the constant appeared in algebraic and complex geometry
TRANSCRIPT
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The β constant appeared in algebraic andcomplex geometry
Min Ru
University of Houston TX, USA
Min Ru The β constant appeared in algebraic and complex geometry
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Nevanlinna theory: Introduction the notations
Let X be a complex projective variety and let D be an effectiveCartier divisor. Let sD be the canonical section of [D] (i.e.[sD = 0] = D) and ‖ ‖ be an hemitian metric, i.e. ‖s‖2 = |sα|2hα.
Let f : C→ X be a holomorphic map. By Poincare-Lelongformula, −ddc [log ‖f ∗sD‖2] = −f ∗D + f ∗c1([D]). Applying∫ t
1dtt
∫|z|<t and use Green-Jensen (Stoke’s theorem), we get the
First Main Theorem:
mf (r ,D) + Nf (r ,D) = Tf ,D(r) + O(1)
where λD(x) = − log ‖sD(x)‖ =− log distance from x to D (Weil
function for D), mf (r ,D) =∫ 2π
0 λD(f (re iθ))dθ2π (Approximation
function). Tf ,L(r) :=∫ r
1dtt
∫|z|<t f
∗c1(L) (Height function).
From First Main Theorem, Nf (r ,D) ≤ Tf ,D(r). The Second MainTheorem (in the spirit of Nevanlinna-Cartan) is to control Tf ,D(r)in terms of Nf (r ,D), or equivalently, to control mf (r ,D) in termsof Tf ,D(r).
Min Ru The β constant appeared in algebraic and complex geometry
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Nevanlinna theory: Introduction the notations
Let X be a complex projective variety and let D be an effectiveCartier divisor. Let sD be the canonical section of [D] (i.e.[sD = 0] = D) and ‖ ‖ be an hemitian metric, i.e. ‖s‖2 = |sα|2hα.Let f : C→ X be a holomorphic map. By Poincare-Lelongformula, −ddc [log ‖f ∗sD‖2] = −f ∗D + f ∗c1([D]).
Applying∫ t1
dtt
∫|z|<t and use Green-Jensen (Stoke’s theorem), we get the
First Main Theorem:
mf (r ,D) + Nf (r ,D) = Tf ,D(r) + O(1)
where λD(x) = − log ‖sD(x)‖ =− log distance from x to D (Weil
function for D), mf (r ,D) =∫ 2π
0 λD(f (re iθ))dθ2π (Approximation
function). Tf ,L(r) :=∫ r
1dtt
∫|z|<t f
∗c1(L) (Height function).
From First Main Theorem, Nf (r ,D) ≤ Tf ,D(r). The Second MainTheorem (in the spirit of Nevanlinna-Cartan) is to control Tf ,D(r)in terms of Nf (r ,D), or equivalently, to control mf (r ,D) in termsof Tf ,D(r).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 4: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/4.jpg)
Nevanlinna theory: Introduction the notations
Let X be a complex projective variety and let D be an effectiveCartier divisor. Let sD be the canonical section of [D] (i.e.[sD = 0] = D) and ‖ ‖ be an hemitian metric, i.e. ‖s‖2 = |sα|2hα.Let f : C→ X be a holomorphic map. By Poincare-Lelongformula, −ddc [log ‖f ∗sD‖2] = −f ∗D + f ∗c1([D]). Applying∫ t
1dtt
∫|z|<t and use Green-Jensen (Stoke’s theorem), we get the
First Main Theorem:
mf (r ,D) + Nf (r ,D) = Tf ,D(r) + O(1)
where λD(x) = − log ‖sD(x)‖ =− log distance from x to D (Weil
function for D), mf (r ,D) =∫ 2π
0 λD(f (re iθ))dθ2π (Approximation
function). Tf ,L(r) :=∫ r
1dtt
∫|z|<t f
∗c1(L) (Height function).
From First Main Theorem, Nf (r ,D) ≤ Tf ,D(r). The Second MainTheorem (in the spirit of Nevanlinna-Cartan) is to control Tf ,D(r)in terms of Nf (r ,D), or equivalently, to control mf (r ,D) in termsof Tf ,D(r).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 5: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/5.jpg)
Nevanlinna theory: Introduction the notations
Let X be a complex projective variety and let D be an effectiveCartier divisor. Let sD be the canonical section of [D] (i.e.[sD = 0] = D) and ‖ ‖ be an hemitian metric, i.e. ‖s‖2 = |sα|2hα.Let f : C→ X be a holomorphic map. By Poincare-Lelongformula, −ddc [log ‖f ∗sD‖2] = −f ∗D + f ∗c1([D]). Applying∫ t
1dtt
∫|z|<t and use Green-Jensen (Stoke’s theorem), we get the
First Main Theorem:
mf (r ,D) + Nf (r ,D) = Tf ,D(r) + O(1)
where λD(x) = − log ‖sD(x)‖ =− log distance from x to D (Weil
function for D), mf (r ,D) =∫ 2π
0 λD(f (re iθ))dθ2π (Approximation
function). Tf ,L(r) :=∫ r
1dtt
∫|z|<t f
∗c1(L) (Height function).
From First Main Theorem, Nf (r ,D) ≤ Tf ,D(r). The Second MainTheorem (in the spirit of Nevanlinna-Cartan) is to control Tf ,D(r)in terms of Nf (r ,D), or equivalently, to control mf (r ,D) in termsof Tf ,D(r).
Min Ru The β constant appeared in algebraic and complex geometry
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Nevanlinna’s SMT for meromorphic functions
The Second Main Theorem(Nevanlinna, 1929). Let f bemeromorphic (non-constant) on C and a1, ..., aq ∈ C ∪ {∞}distinct. Then, for any ε > 0,(q − 2− ε)Tf (r) ≤exc
∑qj=1 Nf (r , aj), or equivalently
q∑j=1
mf (r , aj) ≤exc (2 + ε)Tf (r) ,
where ≤exc means that the inequality holds for r ∈ [0,+∞)outside a set E with finite measure.
This implies the well-knownlittle Picard theorem: If a meromorphic function f on C omitsthree points in C ∪ {∞}, then f must be constant.
Min Ru The β constant appeared in algebraic and complex geometry
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Nevanlinna’s SMT for meromorphic functions
The Second Main Theorem(Nevanlinna, 1929). Let f bemeromorphic (non-constant) on C and a1, ..., aq ∈ C ∪ {∞}distinct. Then, for any ε > 0,(q − 2− ε)Tf (r) ≤exc
∑qj=1 Nf (r , aj), or equivalently
q∑j=1
mf (r , aj) ≤exc (2 + ε)Tf (r) ,
where ≤exc means that the inequality holds for r ∈ [0,+∞)outside a set E with finite measure. This implies the well-knownlittle Picard theorem: If a meromorphic function f on C omitsthree points in C ∪ {∞}, then f must be constant.
Min Ru The β constant appeared in algebraic and complex geometry
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Nevanlinna’s SMT for meromorphic functions
The Second Main Theorem(Nevanlinna, 1929). Let f bemeromorphic (non-constant) on C and a1, ..., aq ∈ C ∪ {∞}distinct. Then, for any ε > 0,(q − 2− ε)Tf (r) ≤exc
∑qj=1 Nf (r , aj), or equivalently
q∑j=1
mf (r , aj) ≤exc (2 + ε)Tf (r) ,
where ≤exc means that the inequality holds for r ∈ [0,+∞)outside a set E with finite measure. This implies the well-knownlittle Picard theorem: If a meromorphic function f on C omitsthree points in C ∪ {∞}, then f must be constant.
Min Ru The β constant appeared in algebraic and complex geometry
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Cartan’s Theorem (1933). Let f : C→ Pn(C) be a linearlynon-degenerate holomorphic map. Let H1, . . . ,Hq be thehyperplanes in general position on Pn(C). Then, for any ε > 0,∑q
j=1 mf (r ,Hj) ≤exc (n + 1 + ε)Tf (r).
In 2004, Ru extended the above result to hypersurfaces forf : C→ Pn(C) with Zariski dense image.∑q
j=11djmf (r ,Dj) ≤exc (n + 1 + ε)Tf (r).
Theorem (Ru, 2009). Let f : C→ X be holo and Zariski dense,D1, . . . ,Dq be divisors in general position in X . Assume thatDj ∼ djA (A being ample). Then, for ∀ ε > 0,q∑
j=1
1
djmf (r ,Dj) ≤exc (dimX + 1 + ε)Tf ,A(r)
Min Ru The β constant appeared in algebraic and complex geometry
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Cartan’s Theorem (1933). Let f : C→ Pn(C) be a linearlynon-degenerate holomorphic map. Let H1, . . . ,Hq be thehyperplanes in general position on Pn(C). Then, for any ε > 0,∑q
j=1 mf (r ,Hj) ≤exc (n + 1 + ε)Tf (r).In 2004, Ru extended the above result to hypersurfaces forf : C→ Pn(C) with Zariski dense image.∑q
j=11djmf (r ,Dj) ≤exc (n + 1 + ε)Tf (r).
