families of varieties of general type: the shafarevich ...kovacs/pdf/colloquium...families examples...

ShafarevichConjecture

Sándor Kovács

Curves

Families

Shafarevich’sConjecture

The Proof

Generalizations

Families of varieties of generaltype: the Shafarevich conjecture

and related problems

Sándor Kovács

University of Washington


Sándor Kovács

Curves

Families


The Proof

Generalizations

Motto

"To me,algebraic geometry

is algebra with a kick"–Solomon Lefschetz


Sándor Kovács

Curves

Families


The Proof

Generalizations

Outline

1 Curves

2 Families

3 Shafarevich’s Conjecture

4 The Proof of Shafarevich’s Conjecture

5 Generalizations


Sándor Kovács

CurvesRiemann Surfaces

Genus

Topology, Arithmetic andDifferential Geometry

Families


The Proof

Generalizations

Outline

1 CurvesRiemann SurfacesGenusTopology, Arithmetic and Differential Geometry


Sándor Kovács


Genus


Families


The Proof

Generalizations

Curves

Smooth ComplexProjective Curve

=Compact Riemann

Surface


Sándor Kovács


Genus


Families


The Proof

Generalizations

Outline



Sándor Kovács


Genus


Families


The Proof

Generalizations

Genus

genus 0

genus 1

genus 2, ...


Sándor Kovács


Genus


Families


The Proof

Generalizations

Outline



Sándor Kovács


Genus


Families


The Proof

Generalizations

Smooth Projective Curves

• C ' P1

• C is a plane cubic

• C has genus ≥ 2


Sándor Kovács


Genus


Families


The Proof

Generalizations


Smooth curves C come in three types:

• C ' P1




Sándor Kovács


Genus


Families


The Proof

Generalizations


Smooth curves C come in three types:

• C ' P1 (rational)

• C is a plane cubic (elliptic)

• C has genus ≥ 2 (general type)


Sándor Kovács


Genus


Families


The Proof

Generalizations

Topology of Curves

Smooth curves C come in three types, and...

• C ' P1 (rational)

• C is a plane cubic (elliptic)

• C has genus ≥ 2 (general type)


Sándor Kovács


Genus


Families


The Proof

Generalizations

Topology of Curves

...their fundamental groups come in three types:

• C ' P1




Sándor Kovács


Genus


Families


The Proof

Generalizations

Topology of Curves


• C ' P1 π1 is trivial




Sándor Kovács


Genus


Families


The Proof

Generalizations

Topology of Curves



• C is a plane cubic π1 is abelian



Sándor Kovács


Genus


Families


The Proof

Generalizations

Topology of Curves



• C is a plane cubic π1 is abelian

• C has genus ≥ 2 π1 is non-commutative


Sándor Kovács


Genus


Families


The Proof

Generalizations

Arithmetic of Curves


• C ' P1




Sándor Kovács


Genus


Families


The Proof

Generalizations


...rational points on them come in three types:

• C ' P1




Sándor Kovács


Genus


Families


The Proof

Generalizations



• C ' P1 lots




Sándor Kovács


Genus


Families


The Proof

Generalizations



• C ' P1 non-finitely generated




Sándor Kovács


Genus


Families


The Proof

Generalizations




• C is a plane cubic finitely generated



Sándor Kovács


Genus


Families


The Proof

Generalizations




• C is a plane cubic finitely generated

• C has genus ≥ 2 finite


Sándor Kovács


Genus


Families


The Proof

Generalizations

Differential Geometry of Curves


• C ' P1




Sándor Kovács


Genus


Families


The Proof

Generalizations


...their curvatures come in three types:

• C ' P1




Sándor Kovács


Genus


Families


The Proof

Generalizations



• C ' P1 positively curved




Sándor Kovács


Genus


Families


The Proof

Generalizations




• C is a plane cubic flat



Sándor Kovács


Genus


Families


The Proof

Generalizations




• C is a plane cubic flat

• C has genus ≥ 2 negatively curved


Sándor Kovács


Genus


Families


The Proof

Generalizations



• C ' P1 parabolic




Sándor Kovács


Genus


Families


The Proof

Generalizations




• C is a plane cubic elliptic



Sándor Kovács


Genus


Families


The Proof

Generalizations




• C is a plane cubic elliptic

• C has genus ≥ 2 hyperbolic


Sándor Kovács

Curves

FamiliesExamples and Definitions

Isotrivial and Non-isotrivial


The Proof

Generalizations

Outline

2 FamiliesExamples and DefinitionsIsotrivial and Non-isotrivial Families


Sándor Kovács

Curves




The Proof

Generalizations

A family of curves of general type

y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.

