L+KX :
Proposition 114. Let X be a Q-factorial projective toric variety and D an effec-
tive Q-divisor on X. Suppose that
1. D = P +N where P is a nef and N ≥ 0;
139
2. let g : X → Y be the semi-ample fibration associated to P . For any effective
Cartier divisor E which is supported by Supp(N), the natural map
OY → g∗O(E),
is an isomorphism.
Then the minimal extremal face of Λeff(X) containing D is generated by vertical
divisors of g and components of N .
Proof. When D is big, then the assertion is trivial. We may assume that dimY <
dimX. Let F be the minimal extremal face of Λeff(X) containing D. Since F is
extremal, it follows that F contains all vertical divisors of g and components of N .
On the other hand, our assumption implies that for general fiber Xy, N |Xy is
a rigid divisor on Xy, and its irreducible components generate an extremal face
F ′ of Λeff(Xy). Let α ∈ NM1(Xy) be a nef cycle supporting F ′ and we consider
Fα = α = 0∩Λeff(X). Since Xy is a general fiber, α ∈ NM1(X) so that Fα is an
extremal face. D · α = 0 and the minimality of F imply that F ⊂ Fα. Let D′ ∈ F
be an effective Q-divisor. D′ ·α = 0 implies that D′ is a sum of vertical divisors of
g and components of N . Thus our assertion follows.
Proposition 115. Let X be a projective toric variety and Y an equivariant com-
pactification of a subtorus of codimension one (possibly singular). Let L be a big
line bundle on X. Then L is weakly balanced with respect to Y .
Proof. Let M be the class of OX(Y ). By applying an equivariant embedded reso-
lution of singularities if necessary, we may assume that X and Y are smooth or at
140
least Q-factorial terminal. Due to a group action of a torus, Y is not rigid so that
a(X,L)L|Y +KY = (a(X,L)L+KX)|Y +M |Y ∈ Λeff(Y ).
Note that a(X,L)L+KX is an effective Q-divisor on X. Thus we have
a(Y, L|Y ) ≤ a(X,L).
Suppose that a(Y, L|Y ) = a(X,L) =: a. Let D = aL + KX + Y and we consider
the Zariski decomposition of D:
Xµ
~~~~~~
~~~~ν
⊃ Y
D ⊂Xf
//___ X ′⊃ D′
where D′ is the strict transformation of D, which is nef, and Y is the strict trans-
formation of Y . We may assume that X and Y are both smooth. Let F be the
minimal extremal face of Λeff(X) containing
aµ∗L+KX + Y .
Since aµ∗L + KX ∈ F , it follows that codimF ≤ b(X,L). Note that since X has
only terminal singularities, we have
aµ∗L+KX + Y = aµ∗L+ µ∗KX +∑i
diEi + Y
where di’s are positive integers and Ei’s are µ-exceptional divisors. Thus it follows
that F is the minimal extremal face containing µ∗D and all µ-exceptional divisors.
141
Let g : X → B be the semi-ample fibration associated to ν∗D′. Note that dimB <
dim X since D is not big. Proposition 114 implies that F is generated by all
vertical divisors of g and all ν-exceptional divisors. We denote the vector space,
generated by F , by VF .
Let F ′ be the minimal extremal face of Λeff(Y ) containing aµ∗L|Y + KY =
(aµ∗L+KX + M)|Y where M be the class of OX(Y ). Then F ′ is also the minimal
extremal face containing µ∗D|Y and all components of (Ei ∩ Y )’s so that F ′ is the
minimal extremal face containing ν∗D′|Y and all components of (Gj ∩ Y )’s where
Gj’s are all ν-exceptional divisors. In particular F ′ contains all vertical divisors
of g|Y : Y → H = g(Y ). Since ν∗D′ admits a section vanishing along Y , H is
a Weil divisor of B, which is a subtoric variety. Let V ′ ⊂ NS(Y ) be a vector
space generated by vertical divisors of g|Y and components of (Gj ∩ Y )’s. Then
b(Y, L) ≤ codimV ′. Consider the following restriction map:
Φ : NS(X)/VF → NS(Y )/V ′.