Theorem (Ru, 2009). Let f : C→ X be holo and Zariski dense,D1, . . . ,Dq be divisors in general position in X . Assume thatDj ∼ djA (A being ample). Then, for ∀ ε > 0,q∑
j=1
1
djmf (r ,Dj) ≤exc (dimX + 1 + ε)Tf ,A(r)
Min Ru The β constant appeared in algebraic and complex geometry
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Cartan’s Theorem (1933). Let f : C→ Pn(C) be a linearlynon-degenerate holomorphic map. Let H1, . . . ,Hq be thehyperplanes in general position on Pn(C). Then, for any ε > 0,∑q
j=1 mf (r ,Hj) ≤exc (n + 1 + ε)Tf (r).In 2004, Ru extended the above result to hypersurfaces forf : C→ Pn(C) with Zariski dense image.∑q
j=11djmf (r ,Dj) ≤exc (n + 1 + ε)Tf (r).
Theorem (Ru, 2009). Let f : C→ X be holo and Zariski dense,D1, . . . ,Dq be divisors in general position in X . Assume thatDj ∼ djA (A being ample). Then, for ∀ ε > 0,q∑
j=1
1
djmf (r ,Dj) ≤exc (dimX + 1 + ε)Tf ,A(r)
Min Ru The β constant appeared in algebraic and complex geometry
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Theorem (Ru-Vojta, Amer. J. Math., 2020). Let X be a smoothcomplex projective variety and let D1, . . . ,Dq be effective Cartierdivisors in general position. Let D = D1 + · · ·+ Dq. Let L be aline sheaf on X with h0(L N) ≥ 1 for N big enough. Letf : C→ X be a holomorphic map with Zariski image. Then, forevery ε > 0,
q∑j=1
βj(L ,Dj)mf (r ,Dj) ≤exc (1 + ε)Tf ,L (r)
where
β(L ,D) = lim supN→+∞
∑m≥1 dimH0(X ,L N(−mD))
N dimH0(X ,L N).
In the case when Dj ∼ A, then β(D,Dj) = qn+1 , where
D = D1 + · · ·+ Dq.
Min Ru The β constant appeared in algebraic and complex geometry
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Theorem (Ru-Vojta, Amer. J. Math., 2020). Let X be a smoothcomplex projective variety and let D1, . . . ,Dq be effective Cartierdivisors in general position. Let D = D1 + · · ·+ Dq. Let L be aline sheaf on X with h0(L N) ≥ 1 for N big enough. Letf : C→ X be a holomorphic map with Zariski image. Then, forevery ε > 0,
q∑j=1
βj(L ,Dj)mf (r ,Dj) ≤exc (1 + ε)Tf ,L (r)
where
β(L ,D) = lim supN→+∞
∑m≥1 dimH0(X ,L N(−mD))
N dimH0(X ,L N).
In the case when Dj ∼ A, then β(D,Dj) = qn+1 , where
D = D1 + · · ·+ Dq.
Min Ru The β constant appeared in algebraic and complex geometry
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The proof is based on the following basic theorem, which isbasically a reformulation of Cartan’s theorem above:
The Basic Theorem. Let X be a complex projective variety and letL be a line sheaf on X with dimH0(X ,L) ≥ 1. Lets1, . . . , sq ∈ H0(X ,L). Let f : C→ X be a holomorphic map withZariski-dense image. Then, for any ε > 0,∫ 2π
0maxJ
∑j∈J
λsj (f (re iθ))dθ
2π≤exc (dimH0(X ,L) + ε)Tf ,L(r)
where the set J ranges over all subsets of {1, . . . , q} such that thesections (sj)j∈J are linearly independent. Note: The D ∼Q L is ofm-basis type if D := 1
mNm
∑s∈B(s), where B is a basis of
H0(X ,L⊗m), where Nm = dimH0(X ,L⊗m).
Min Ru The β constant appeared in algebraic and complex geometry
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The proof is based on the following basic theorem, which isbasically a reformulation of Cartan’s theorem above:The Basic Theorem. Let X be a complex projective variety and letL be a line sheaf on X with dimH0(X ,L) ≥ 1. Lets1, . . . , sq ∈ H0(X ,L). Let f : C→ X be a holomorphic map withZariski-dense image. Then, for any ε > 0,∫ 2π
0maxJ
∑j∈J
λsj (f (re iθ))dθ
2π≤exc (dimH0(X ,L) + ε)Tf ,L(r)
where the set J ranges over all subsets of {1, . . . , q} such that thesections (sj)j∈J are linearly independent.
Note: The D ∼Q L is ofm-basis type if D := 1
mNm
∑s∈B(s), where B is a basis of
H0(X ,L⊗m), where Nm = dimH0(X ,L⊗m).
Min Ru The β constant appeared in algebraic and complex geometry
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The proof is based on the following basic theorem, which isbasically a reformulation of Cartan’s theorem above:The Basic Theorem. Let X be a complex projective variety and letL be a line sheaf on X with dimH0(X ,L) ≥ 1. Lets1, . . . , sq ∈ H0(X ,L). Let f : C→ X be a holomorphic map withZariski-dense image. Then, for any ε > 0,∫ 2π
0maxJ
∑j∈J
λsj (f (re iθ))dθ
2π≤exc (dimH0(X ,L) + ε)Tf ,L(r)
where the set J ranges over all subsets of {1, . . . , q} such that thesections (sj)j∈J are linearly independent. Note: The D ∼Q L is ofm-basis type if D := 1
mNm
∑s∈B(s), where B is a basis of
H0(X ,L⊗m), where Nm = dimH0(X ,L⊗m).
Min Ru The β constant appeared in algebraic and complex geometry
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Theorem (Weak version of Ru-Vojta). Let X be a complexprojective variety and let D1, . . . ,Dq be effective Cartier divisorssuch that at most ` of such divisors meet at any point of X . Let L
be a line sheaf on X with h0(LN) ≥ 1 for N big enough. Letf : C→ X be a holomorphic map with Zariski-dense image. Then,for every ε > 0,
∑qj=1 β(L,Dj)mf (r ,Dj) ≤exc ` (1 + ε)Tf ,L(r).
The proof is using the Basic Theorem by choosing a a suitablem-basis of H0(X ,Lm) through a filtration.
Min Ru The β constant appeared in algebraic and complex geometry
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Theorem (Weak version of Ru-Vojta). Let X be a complexprojective variety and let D1, . . . ,Dq be effective Cartier divisorssuch that at most ` of such divisors meet at any point of X . Let L
be a line sheaf on X with h0(LN) ≥ 1 for N big enough. Letf : C→ X be a holomorphic map with Zariski-dense image. Then,for every ε > 0,
∑qj=1 β(L,Dj)mf (r ,Dj) ≤exc ` (1 + ε)Tf ,L(r).
The proof is using the Basic Theorem by choosing a a suitablem-basis of H0(X ,Lm) through a filtration.
Min Ru The β constant appeared in algebraic and complex geometry
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Outline of the proof:
For each f (z) = x ∈ X , from the condition that at most ` ofDj , 1 ≤ j ≤ q, meet at x , we have∑q
j=1 βjλDj(x) ≤ `βi0λDi0
(x) + O(1).
Consider the following filtration of H0(X ,LN):
H0(X ,LN) ⊇ H0(X ,LN(−Di0)) ⊇ · · · ⊇ H0(X ,LN(−mDi0)) ⊇ · · ·
and choose a basis s1, · · · , sl ∈ H0(X ,LN), where l = h0(LN)according to this filtration. Notice that for any sections ∈ H0(X ,LN(−mDi0)), we have (s) ≥ mDi0 , so
l∑j=1
(sj) ≥
( ∞∑m=0
m[h0(LN(−mDi0))− h0(LN(−(m + 1)Di0))]
)Di0
=
( ∞∑m=1
h0(LN(−mDi0))
)Di0 .
Min Ru The β constant appeared in algebraic and complex geometry
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Outline of the proof:
For each f (z) = x ∈ X , from the condition that at most ` ofDj , 1 ≤ j ≤ q, meet at x ,
we have∑qj=1 βjλDj
(x) ≤ `βi0λDi0(x) + O(1).