X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ

y

π

A1λ


Sándor Kovács

Curves




The Proof

Generalizations



X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ

y

π

A1λ

X

y

π

A1λ


Sándor Kovács

Curves




The Proof

Generalizations



X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ

y

π

A1λ

A2x ,y ⊇ Xλ = π−1(λ) ⊆ X

y

π

λ ∈ A1λ


Sándor Kovács

Curves




The Proof

Generalizations

Notation

B – smooth projective curve of genus g,∆ ⊆ B – finite set of points.

A family over B \ ∆, f : X → B \ ∆, is asurjective (flat) map with equidimensional,connected fibers.

A family is smooth if Xb = f−1(b) is smooth for∀b ∈ B.


Sándor Kovács

Curves




The Proof

Generalizations

Outline

2 FamiliesExamples and DefinitionsIsotrivial and Non-isotrivial Families


Sándor Kovács

Curves




The Proof

Generalizations

Isotrivial Family


Sándor Kovács

Curves




The Proof

Generalizations

Isotrivial Family

All smooth fibers are isomorphic to P1.


Sándor Kovács

Curves




The Proof

Generalizations

Non-isotrivial families

The family defined by y2 = x5 − 5λx + 4λ is notisotrivial, because

Xλ1 6' Xλ2

for λ1 6= λ2.

A family f : X → B is isotrivial if there exists afinite set ∆ ⊂ B such that Xb ' Xb′ for allb, b′ ∈ B \ ∆.


Sándor Kovács

Curves




The Proof

Generalizations



Xλ1 6' Xλ2

for λ1 6= λ2.


Warm-Up Question: Let q ∈ Z. How manyisotrivial families of curves of genus q over Bare there?


Sándor Kovács

Curves




The Proof

Generalizations



Xλ1 6' Xλ2

for λ1 6= λ2.


Warm-Up Question: Let q ∈ Z. How manyisotrivial families of curves of genus q over Bare there? ∞


Sándor Kovács

Curves

Families

Shafarevich’sConjectureStatements

Results and Connections

The Proof

Generalizations

Outline

3 Shafarevich’s ConjectureStatement and DefinitionsResults and Connections


Sándor Kovács

Curves

Families



The Proof

Generalizations

Shafarevich’s Conjecture (1962)

Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then

(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.

(II) If2g(B) − 2 + #∆ ≤ 0,

then there aren’t any admissible families.


Sándor Kovács

Curves

Families



The Proof

Generalizations



(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.These will be called “admissible families”.

(II) If2g(B) − 2 + #∆ ≤ 0,



Sándor Kovács

Curves

Families



The Proof

Generalizations



(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.These will be called “admissible families”.

(II) If2g(B) − 2 + #∆ ≤ 0,


Note:

2g(B) − 2 + #∆ ≤ 0 ⇔

{

g(B) = 0 & #∆ ≤ 2g(B) = 1 & ∆ = ∅


Sándor Kovács

Curves

Families



The Proof

Generalizations

Hyperbolicity

A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic implies that

• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any

complex torus T = Cn/Z2n.


Sándor Kovács

Curves

Families



The Proof

Generalizations

Hyperbolicity

A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic is equivalent to




Sándor Kovács

Curves

Families



The Proof

Generalizations

Hyperbolicity




An algebraic variety X is algebraicallyhyperbolic if


Sándor Kovács

Curves

Families



The Proof

Generalizations

Hyperbolicity





• ∀ A1 \ {0} → X regular map is constant


Sándor Kovács

Curves

Families



The Proof

Generalizations

Hyperbolicity





• ∀ A1 \ {0} → X regular map is constant, and• ∀ A → X regular map is constant, for any

abelian variety (projective algebraic group) A.


Sándor Kovács

Curves

Families



The Proof

Generalizations

Hyperbolicity: Examples

B – smooth projective curve,∆ ⊆ B – finite subset.