We claim that Φ is surjective. Let N be an irreducible component of the boundary
divisor of Y which dominates H. There exists an irreducible component N ′ of the
boundary divisor of X such that N ′ contains N . Then N ′ also dominates B. As
in the proof of Lemma 106 one can prove that
N ′ ∩ Y = mN + (vertical divisors of g|Y ).
Our claim follows from this. Hence we conclude that
b(Y, L) ≤ codimV ′ ≤ codimVF ≤ b(X,L).
142
Proposition 116. Let X be a Q-factorial terminal projective toric variety and L
a big line bundle on X. Suppose that the positive part of Zariski decomposition of
D := a(X,L)L+KX is nontrivial. Then L is not balanced.
Proof. After applying blowing up if necessary, we may assume that D itself admits
the Zariski decomposition i.e.,
D = P +N
where P is a nef Q-divisor and N ≥ 0 is the negative part. Let g : X → Y
be the semi-ample fibration associated to P . We consider a general fiber Xy of
g. Since a(X,L)L|Xy + KXy = N |Xy is a rigid effective divisor, we conclude that
a(X,L) = a(Xy, L|Xy). Let V ⊂ NS(X) be a vector space generated by vertical
divisors of g and components of N and V ′ ⊂ NS(Xy) a vector space generated by
components of N |Xy . We consider the following restriction map:
Φ : NS(X)/V → NS(Xy)/V′.
Lemma 106 implies that Φ is surjective. On the other hand, let T be the big torus
of Y . Then the preimage g−1(T ) of T is a product of T and a general fiber Xy. It
follows that Φ is injective. Thus we have
b(X,L) = b(Xy, L|Xy).
Hence L is not balanced on X.
Alternative proof of Theorem 108 for toric varieties is provided below:
143
Proposition 117. Let X be a Q-factorial terminal projective toric variety, L a big
line bundle on X, and Y an equivariant compactification of a subtorus of codimen-
sion one (possibly singular). Suppose that the positive part of Zariski decomposition
of a(X,L)L+KX is trivial. Then L is balanced with respect to Y .
Proof. We follow the notations in the proof of Proposition 115. Only part where
we need to explain is why b(Y, L|Y ) < b(X,L) holds when a(Y, L|Y ) = a(X,L).
Since aµ∗L+KX is rigid, it follows that
codimVF < b(X,L).
Thus our assertion follows.
Corollary 118. Let X be a Q-factorial terminal projective toric variety. A big line
bundle L is balanced with respect to all subtoric varieties if and only if a(X,L)L+
KX is rigid.
Remark 119. Propositions 115 and 117 hold when X is Q-factorial terminal and
Y a general smooth divisor. The proofs work with small modifications. However
we still do not know whether they hold for singular divisors.
Example 120. Consider the standard action of G3m = (t0, t1, t2) on P3 by
(t0, t1, t2) · (x0 : x1 : x2 : x3) 7→ (t0x0 : t1x1 : t2x2 : x3).
Consider the subtorus
M = (t0, t1, (t0t1)−1) ⊂ G3m,
144
and let S be the equivariant compactification of M defined by
x0x1x2 = x33.
This is a singular cubic surface with three isolated singularities of type A2. We
denote them by p1, p2, p3 ∈ P3. Since they are fixed under the action of G3m on P3,
the blowup B := Blp1,p2,p3(P3) is an equivariant compactification of G3m. Moreover,
the closure S of M in B is the minimal desingularization of S and the class of S
in Pic(B) is ample. Put X := B × P1 and Y := S × P1. We have a diagram
X
π
Y?_oo
B S,? _oo
Then Y is a nef divisor, and we have
rk NS(Y ) = 8 > rk NS(X) = 5.
However the anticanonical class −KX is still balanced with respect to Y since
a(Y,−KX |Y ) = a(X,−KX) = 1
b(Y,−KX |Y ) = 1 < b(X,−KX) = 5.
This shows that, in general, we cannot expect to control the subgroup of NS(X)
generated by vertical divisors. In the proof of Proposition 115, we were able to
control the quotient by this subgroup.
145
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