Consider the following filtration of H0(X ,LN):
H0(X ,LN) ⊇ H0(X ,LN(−Di0)) ⊇ · · · ⊇ H0(X ,LN(−mDi0)) ⊇ · · ·
and choose a basis s1, · · · , sl ∈ H0(X ,LN), where l = h0(LN)according to this filtration. Notice that for any sections ∈ H0(X ,LN(−mDi0)), we have (s) ≥ mDi0 , so
l∑j=1
(sj) ≥
( ∞∑m=0
m[h0(LN(−mDi0))− h0(LN(−(m + 1)Di0))]
)Di0
=
( ∞∑m=1
h0(LN(−mDi0))
)Di0 .
Min Ru The β constant appeared in algebraic and complex geometry
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Outline of the proof:
For each f (z) = x ∈ X , from the condition that at most ` ofDj , 1 ≤ j ≤ q, meet at x , we have∑q
j=1 βjλDj(x) ≤ `βi0λDi0
(x) + O(1).
Consider the following filtration of H0(X ,LN):
H0(X ,LN) ⊇ H0(X ,LN(−Di0)) ⊇ · · · ⊇ H0(X ,LN(−mDi0)) ⊇ · · ·
and choose a basis s1, · · · , sl ∈ H0(X ,LN), where l = h0(LN)according to this filtration. Notice that for any sections ∈ H0(X ,LN(−mDi0)), we have (s) ≥ mDi0 , so
l∑j=1
(sj) ≥
( ∞∑m=0
m[h0(LN(−mDi0))− h0(LN(−(m + 1)Di0))]
)Di0
=
( ∞∑m=1
h0(LN(−mDi0))
)Di0 .
Min Ru The β constant appeared in algebraic and complex geometry
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Outline of the proof:
For each f (z) = x ∈ X , from the condition that at most ` ofDj , 1 ≤ j ≤ q, meet at x , we have∑q
j=1 βjλDj(x) ≤ `βi0λDi0
(x) + O(1).
Consider the following filtration of H0(X ,LN):
H0(X ,LN) ⊇ H0(X ,LN(−Di0)) ⊇ · · · ⊇ H0(X ,LN(−mDi0)) ⊇ · · ·
and choose a basis s1, · · · , sl ∈ H0(X ,LN), where l = h0(LN)according to this filtration.
Notice that for any sections ∈ H0(X ,LN(−mDi0)), we have (s) ≥ mDi0 , so
l∑j=1
(sj) ≥
( ∞∑m=0
m[h0(LN(−mDi0))− h0(LN(−(m + 1)Di0))]
)Di0
=
( ∞∑m=1
h0(LN(−mDi0))
)Di0 .
Min Ru The β constant appeared in algebraic and complex geometry
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Outline of the proof:
For each f (z) = x ∈ X , from the condition that at most ` ofDj , 1 ≤ j ≤ q, meet at x , we have∑q
j=1 βjλDj(x) ≤ `βi0λDi0
(x) + O(1).
Consider the following filtration of H0(X ,LN):
H0(X ,LN) ⊇ H0(X ,LN(−Di0)) ⊇ · · · ⊇ H0(X ,LN(−mDi0)) ⊇ · · ·
and choose a basis s1, · · · , sl ∈ H0(X ,LN), where l = h0(LN)according to this filtration. Notice that for any sections ∈ H0(X ,LN(−mDi0)), we have (s) ≥ mDi0 ,
so
l∑j=1
(sj) ≥
( ∞∑m=0
m[h0(LN(−mDi0))− h0(LN(−(m + 1)Di0))]
)Di0
=
( ∞∑m=1
h0(LN(−mDi0))
)Di0 .
Min Ru The β constant appeared in algebraic and complex geometry
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Outline of the proof:
For each f (z) = x ∈ X , from the condition that at most ` ofDj , 1 ≤ j ≤ q, meet at x , we have∑q
j=1 βjλDj(x) ≤ `βi0λDi0
(x) + O(1).
Consider the following filtration of H0(X ,LN):
H0(X ,LN) ⊇ H0(X ,LN(−Di0)) ⊇ · · · ⊇ H0(X ,LN(−mDi0)) ⊇ · · ·
and choose a basis s1, · · · , sl ∈ H0(X ,LN), where l = h0(LN)according to this filtration. Notice that for any sections ∈ H0(X ,LN(−mDi0)), we have (s) ≥ mDi0 , so
l∑j=1
(sj) ≥
( ∞∑m=0
m[h0(LN(−mDi0))− h0(LN(−(m + 1)Di0))]
)Di0
=
( ∞∑m=1
h0(LN(−mDi0))
)Di0 .
Min Ru The β constant appeared in algebraic and complex geometry
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Hence the m-basis
1
Nh0(LN)
h0(LN)∑j=1
(sj)
≥∑∞
m=1 h0(LN(−mDi0)
Nh0(LN)Di0 .
It then follows from the Basic Theorem.
In summary: The proofis about estimate the order of the m-basis coming from thefiltration, and then apply the basic Theroem.
Min Ru The β constant appeared in algebraic and complex geometry
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Hence the m-basis
1
Nh0(LN)
h0(LN)∑j=1
(sj)
≥∑∞
m=1 h0(LN(−mDi0)
Nh0(LN)Di0 .
It then follows from the Basic Theorem. In summary: The proofis about estimate the order of the m-basis coming from thefiltration, and then apply the basic Theroem.
Min Ru The β constant appeared in algebraic and complex geometry
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Diophantine approximation
Roth’s theorem states that every irrational algebraic number αhas approximation exponent equal to 2, i.e.,Theorem (Roth, 1955). Let α be an algebraic number of
degree≥ 2. Then, for any given ε > 0, we have∣∣∣α− p
q
∣∣∣ > 1q2+ε for
all, but finitely many, coprime integers p and q.Roth’s Theorem. k=number field and S=finite set of places on k .a1, . . . , aq distinct in P1(k). Then
q∑j=1
∑υ∈S
log+ 1
‖x − aj‖υ≤ (2 + ε)h(x)
holds for ∀ x ∈ P1(k) except for finitely many points.Denote by
mS(x , a) :=∑υ∈S
log+ 1
‖x − a‖υ.
Then∑q
j=1 mS(x , aj) ≤exc (2 + ε)h(x).
Min Ru The β constant appeared in algebraic and complex geometry
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Diophantine approximation
Roth’s theorem states that every irrational algebraic number αhas approximation exponent equal to 2, i.e.,
Theorem (Roth, 1955). Let α be an algebraic number of
degree≥ 2. Then, for any given ε > 0, we have∣∣∣α− p
q
∣∣∣ > 1q2+ε for
all, but finitely many, coprime integers p and q.Roth’s Theorem. k=number field and S=finite set of places on k .a1, . . . , aq distinct in P1(k). Then
q∑j=1
∑υ∈S
log+ 1
‖x − aj‖υ≤ (2 + ε)h(x)
holds for ∀ x ∈ P1(k) except for finitely many points.Denote by
mS(x , a) :=∑υ∈S
log+ 1
‖x − a‖υ.
Then∑q
j=1 mS(x , aj) ≤exc (2 + ε)h(x).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 29: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/29.jpg)
Diophantine approximation
Roth’s theorem states that every irrational algebraic number αhas approximation exponent equal to 2, i.e.,Theorem (Roth, 1955). Let α be an algebraic number of
degree≥ 2. Then, for any given ε > 0, we have∣∣∣α− p
q
∣∣∣ > 1q2+ε for
all, but finitely many, coprime integers p and q.
Roth’s Theorem. k=number field and S=finite set of places on k .a1, . . . , aq distinct in P1(k). Then
q∑j=1
∑υ∈S
log+ 1
‖x − aj‖υ≤ (2 + ε)h(x)
holds for ∀ x ∈ P1(k) except for finitely many points.Denote by
mS(x , a) :=∑υ∈S
log+ 1
‖x − a‖υ.
Then∑q
j=1 mS(x , aj) ≤exc (2 + ε)h(x).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 30: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/30.jpg)
Diophantine approximation
Roth’s theorem states that every irrational algebraic number αhas approximation exponent equal to 2, i.e.,Theorem (Roth, 1955). Let α be an algebraic number of
degree≥ 2. Then, for any given ε > 0, we have∣∣∣α− p
q
∣∣∣ > 1q2+ε for
all, but finitely many, coprime integers p and q.Roth’s Theorem. k=number field and S=finite set of places on k .a1, . . . , aq distinct in P1(k). Then
q∑j=1
∑υ∈S
log+ 1
‖x − aj‖υ≤ (2 + ε)h(x)
holds for ∀ x ∈ P1(k) except for finitely many points.Denote by
mS(x , a) :=∑υ∈S
log+ 1
‖x − a‖υ.