• If g(B) ≥ 2, thenB \ ∆ is hyperbolic for arbitrary ∆.

• If g(B) = 1, thenB \ ∆ is hyperbolic ⇔ ∆ 6= ∅.

• If g(B) = 0, thenB \ ∆ is hyperbolic ⇔ #∆ ≥ 3.

B \ ∆ is hyperbolic ⇔ 2g(B) − 2 + #∆ > 0.


Sándor Kovács

Curves

Families



The Proof

Generalizations

Shafarevich’s Conjecture: Take Two


(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q overB \ ∆. “admissible families”.

(II) If2g(B) − 2 + #∆ ≤ 0,


(II∗) If there exists any admissible families, thenB \ ∆ is hyperbolic.


Sándor Kovács

Curves

Families



The Proof

Generalizations

Shafarevich’s Conjecture: Take Two


(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q overB \ ∆, “admissible families”.

(II) If2g(B) − 2 + #∆ ≤ 0,


(II∗) If there exists any admissible families, thenB \ ∆ is hyperbolic.


Sándor Kovács

Curves

Families



The Proof

Generalizations

Outline

3 Shafarevich’s ConjectureStatement and DefinitionsResults and Connections


Sándor Kovács

Curves

Families



The Proof

Generalizations

History

The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.

• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2

defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).

• The Shafarevich Conjecture has a number fieldversion as well.


Sándor Kovács

Curves

Families



The Proof

Generalizations

Geometric Mordell Conjecture

Let f : X → B be a non-isotrivial family ofprojective curves of genus ≥ 2.Then there are only finitely many sections of f ,i.e., σ : B → X , such that f ◦ σ = idB.

This was proved by MANIN (1963).

And again by PARSHIN using

“Parshin’s Covering Trick” to prove that

The Shafarevich Conjectureimplies

The Mordell Conjecture.


Sándor Kovács

Curves

Families



The Proof

Generalizations




And again by PARSHIN (1968). using





Sándor Kovács

Curves

Families



The Proof

Generalizations




And again by PARSHIN (1968) using





Sándor Kovács

Curves

Families


The ProofDeformation Theory

Arakelov-Parshin method

Generalizations

Outline

4 The Proof of Shafarevich’s ConjectureDeformation TheoryThe Arakelov-Parshin method


Sándor Kovács

Curves

Families




Generalizations

Deformations

A deformation of an algebraic variety X is afamily F : X → T such that there exists a t ∈ Tthat X ' Xt .


Sándor Kovács

Curves

Families




Generalizations

Deformations

A deformation of an algebraic variety X is afamily F : X → T such that there exists a t ∈ Tthat X ' Xt .

X

T

f

Xt

t


Sándor Kovács

Curves

Families




Generalizations

Deformation Type

Two algebraic varieties X1 and X2 have thesame deformation type if there exists a familyF : X → T , t1, t2 ∈ T such that X1 ' Xt1 andX2 ' Xt2 .


Sándor Kovács

Curves

Families




Generalizations

Deformation Type

Two algebraic varieties X1 and X2 have thesame deformation type if there exists a familyF : X → T , t1, t2 ∈ T such that X1 ' Xt1 andX2 ' Xt2 .

X

T

ft1

t2

Xt1

Xt2


Sándor Kovács

Curves

Families




Generalizations

Deformations of Families

A deformation of a family X → B is a familyF : X → B × T such that there exists a t ∈ Tthat (X → B) ' (Xt → B × {t}), whereXt = F−1(B × {t}).


Sándor Kovács

Curves

Families




Generalizations

Deformations of Families

A deformation of a family X → B is a familyF : X → B × T such that there exists a t ∈ Tthat (X → B) ' (Xt → B × {t}), whereXt = F−1(B × {t}).

F

X

T×B

Xt

{ }B t×


Sándor Kovács

Curves

Families




Generalizations

Deformation Type of Families

Two families X1 → B and X2 → B have thesame deformation type if there exists a familyF : X → B × T , t1, t2 ∈ T such that(X1 → B) ' (Xt1 → B × {t}) and(X2 → B) ' (Xt2 → B × {t}).