Then∑q
j=1 mS(x , aj) ≤exc (2 + ε)h(x).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 31: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/31.jpg)
Let L be a big line bundle on X and D an effective divisor.
Define
β(L,D) := lim supN→∞
∑m≥1 h
0(LN(−mD))
Nh0(LN).
Theorem (Ru-Vojta, 2020) [Arithmetic Part] Let X be a projectivevariety over a number field k, and D1, . . . ,Dq be effective Cartierdivisors intersecting properly on X . Let S ⊂ Mk be a finite set ofplaces. Then, for every ε > 0, the inequality
q∑i=1
β(L,Dj)mS(x ,Dj) ≤ (1 + ε)hL(x)
holds for all k-rational points outside a proper Zariski-closedsubset of X .
Min Ru The β constant appeared in algebraic and complex geometry
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Let L be a big line bundle on X and D an effective divisor. Define
β(L,D) := lim supN→∞
∑m≥1 h
0(LN(−mD))
Nh0(LN).
Theorem (Ru-Vojta, 2020) [Arithmetic Part] Let X be a projectivevariety over a number field k, and D1, . . . ,Dq be effective Cartierdivisors intersecting properly on X . Let S ⊂ Mk be a finite set ofplaces. Then, for every ε > 0, the inequality
q∑i=1
β(L,Dj)mS(x ,Dj) ≤ (1 + ε)hL(x)
holds for all k-rational points outside a proper Zariski-closedsubset of X .
Min Ru The β constant appeared in algebraic and complex geometry
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The volume function
One studies the asymptotic behavior H0(X ,mL) as m→∞.
Perhaps the most important important asymptotic invariant for aline bundle (divisor) L is the volume:
Vol(L) = lim supm→∞
dimH0(X ,mL)
mn/n!
or
h0(mL) =Vol(L)
n!mn + O(mn−1).
Notice that Vol(kL) = knVol(L) so the volume function can beextended to Q-divisors. Also note that Vol( ) depends only on thenumerical class of L, so it is defined onNS(X ) := Div(X )/Num(X ) and extends uniquely to a continuousfunction on NS(X )R. The volume function lies at the intersectionof many fields of mathematics and has a variety of interestingapplications (bi-rational geometry, complex geometry, numbertheory etc.)
Min Ru The β constant appeared in algebraic and complex geometry
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The volume function
One studies the asymptotic behavior H0(X ,mL) as m→∞.Perhaps the most important important asymptotic invariant for aline bundle (divisor) L is the volume:
Vol(L) = lim supm→∞
dimH0(X ,mL)
mn/n!
or
h0(mL) =Vol(L)
n!mn + O(mn−1).
Notice that Vol(kL) = knVol(L) so the volume function can beextended to Q-divisors. Also note that Vol( ) depends only on thenumerical class of L, so it is defined onNS(X ) := Div(X )/Num(X ) and extends uniquely to a continuousfunction on NS(X )R. The volume function lies at the intersectionof many fields of mathematics and has a variety of interestingapplications (bi-rational geometry, complex geometry, numbertheory etc.)
Min Ru The β constant appeared in algebraic and complex geometry
![Page 35: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/35.jpg)
The volume function
One studies the asymptotic behavior H0(X ,mL) as m→∞.Perhaps the most important important asymptotic invariant for aline bundle (divisor) L is the volume:
Vol(L) = lim supm→∞
dimH0(X ,mL)
mn/n!
or
h0(mL) =Vol(L)
n!mn + O(mn−1).
Notice that Vol(kL) = knVol(L) so the volume function can beextended to Q-divisors. Also note that Vol( ) depends only on thenumerical class of L, so it is defined onNS(X ) := Div(X )/Num(X ) and extends uniquely to a continuousfunction on NS(X )R. The volume function lies at the intersectionof many fields of mathematics and has a variety of interestingapplications (bi-rational geometry, complex geometry, numbertheory etc.)
Min Ru The β constant appeared in algebraic and complex geometry
![Page 36: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/36.jpg)
Recall
β(L,D) := lim supN→∞
∑m≥1 h
0(LN(−mD))
Nh0(LN).
So we can express the above constant through the notion ofVol(L),
β(L,D) =1
Vol(L)
∫ ∞0
Vol(L− tD)dt.
This can be proved by using the theory of Okounkov body.
Min Ru The β constant appeared in algebraic and complex geometry
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Recall
β(L,D) := lim supN→∞
∑m≥1 h
0(LN(−mD))
Nh0(LN).
So we can express the above constant through the notion ofVol(L),
β(L,D) =1
Vol(L)
∫ ∞0
Vol(L− tD)dt.
This can be proved by using the theory of Okounkov body.
Min Ru The β constant appeared in algebraic and complex geometry
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Okounkove body
Let L be a big line bindule on X . An Okounkov body ∆(L) ⊂ Rn
(where n = dimX ) is a compact convex set designed to study theasymptotic behavior of H0(X ,mL), as m→∞. They have thecrucial property that the Eulidean volume
Vol(∆) = limm→∞dimH0(X ,mL)
mn =Vol(L)
n! . Here is the detaileddescription. Fix a system z = (z1, . . . , zn) of parameters centeredat a regular closed point ξ of X . This defines a real rank-nvaluation ordz : OX ,ξ\{0} → Nn byf 7→ ordz(f ) := minlex{α ∈ Nn | aα 6= 0}. LetΓm := ordz
(H0(X ,mL)\{0}
)⊂ Nn,then #Γm = dimH0(X ,mL).
Let Σ be the closed convex cone generated by{(m, α) ∈ Nn+1 | α ∈ Γm}. The Okounkov body of L is
∆ = Σ ∩ ({1} × Rn) ⊂ Rn.
Min Ru The β constant appeared in algebraic and complex geometry
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Okounkove body
Let L be a big line bindule on X . An Okounkov body ∆(L) ⊂ Rn
(where n = dimX ) is a compact convex set designed to study theasymptotic behavior of H0(X ,mL), as m→∞. They have thecrucial property that the Eulidean volume
Vol(∆) = limm→∞dimH0(X ,mL)
mn =Vol(L)
n! .
Here is the detaileddescription. Fix a system z = (z1, . . . , zn) of parameters centeredat a regular closed point ξ of X . This defines a real rank-nvaluation ordz : OX ,ξ\{0} → Nn byf 7→ ordz(f ) := minlex{α ∈ Nn | aα 6= 0}. LetΓm := ordz
(H0(X ,mL)\{0}
)⊂ Nn,then #Γm = dimH0(X ,mL).
Let Σ be the closed convex cone generated by{(m, α) ∈ Nn+1 | α ∈ Γm}. The Okounkov body of L is
∆ = Σ ∩ ({1} × Rn) ⊂ Rn.
Min Ru The β constant appeared in algebraic and complex geometry
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Okounkove body
Let L be a big line bindule on X . An Okounkov body ∆(L) ⊂ Rn
(where n = dimX ) is a compact convex set designed to study theasymptotic behavior of H0(X ,mL), as m→∞. They have thecrucial property that the Eulidean volume
Vol(∆) = limm→∞dimH0(X ,mL)
mn =Vol(L)
n! . Here is the detaileddescription.
Fix a system z = (z1, . . . , zn) of parameters centeredat a regular closed point ξ of X . This defines a real rank-nvaluation ordz : OX ,ξ\{0} → Nn byf 7→ ordz(f ) := minlex{α ∈ Nn | aα 6= 0}. LetΓm := ordz
(H0(X ,mL)\{0}
)⊂ Nn,then #Γm = dimH0(X ,mL).
Let Σ be the closed convex cone generated by{(m, α) ∈ Nn+1 | α ∈ Γm}. The Okounkov body of L is
∆ = Σ ∩ ({1} × Rn) ⊂ Rn.
Min Ru The β constant appeared in algebraic and complex geometry
![Page 41: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/41.jpg)
Okounkove body
Let L be a big line bindule on X . An Okounkov body ∆(L) ⊂ Rn
(where n = dimX ) is a compact convex set designed to study theasymptotic behavior of H0(X ,mL), as m→∞. They have thecrucial property that the Eulidean volume
Vol(∆) = limm→∞dimH0(X ,mL)
mn =Vol(L)
n! . Here is the detaileddescription. Fix a system z = (z1, . . . , zn) of parameters centeredat a regular closed point ξ of X . This defines a real rank-nvaluation ordz : OX ,ξ\{0} → Nn byf 7→ ordz(f ) := minlex{α ∈ Nn | aα 6= 0}.
LetΓm := ordz
(H0(X ,mL)\{0}
)⊂ Nn,then #Γm = dimH0(X ,mL).
Let Σ be the closed convex cone generated by{(m, α) ∈ Nn+1 | α ∈ Γm}. The Okounkov body of L is
∆ = Σ ∩ ({1} × Rn) ⊂ Rn.
Min Ru The β constant appeared in algebraic and complex geometry
![Page 42: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/42.jpg)
Okounkove body
Let L be a big line bindule on X . An Okounkov body ∆(L) ⊂ Rn
(where n = dimX ) is a compact convex set designed to study theasymptotic behavior of H0(X ,mL), as m→∞. They have thecrucial property that the Eulidean volume
Vol(∆) = limm→∞dimH0(X ,mL)
mn =Vol(L)
n! . Here is the detaileddescription. Fix a system z = (z1, . . . , zn) of parameters centeredat a regular closed point ξ of X . This defines a real rank-nvaluation ordz : OX ,ξ\{0} → Nn byf 7→ ordz(f ) := minlex{α ∈ Nn | aα 6= 0}. LetΓm := ordz
(H0(X ,mL)\{0}
)⊂ Nn,then #Γm = dimH0(X ,mL).
Let Σ be the closed convex cone generated by{(m, α) ∈ Nn+1 | α ∈ Γm}. The Okounkov body of L is
∆ = Σ ∩ ({1} × Rn) ⊂ Rn.
Min Ru The β constant appeared in algebraic and complex geometry
![Page 43: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/43.jpg)
Okounkove body
Let L be a big line bindule on X . An Okounkov body ∆(L) ⊂ Rn
(where n = dimX ) is a compact convex set designed to study theasymptotic behavior of H0(X ,mL), as m→∞. They have thecrucial property that the Eulidean volume
Vol(∆) = limm→∞dimH0(X ,mL)
mn =Vol(L)
n! . Here is the detaileddescription. Fix a system z = (z1, . . . , zn) of parameters centeredat a regular closed point ξ of X . This defines a real rank-nvaluation ordz : OX ,ξ\{0} → Nn byf 7→ ordz(f ) := minlex{α ∈ Nn | aα 6= 0}. LetΓm := ordz
(H0(X ,mL)\{0}
)⊂ Nn,then #Γm = dimH0(X ,mL).
Let Σ be the closed convex cone generated by{(m, α) ∈ Nn+1 | α ∈ Γm}. The Okounkov body of L is
∆ = Σ ∩ ({1} × Rn) ⊂ Rn.
Min Ru The β constant appeared in algebraic and complex geometry
![Page 44: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/44.jpg)
We can also construct a Okounkov body for a linear seriesVm ⊂ H0(X ,mL).
Write V• :=⊕
m Vm. According tolazarsfeld-Mustata (2009), textcolorbluethe Eucldean volumeVol(∆(V•)) is equal to limm→∞m−n dimVm.The Vanishing sum: Given a filtration F (for exampleFtm := H0(mL− tD)), consider the jumping numbers
0 ≤ am,1 ≤ · · · ≤ am,Nm , defined by,am,j = aFm,j = inf{t ∈ R+ | codimFt
m ≥ j} for 1 ≤ j ≤ Nm.
Define a positive (Duistermaat-Heckman) measure µm = µFm onR+ by µm = 1
mn
∑Nmj=1 δm−1am,j . Then, from Boucksom-Chen
(2011), we havelim
m→+∞µm = µ
in the weak sense of measures on R+, where µ = (GF)∗λ,GF : ∆(V•)→ [−∞,+∞), GF(x) := sup{t ∈ R, x ∈ ∆(V t
• )}.
Min Ru The β constant appeared in algebraic and complex geometry
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We can also construct a Okounkov body for a linear seriesVm ⊂ H0(X ,mL). Write V• :=
⊕m Vm. According to
lazarsfeld-Mustata (2009), textcolorbluethe Eucldean volumeVol(∆(V•)) is equal to limm→∞m−n dimVm.
The Vanishing sum: Given a filtration F (for exampleFtm := H0(mL− tD)), consider the jumping numbers
0 ≤ am,1 ≤ · · · ≤ am,Nm , defined by,am,j = aFm,j = inf{t ∈ R+ | codimFt
m ≥ j} for 1 ≤ j ≤ Nm.
Define a positive (Duistermaat-Heckman) measure µm = µFm onR+ by µm = 1
mn
∑Nmj=1 δm−1am,j . Then, from Boucksom-Chen
(2011), we havelim
m→+∞µm = µ
in the weak sense of measures on R+, where µ = (GF)∗λ,GF : ∆(V•)→ [−∞,+∞), GF(x) := sup{t ∈ R, x ∈ ∆(V t
• )}.
Min Ru The β constant appeared in algebraic and complex geometry
![Page 46: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/46.jpg)
We can also construct a Okounkov body for a linear seriesVm ⊂ H0(X ,mL). Write V• :=
⊕m Vm. According to
lazarsfeld-Mustata (2009), textcolorbluethe Eucldean volumeVol(∆(V•)) is equal to limm→∞m−n dimVm.The Vanishing sum: Given a filtration F (for exampleFtm := H0(mL− tD)), consider the jumping numbers
0 ≤ am,1 ≤ · · · ≤ am,Nm , defined by,am,j = aFm,j = inf{t ∈ R+ | codimFt
m ≥ j} for 1 ≤ j ≤ Nm.
Define a positive (Duistermaat-Heckman) measure µm = µFm onR+ by µm = 1
mn
∑Nmj=1 δm−1am,j . Then, from Boucksom-Chen
(2011), we havelim
m→+∞µm = µ
in the weak sense of measures on R+, where µ = (GF)∗λ,GF : ∆(V•)→ [−∞,+∞), GF(x) := sup{t ∈ R, x ∈ ∆(V t
• )}.
Min Ru The β constant appeared in algebraic and complex geometry
![Page 47: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/47.jpg)
We can also construct a Okounkov body for a linear seriesVm ⊂ H0(X ,mL). Write V• :=
⊕m Vm. According to
lazarsfeld-Mustata (2009), textcolorbluethe Eucldean volumeVol(∆(V•)) is equal to limm→∞m−n dimVm.The Vanishing sum: Given a filtration F (for exampleFtm := H0(mL− tD)), consider the jumping numbers
0 ≤ am,1 ≤ · · · ≤ am,Nm , defined by,am,j = aFm,j = inf{t ∈ R+ | codimFt
m ≥ j} for 1 ≤ j ≤ Nm.
Define a positive (Duistermaat-Heckman) measure µm = µFm onR+ by µm = 1
mn
∑Nmj=1 δm−1am,j .
Then, from Boucksom-Chen(2011), we have
limm→+∞
µm = µ
in the weak sense of measures on R+, where µ = (GF)∗λ,GF : ∆(V•)→ [−∞,+∞), GF(x) := sup{t ∈ R, x ∈ ∆(V t
• )}.
Min Ru The β constant appeared in algebraic and complex geometry
![Page 48: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/48.jpg)
We can also construct a Okounkov body for a linear seriesVm ⊂ H0(X ,mL). Write V• :=
⊕m Vm. According to
lazarsfeld-Mustata (2009), textcolorbluethe Eucldean volumeVol(∆(V•)) is equal to limm→∞m−n dimVm.The Vanishing sum: Given a filtration F (for exampleFtm := H0(mL− tD)), consider the jumping numbers
0 ≤ am,1 ≤ · · · ≤ am,Nm , defined by,am,j = aFm,j = inf{t ∈ R+ | codimFt
m ≥ j} for 1 ≤ j ≤ Nm.
Define a positive (Duistermaat-Heckman) measure µm = µFm onR+ by µm = 1
mn
∑Nmj=1 δm−1am,j . Then, from Boucksom-Chen
(2011), we havelim
m→+∞µm = µ
in the weak sense of measures on R+, where µ = (GF)∗λ,GF : ∆(V•)→ [−∞,+∞), GF(x) := sup{t ∈ R, x ∈ ∆(V t
• )}.
Min Ru The β constant appeared in algebraic and complex geometry
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K-stablility
The notion of the K-stability of Fano varieties is analgebro-geometric stability condition originally motivated by studiesof Kahler metrics.
Indeed, as expected, when the base field is thecomplex number field, it is recently established that the existenceof positive scalar curvature Kahler-Einstein metrics, i.e., Kahlermetrics with constant Ricci curvature, is actually equivalent to thealgebro-geometric condition “K-stability”, by the works of Tian,Donaldson, and Chen-Donaldson-Sun. This equivalence had beenknown before as the Yau-Tian-Donaldson conjecture (for the caseof Fano varieties). An important problem in algebraic geometry isto find a simple criterion to test the K -stability of the variety X .