Sándor Kovács

Curves

Families




Generalizations

Deformation Type of Families

Two families X1 → B and X2 → B have thesame deformation type if there exists a familyF : X → B × T , t1, t2 ∈ T such that(X1 → B) ' (Xt1 → B × {t}) and(X2 → B) ' (Xt2 → B × {t}).

F

XXt

1

2{ }B t×

1{ }B t×

Xt2

T×B


Sándor Kovács

Curves

Families




Generalizations

Outline

4 The Proof of Shafarevich’s ConjectureDeformation TheoryThe Arakelov-Parshin method


Sándor Kovács

Curves

Families




Generalizations

The Arakelov-Parshin method

ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:


Sándor Kovács

Curves

Families




Generalizations


ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:(B) Boundedness: There are only finitely many

deformation types of admissible families.


Sándor Kovács

Curves

Families




Generalizations



deformation types of admissible families.(R) Rigidity: Admissible families do not admit

non-trivial deformations.


Sándor Kovács

Curves

Families




Generalizations




non-trivial deformations.

Note:• (B) and (R) together imply (I).


Sándor Kovács

Curves

Families




Generalizations




non-trivial deformations.(H) Hyperbolicity: If there exist admissible families,

then B \ ∆ is hyperbolic.

Note:• (B) and (R) together imply (I).


Sándor Kovács

Curves

Families




Generalizations




non-trivial deformations.(H) Hyperbolicity: If there exist admissible families,

then B \ ∆ is hyperbolic.

Note:• (B) and (R) together imply (I).• (H) = (II∗) and hence is equivalent to (II).


Sándor Kovács

Curves

Families


The Proof

GeneralizationsHigher Dimensional Fibers

Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results

Higher Dimensional Bases

Rigidity

Where Next?

Outline

5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Higher dimensional fibers

First order of business:

Generalize the setup.

Find a replacement for “genus” and “q ≥ 2”.

• Hilbert polynomial, h.• Kodaira dimension, κ = −∞, 0, 1, . . . , dim.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Outline



Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Kodaira dimension

Plane curves:Kodaira

type degree genus dimension


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Kodaira dimension



P1 deg = 1, 2 g = 0 κ = −∞


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Kodaira dimension



P1 deg = 1, 2 g = 0 κ = −∞

elliptic deg = 3 g = 1 κ = 0


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Kodaira dimension



P1 deg = 1, 2 g = 0 κ = −∞

elliptic deg = 3 g = 1 κ = 0

general deg ≥ 4 g ≥ 2 κ = 1type


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Kodaira dimension

Hypersurfaces in Pn:Kodaira

type degree dimension


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Kodaira dimension



deg < n + 1 κ = −∞


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Kodaira dimension



deg < n + 1 κ = −∞

deg = n + 1 κ = 0


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Kodaira dimension



deg < n + 1 κ = −∞

deg = n + 1 κ = 0

deg > n + 1 κ = dim


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Kodaira dimension



Fano deg < n + 1 κ = −∞

deg = n + 1 κ = 0



Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Kodaira dimension



Fano deg < n + 1 κ = −∞

Calabi-Yau deg = n + 1 κ = 0



Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Kodaira dimension



Fano deg < n + 1 κ = −∞


general type deg > n + 1 κ = dim


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Kodaira dimension



Fano deg < n + 1 κ = −∞


general type deg > n + 1 κ = dim

Products:

dim(X × Y ) = dim X + dim Y

κ(X × Y ) = κ(X ) + κ(Y )


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Outline



Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Varieties of General Type

X is of general type if κ(X ) = dim X .

Recall: A curve C is of general type iff g(C) ≥ 2.

Genus is replaced with the Hilbert polynomial.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Varieties of General Type

X is of general type if κ(X ) = dim X .

Recall: A curve C is of general type iff g(C) ≥ 2.

Genus is replaced with the Hilbert polynomial.

For curves, the Hilbert polynomial is determinedby the genus, so this is a natural generalization.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

The Shafarevich Conjecture

Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if

• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective

variety, canonically embedded with Hilbertpolynomial h.

“Canonically embedded” means that it isembedded by the global sections of ωm

Xb.

In particular, Xb is of general type.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Outline



Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Shafarevich Conjecture


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?


(B) Boundedness: There are only finitely manydeformation types of admissible families.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?