Min Ru The β constant appeared in algebraic and complex geometry
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K-stablility
The notion of the K-stability of Fano varieties is analgebro-geometric stability condition originally motivated by studiesof Kahler metrics. Indeed, as expected, when the base field is thecomplex number field, it is recently established that the existenceof positive scalar curvature Kahler-Einstein metrics, i.e., Kahlermetrics with constant Ricci curvature, is actually equivalent to thealgebro-geometric condition “K-stability”, by the works of Tian,Donaldson, and Chen-Donaldson-Sun. This equivalence had beenknown before as the Yau-Tian-Donaldson conjecture (for the caseof Fano varieties).
An important problem in algebraic geometry isto find a simple criterion to test the K -stability of the variety X .
Min Ru The β constant appeared in algebraic and complex geometry
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K-stablility
The notion of the K-stability of Fano varieties is analgebro-geometric stability condition originally motivated by studiesof Kahler metrics. Indeed, as expected, when the base field is thecomplex number field, it is recently established that the existenceof positive scalar curvature Kahler-Einstein metrics, i.e., Kahlermetrics with constant Ricci curvature, is actually equivalent to thealgebro-geometric condition “K-stability”, by the works of Tian,Donaldson, and Chen-Donaldson-Sun. This equivalence had beenknown before as the Yau-Tian-Donaldson conjecture (for the caseof Fano varieties). An important problem in algebraic geometry isto find a simple criterion to test the K -stability of the variety X .
Min Ru The β constant appeared in algebraic and complex geometry
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In 2015, Fujita showed that if (Fano) X is K -(semi) stable, thenβ(−KX ,D) < 1 (resp. β(−KX ,D) ≤ 1) for any nonzero effectivedivisors on X .
After the Annals paper (2014) by C. Xu and C. Lientitled ”Special test configuration and K -stablity of Fanovarieties”, Fujita and C. Li independently proved that it is indeedan equivalence condition if one goes to the birational model, i.e.The Q-fano varietie X is K -(semi) stable if and only if
AX (E)β(−KX ,E) > 1 for any prime divisors E over X (i.e. E is a prime
divisor on a birational model π : X → X ), whereAX (E ) := 1 + ordE (KY /X ) and is called the log discrepancy. X issaid to have klt singularities if AX (E ) > 0 for all prime divisors
over X . We call δ(L) = infEAX (E)β(L,E) the stability threshold. So X
is K -(semi) stable iff δ(−KX ) > 1.
Min Ru The β constant appeared in algebraic and complex geometry
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In 2015, Fujita showed that if (Fano) X is K -(semi) stable, thenβ(−KX ,D) < 1 (resp. β(−KX ,D) ≤ 1) for any nonzero effectivedivisors on X . After the Annals paper (2014) by C. Xu and C. Lientitled ”Special test configuration and K -stablity of Fanovarieties”, Fujita and C. Li independently proved that it is indeedan equivalence condition if one goes to the birational model, i.e.The Q-fano varietie X is K -(semi) stable if and only if
AX (E)β(−KX ,E) > 1 for any prime divisors E over X (i.e. E is a prime
divisor on a birational model π : X → X ), whereAX (E ) := 1 + ordE (KY /X ) and is called the log discrepancy.
X issaid to have klt singularities if AX (E ) > 0 for all prime divisors
over X . We call δ(L) = infEAX (E)β(L,E) the stability threshold. So X
is K -(semi) stable iff δ(−KX ) > 1.
Min Ru The β constant appeared in algebraic and complex geometry
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In 2015, Fujita showed that if (Fano) X is K -(semi) stable, thenβ(−KX ,D) < 1 (resp. β(−KX ,D) ≤ 1) for any nonzero effectivedivisors on X . After the Annals paper (2014) by C. Xu and C. Lientitled ”Special test configuration and K -stablity of Fanovarieties”, Fujita and C. Li independently proved that it is indeedan equivalence condition if one goes to the birational model, i.e.The Q-fano varietie X is K -(semi) stable if and only if
AX (E)β(−KX ,E) > 1 for any prime divisors E over X (i.e. E is a prime
divisor on a birational model π : X → X ), whereAX (E ) := 1 + ordE (KY /X ) and is called the log discrepancy. X issaid to have klt singularities if AX (E ) > 0 for all prime divisors
over X .
We call δ(L) = infEAX (E)β(L,E) the stability threshold. So X
is K -(semi) stable iff δ(−KX ) > 1.
Min Ru The β constant appeared in algebraic and complex geometry
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In 2015, Fujita showed that if (Fano) X is K -(semi) stable, thenβ(−KX ,D) < 1 (resp. β(−KX ,D) ≤ 1) for any nonzero effectivedivisors on X . After the Annals paper (2014) by C. Xu and C. Lientitled ”Special test configuration and K -stablity of Fanovarieties”, Fujita and C. Li independently proved that it is indeedan equivalence condition if one goes to the birational model, i.e.The Q-fano varietie X is K -(semi) stable if and only if
AX (E)β(−KX ,E) > 1 for any prime divisors E over X (i.e. E is a prime
divisor on a birational model π : X → X ), whereAX (E ) := 1 + ordE (KY /X ) and is called the log discrepancy. X issaid to have klt singularities if AX (E ) > 0 for all prime divisors
over X . We call δ(L) = infEAX (E)β(L,E) the stability threshold. So X
is K -(semi) stable iff δ(−KX ) > 1.
Min Ru The β constant appeared in algebraic and complex geometry
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Blum-Jonsson used m-basis type to describe the stability thresholdδ(L):
they proved δ(L) = lim δm(L), whereδm(L) := inf{lct(D) | D ∼Q L of m-basis type}. (through m-basis).Algebraic geometry definition of “log canonical threshold”:
lct(D) = minE
AX (E )
ordE (D),
where the minimal is taken over all primes E over X .
Min Ru The β constant appeared in algebraic and complex geometry
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Blum-Jonsson used m-basis type to describe the stability thresholdδ(L): they proved δ(L) = lim δm(L), whereδm(L) := inf{lct(D) | D ∼Q L of m-basis type}. (through m-basis).Algebraic geometry definition of “log canonical threshold”:
lct(D) = minE
AX (E )
ordE (D),
where the minimal is taken over all primes E over X .
Min Ru The β constant appeared in algebraic and complex geometry
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The log canonical threshold through singular metric
Tian in 1987 introduced α(L) the log canonical threshold of L asfollows: Let h = e−φ be a singular metric with ΘL,h ≥ 0, where
ΘL,h =√−1π ∂∂ log φ. Define cp(h) = sup{c | e−2cφ is locally
integrable at p }. Define, for p ∈ X , αp(L) = infh:ΘL,h≥0 cp(h) andα(L) = infp∈X αp(L).
Tian proved that if α(−KX ) > nn+1 , then
X is K -stable. Let D be an effective Cartier divisor, then thecanonical section sD of [D] gives a singular metric on [D] withφ := log |sD |. We denote lctp(D) := cp(h) andlct(D) := infp∈X lctp(D) with such metric. According toDemailly,
α(L) = inf{lct(D) | D is effective,D ∼Q L}.
Use the fact that, for φ = log |f |, e−2cφ = 1|f |2c , and the fact
that∫
1|z|a2λ <∞ iff λa− 1 < 0, i.e. λ < 1
a , this links with the
(algebraic geometry) definition for lct(D).
Min Ru The β constant appeared in algebraic and complex geometry
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The log canonical threshold through singular metric
Tian in 1987 introduced α(L) the log canonical threshold of L asfollows: Let h = e−φ be a singular metric with ΘL,h ≥ 0, where
ΘL,h =√−1π ∂∂ log φ. Define cp(h) = sup{c | e−2cφ is locally
integrable at p }. Define, for p ∈ X , αp(L) = infh:ΘL,h≥0 cp(h) andα(L) = infp∈X αp(L). Tian proved that if α(−KX ) > n
n+1 , thenX is K -stable.
Let D be an effective Cartier divisor, then thecanonical section sD of [D] gives a singular metric on [D] withφ := log |sD |. We denote lctp(D) := cp(h) andlct(D) := infp∈X lctp(D) with such metric. According toDemailly,
α(L) = inf{lct(D) | D is effective,D ∼Q L}.
Use the fact that, for φ = log |f |, e−2cφ = 1|f |2c , and the fact
that∫
1|z|a2λ <∞ iff λa− 1 < 0, i.e. λ < 1
a , this links with the
(algebraic geometry) definition for lct(D).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 60: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/60.jpg)
The log canonical threshold through singular metric
Tian in 1987 introduced α(L) the log canonical threshold of L asfollows: Let h = e−φ be a singular metric with ΘL,h ≥ 0, where
ΘL,h =√−1π ∂∂ log φ. Define cp(h) = sup{c | e−2cφ is locally
integrable at p }. Define, for p ∈ X , αp(L) = infh:ΘL,h≥0 cp(h) andα(L) = infp∈X αp(L). Tian proved that if α(−KX ) > n
n+1 , thenX is K -stable. Let D be an effective Cartier divisor, then thecanonical section sD of [D] gives a singular metric on [D] withφ := log |sD |. We denote lctp(D) := cp(h) andlct(D) := infp∈X lctp(D) with such metric.
According toDemailly,
α(L) = inf{lct(D) | D is effective,D ∼Q L}.
Use the fact that, for φ = log |f |, e−2cφ = 1|f |2c , and the fact
that∫
1|z|a2λ <∞ iff λa− 1 < 0, i.e. λ < 1
a , this links with the
(algebraic geometry) definition for lct(D).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 61: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/61.jpg)
The log canonical threshold through singular metric
Tian in 1987 introduced α(L) the log canonical threshold of L asfollows: Let h = e−φ be a singular metric with ΘL,h ≥ 0, where
ΘL,h =√−1π ∂∂ log φ. Define cp(h) = sup{c | e−2cφ is locally
integrable at p }. Define, for p ∈ X , αp(L) = infh:ΘL,h≥0 cp(h) andα(L) = infp∈X αp(L). Tian proved that if α(−KX ) > n
n+1 , thenX is K -stable. Let D be an effective Cartier divisor, then thecanonical section sD of [D] gives a singular metric on [D] withφ := log |sD |. We denote lctp(D) := cp(h) andlct(D) := infp∈X lctp(D) with such metric. According toDemailly,
α(L) = inf{lct(D) | D is effective,D ∼Q L}.
Use the fact that, for φ = log |f |, e−2cφ = 1|f |2c , and the fact
that∫
1|z|a2λ <∞ iff λa− 1 < 0, i.e. λ < 1
a , this links with the
(algebraic geometry) definition for lct(D).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 62: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/62.jpg)
The log canonical threshold through singular metric
Tian in 1987 introduced α(L) the log canonical threshold of L asfollows: Let h = e−φ be a singular metric with ΘL,h ≥ 0, where
ΘL,h =√−1π ∂∂ log φ. Define cp(h) = sup{c | e−2cφ is locally
integrable at p }. Define, for p ∈ X , αp(L) = infh:ΘL,h≥0 cp(h) andα(L) = infp∈X αp(L). Tian proved that if α(−KX ) > n
n+1 , thenX is K -stable. Let D be an effective Cartier divisor, then thecanonical section sD of [D] gives a singular metric on [D] withφ := log |sD |. We denote lctp(D) := cp(h) andlct(D) := infp∈X lctp(D) with such metric. According toDemailly,
α(L) = inf{lct(D) | D is effective,D ∼Q L}.
Use the fact that, for φ = log |f |, e−2cφ = 1|f |2c ,
and the fact
that∫
1|z|a2λ <∞ iff λa− 1 < 0, i.e. λ < 1
a , this links with the
(algebraic geometry) definition for lct(D).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 63: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/63.jpg)
The log canonical threshold through singular metric
Tian in 1987 introduced α(L) the log canonical threshold of L asfollows: Let h = e−φ be a singular metric with ΘL,h ≥ 0, where
ΘL,h =√−1π ∂∂ log φ. Define cp(h) = sup{c | e−2cφ is locally
integrable at p }. Define, for p ∈ X , αp(L) = infh:ΘL,h≥0 cp(h) andα(L) = infp∈X αp(L). Tian proved that if α(−KX ) > n
n+1 , thenX is K -stable. Let D be an effective Cartier divisor, then thecanonical section sD of [D] gives a singular metric on [D] withφ := log |sD |. We denote lctp(D) := cp(h) andlct(D) := infp∈X lctp(D) with such metric. According toDemailly,
α(L) = inf{lct(D) | D is effective,D ∼Q L}.
Use the fact that, for φ = log |f |, e−2cφ = 1|f |2c , and the fact
that∫
1|z|a2λ <∞ iff λa− 1 < 0, i.e. λ < 1
a ,
this links with the
(algebraic geometry) definition for lct(D).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 64: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/64.jpg)
The log canonical threshold through singular metric
Tian in 1987 introduced α(L) the log canonical threshold of L asfollows: Let h = e−φ be a singular metric with ΘL,h ≥ 0, where
ΘL,h =√−1π ∂∂ log φ. Define cp(h) = sup{c | e−2cφ is locally
integrable at p }. Define, for p ∈ X , αp(L) = infh:ΘL,h≥0 cp(h) andα(L) = infp∈X αp(L). Tian proved that if α(−KX ) > n
n+1 , thenX is K -stable. Let D be an effective Cartier divisor, then thecanonical section sD of [D] gives a singular metric on [D] withφ := log |sD |. We denote lctp(D) := cp(h) andlct(D) := infp∈X lctp(D) with such metric. According toDemailly,
α(L) = inf{lct(D) | D is effective,D ∼Q L}.
Use the fact that, for φ = log |f |, e−2cφ = 1|f |2c , and the fact
that∫
1|z|a2λ <∞ iff λa− 1 < 0, i.e. λ < 1
a , this links with the
(algebraic geometry) definition for lct(D).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 65: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/65.jpg)
The log canonical threshold through singular metric
Tian in 1987 introduced α(L) the log canonical threshold of L asfollows: Let h = e−φ be a singular metric with ΘL,h ≥ 0, where
ΘL,h =√−1π ∂∂ log φ. Define cp(h) = sup{c | e−2cφ is locally
integrable at p }. Define, for p ∈ X , αp(L) = infh:ΘL,h≥0 cp(h) andα(L) = infp∈X αp(L). Tian proved that if α(−KX ) > n
n+1 , thenX is K -stable. Let D be an effective Cartier divisor, then thecanonical section sD of [D] gives a singular metric on [D] withφ := log |sD |. We denote lctp(D) := cp(h) andlct(D) := infp∈X lctp(D) with such metric. According toDemailly,
α(L) = inf{lct(D) | D is effective,D ∼Q L}.
Use the fact that, for φ = log |f |, e−2cφ = 1|f |2c , and the fact
that∫
1|z|a2λ <∞ iff λa− 1 < 0, i.e. λ < 1
a , this links with the
(algebraic geometry) definition for lct(D).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 66: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/66.jpg)
Proof of Blum-Jonsson’s result
To see Blum-Jonsson’s result: limm→∞ δm(L) = δ(L), where
δ(L) = infEAX (E)β(L,E) , δm(L) := inf{lct(D) | D ∼Q L of m-basis type},
lct(D) = minEAX (E)
ordE (D) ,
we need to choose an m-basis. The
m-basis comes from the filtration Ftm := H0(X ,mL− tE ), t ≥ 0, of
H0(X ,mL).
Min Ru The β constant appeared in algebraic and complex geometry
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Proof of Blum-Jonsson’s result
To see Blum-Jonsson’s result: limm→∞ δm(L) = δ(L), where
δ(L) = infEAX (E)β(L,E) , δm(L) := inf{lct(D) | D ∼Q L of m-basis type},
lct(D) = minEAX (E)
ordE (D) , we need to choose an m-basis. The
m-basis comes from the filtration Ftm := H0(X ,mL− tE ), t ≥ 0, of
H0(X ,mL).
Min Ru The β constant appeared in algebraic and complex geometry
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The choice of m-basis
Let E be an effective Cartier divisor. The m-basis comes from thefiltration Ft
m = H0(X ,mL− tE ), t ≥ 0 of H0(X ,mL). The m-basisis D := 1
mNm
∑s∈B(s). Notice that, for any
s ∈Wt := H0(X ,mL− tE ), ordE (s) ≥ t, so ordE (D) =
1
mNm
∑s∈B
ordE (s) ≥ 1
mNm
( ∞∑t=0
t(dimWt − dimWt+1)
)
=1
mNm
( ∞∑t=1
dimWt
)→ β(L,E ) as m→∞.
Indeed: βm(L,E ) := inf{lct(D) | D ∼Q L of m-basis type}= maxsj
1Nm
∑Nmj=1 ordE (sj),
where the maximum is over all bases
s1, . . . , sNm of H0(X ,mL), so δm(L)→ δ(L) := infEAX (E)β(L,E) .
Min Ru The β constant appeared in algebraic and complex geometry
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The choice of m-basis
Let E be an effective Cartier divisor. The m-basis comes from thefiltration Ft
m = H0(X ,mL− tE ), t ≥ 0 of H0(X ,mL). The m-basisis D := 1
mNm
∑s∈B(s). Notice that, for any
s ∈Wt := H0(X ,mL− tE ), ordE (s) ≥ t, so ordE (D) =
1
mNm
∑s∈B
ordE (s) ≥ 1
mNm
( ∞∑t=0
t(dimWt − dimWt+1)
)
=1
mNm
( ∞∑t=1
dimWt
)→ β(L,E ) as m→∞.
Indeed: βm(L,E ) := inf{lct(D) | D ∼Q L of m-basis type}= maxsj
1Nm
∑Nmj=1 ordE (sj), where the maximum is over all bases
s1, . . . , sNm of H0(X ,mL), so δm(L)→ δ(L) := infEAX (E)β(L,E) .
Min Ru The β constant appeared in algebraic and complex geometry
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By taking Ftm = H0(X ,mL− tD), t ≥ 0, we can show that, for any
effective divisor D, δ(L) ≤ 1β(L,D) lct(D).
Note: In stability part,
one is concerned about the lower bound of δ(L) (in the Fano casewe need δ(−KX ) > 1), and in Nevanlinna theory, we basically tryto find the upper bound of δ(L). So they are just opposite,although concepts and some methods are similar. With thefiltration in Ru-Vojta, we can prove thatTheorem.
δ(L) ≤ 1
max1≤i≤q β(Di , L)lct(D),
for any divisor D = D1 + · · ·+ Dq with D1, . . . ,Dq are in generalposition on X . Ru-Vojta theorem is just above result plus theBasic Theorem.
Min Ru The β constant appeared in algebraic and complex geometry
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By taking Ftm = H0(X ,mL− tD), t ≥ 0, we can show that, for any
effective divisor D, δ(L) ≤ 1β(L,D) lct(D). Note: In stability part,
one is concerned about the lower bound of δ(L) (in the Fano casewe need δ(−KX ) > 1), and in Nevanlinna theory, we basically tryto find the upper bound of δ(L).
So they are just opposite,although concepts and some methods are similar. With thefiltration in Ru-Vojta, we can prove thatTheorem.
δ(L) ≤ 1
max1≤i≤q β(Di , L)lct(D),
for any divisor D = D1 + · · ·+ Dq with D1, . . . ,Dq are in generalposition on X . Ru-Vojta theorem is just above result plus theBasic Theorem.
Min Ru The β constant appeared in algebraic and complex geometry
![Page 72: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/72.jpg)
By taking Ftm = H0(X ,mL− tD), t ≥ 0, we can show that, for any
effective divisor D, δ(L) ≤ 1β(L,D) lct(D). Note: In stability part,
one is concerned about the lower bound of δ(L) (in the Fano casewe need δ(−KX ) > 1), and in Nevanlinna theory, we basically tryto find the upper bound of δ(L). So they are just opposite,although concepts and some methods are similar.
With thefiltration in Ru-Vojta, we can prove thatTheorem.
δ(L) ≤ 1
max1≤i≤q β(Di , L)lct(D),
for any divisor D = D1 + · · ·+ Dq with D1, . . . ,Dq are in generalposition on X . Ru-Vojta theorem is just above result plus theBasic Theorem.
Min Ru The β constant appeared in algebraic and complex geometry
![Page 73: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/73.jpg)
By taking Ftm = H0(X ,mL− tD), t ≥ 0, we can show that, for any
effective divisor D, δ(L) ≤ 1β(L,D) lct(D). Note: In stability part,
one is concerned about the lower bound of δ(L) (in the Fano casewe need δ(−KX ) > 1), and in Nevanlinna theory, we basically tryto find the upper bound of δ(L). So they are just opposite,although concepts and some methods are similar. With thefiltration in Ru-Vojta, we can prove thatTheorem.
δ(L) ≤ 1
max1≤i≤q β(Di , L)lct(D),
for any divisor D = D1 + · · ·+ Dq with D1, . . . ,Dq are in generalposition on X .
Ru-Vojta theorem is just above result plus theBasic Theorem.
Min Ru The β constant appeared in algebraic and complex geometry
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By taking Ftm = H0(X ,mL− tD), t ≥ 0, we can show that, for any
effective divisor D, δ(L) ≤ 1β(L,D) lct(D). Note: In stability part,
one is concerned about the lower bound of δ(L) (in the Fano casewe need δ(−KX ) > 1), and in Nevanlinna theory, we basically tryto find the upper bound of δ(L). So they are just opposite,although concepts and some methods are similar. With thefiltration in Ru-Vojta, we can prove thatTheorem.
δ(L) ≤ 1
max1≤i≤q β(Di , L)lct(D),
for any divisor D = D1 + · · ·+ Dq with D1, . . . ,Dq are in generalposition on X . Ru-Vojta theorem is just above result plus theBasic Theorem.
Min Ru The β constant appeared in algebraic and complex geometry
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Three interesting constants
Let L be ample, we define
Seshadri constant ε(L,D)):
ε(L,D) = sup{γ ∈ Q : L− γD is nef}.
T (L,D) = sup{γ ∈ Q : L−γD is effective or pseudo-effective}.
Then we have (Blum-Jonsson) ε(L,D) ≤ T (L,D) and1
n+1T (L,D) ≤ β(L,D) ≤ T (L,D).
Furthermore, α(L) = infEA(E)
T (L,E) . This gives (B) (as above)
α(L) ≤ δ(L) ≤ (n + 1)α(L).
Min Ru The β constant appeared in algebraic and complex geometry
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Three interesting constants
Let L be ample, we define
Seshadri constant ε(L,D)):
ε(L,D) = sup{γ ∈ Q : L− γD is nef}.
T (L,D) = sup{γ ∈ Q : L−γD is effective or pseudo-effective}.
Then we have (Blum-Jonsson) ε(L,D) ≤ T (L,D) and1
n+1T (L,D) ≤ β(L,D) ≤ T (L,D).
Furthermore, α(L) = infEA(E)
T (L,E) . This gives (B) (as above)
α(L) ≤ δ(L) ≤ (n + 1)α(L).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 77: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/77.jpg)
Three interesting constants
Let L be ample, we define
Seshadri constant ε(L,D)):
ε(L,D) = sup{γ ∈ Q : L− γD is nef}.
T (L,D) = sup{γ ∈ Q : L−γD is effective or pseudo-effective}.
Then we have (Blum-Jonsson) ε(L,D) ≤ T (L,D)
and1
n+1T (L,D) ≤ β(L,D) ≤ T (L,D).
Furthermore, α(L) = infEA(E)
T (L,E) . This gives (B) (as above)
α(L) ≤ δ(L) ≤ (n + 1)α(L).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 78: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/78.jpg)
Three interesting constants
Let L be ample, we define
Seshadri constant ε(L,D)):
ε(L,D) = sup{γ ∈ Q : L− γD is nef}.
T (L,D) = sup{γ ∈ Q : L−γD is effective or pseudo-effective}.
Then we have (Blum-Jonsson) ε(L,D) ≤ T (L,D) and1
n+1T (L,D) ≤ β(L,D) ≤ T (L,D).
Furthermore, α(L) = infEA(E)
T (L,E) . This gives (B) (as above)
α(L) ≤ δ(L) ≤ (n + 1)α(L).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 79: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/79.jpg)
Three interesting constants
Let L be ample, we define
Seshadri constant ε(L,D)):
ε(L,D) = sup{γ ∈ Q : L− γD is nef}.
T (L,D) = sup{γ ∈ Q : L−γD is effective or pseudo-effective}.
Then we have (Blum-Jonsson) ε(L,D) ≤ T (L,D) and1
n+1T (L,D) ≤ β(L,D) ≤ T (L,D).
Furthermore, α(L) = infEA(E)
T (L,E) .
This gives (B) (as above)
α(L) ≤ δ(L) ≤ (n + 1)α(L).
Min Ru The β constant appeared in algebraic and complex geometry
![Page 80: The constant appeared in algebraic and complex geometry](https://reader031.vdocuments.net/reader031/viewer/2022012407/616a231311a7b741a34f33f2/html5/thumbnails/80.jpg)
Three interesting constants
Let L be ample, we define
Seshadri constant ε(L,D)):
ε(L,D) = sup{γ ∈ Q : L− γD is nef}.
T (L,D) = sup{γ ∈ Q : L−γD is effective or pseudo-effective}.
Then we have (Blum-Jonsson) ε(L,D) ≤ T (L,D) and1
n+1T (L,D) ≤ β(L,D) ≤ T (L,D).
Furthermore, α(L) = infEA(E)
T (L,E) . This gives (B) (as above)
α(L) ≤ δ(L) ≤ (n + 1)α(L).
Min Ru The β constant appeared in algebraic and complex geometry