(R) Rigidity: Admissible families do not admitnon-trivial deformations.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?




(H) Hyperbolicity: If there exist admissible families,then B \ ∆ is hyperbolic.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?




(H) Hyperbolicity: If there exist admissible families,then B \ ∆ is hyperbolic.

(WB) Weak Boundedness:If f : X → B is an admissible family, thendeg f∗ωm

X/B is bounded in terms of B, ∆, h, m.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Weak Boundedness

Moduli Theory + (WB) ⇒ (B)


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Weak Boundedness


(WB) ⇒ (H)


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Weak Boundedness


(K , 2002) (WB) ⇒ (H)


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Weak Boundedness


(K , 2002) (WB) ⇒ (H)

Hence Weak Boundedness is the key statementtowards proving (B) and (H).


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Weak Boundedness


(K , 2002) (WB) ⇒ (H)

Hence Weak Boundedness is the key statementtowards proving (B) and (H).

(WB) Weak Boundedness:If f : X → B is an admissible family, thendeg f∗ωm

X/B is bounded in terms of B, ∆, h, m.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Outline



Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Modern History


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Modern History

“modern” = last 25 years


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Modern History

“modern” ≈ last 10 years


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Modern History: Hyperbolicity


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?


BEAUVILLE (1981):(H) holds for dim(X/B) = 1.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?



CATANESE-SCHNEIDER (1994):(H) can be used for giving upper bounds on thesize of automorphisms groups of varieties ofgeneral type.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?



CATANESE-SCHNEIDER (1994):(H) can be used for giving upper bounds on thesize of automorphisms groups of varieties ofgeneral type.

SHOKUROV (1995):(H) can be used to prove quasi-projectivity ofmoduli spaces of varieties of general type.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?


MIGLIORINI (1995):(H) holds for dim(X/B) = 2 and g(B) = 1.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?



K (1996):(H) holds for dim(X/B) arbitrary and g(B) = 1.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?




K (1997):(H) holds for dim(X/B) = 2.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?





K (2000):(H) holds for dim(X/B) arbitrary.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?





K (2000):(H) holds for dim(X/B) arbitrary.

VIEHWEG AND ZUO (2002):Brody hyperbolicity holds as well.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Modern History:(Weak) Boundedness

BEDULEV AND VIEHWEG (2000):


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?


BEDULEV AND VIEHWEG (2000):• (B) holds for dim(X/B) = 2, and


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?


BEDULEV AND VIEHWEG (2000):• (B) holds for dim(X/B) = 2, and• (WB) holds for dim(X/B) arbitrary.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?



As a byproduct of their proof they also obtainedthat (H) holds in these cases.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?




K (2002), VIEHWEG AND ZUO (2002):(WB) holds under more general assumptions.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?




K (2002), VIEHWEG AND ZUO (2002):(WB) holds under more general assumptions.

K (2002):(WB) implies (H).


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

More History

...more related results by

FALTINGS (1983)

ZHANG (1997)

OGUISO AND VIEHWEG (2001)

..and more.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Outline



Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Higher dimensional bases

Many details change:


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?



Instead of ∆ being a finite subset,


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?



Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?




A family being non-isotrivial is no longer a goodassumption.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?




A family being non-isotrivial is no longer a goodassumption. It has to be replaced by a familyhaving maximal variation in moduli.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?





Viehweg’s Conjecture (2001): If there exists anadmissible family with maximal variation inmoduli, then (B, ∆) is of log general type.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?





Viehweg’s Conjecture (2001): If there exists anadmissible family with maximal variation inmoduli, then (B, ∆) is of log general type.That is, the log-Kodaira dimension of B ismaximal: κ(B, ∆) = dim B.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?




Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?



K (2003), VIEHWEG-ZUO, (2003):Viehweg’s conjecture holds for B = Pn.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?



K (2003), VIEHWEG-ZUO, (2003):Viehweg’s conjecture holds for B = Pn.

K (1997):If f : X → B is admissible and B is an abelianvariety, then ∆ 6= ∅.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Rigidity

But, what about Rigidity?


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Outline



Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Rigidity – An Example


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?


Let Y → B be an arbitrary non-isotrivial familyof curves of genus ≥ 2, and


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?



C a smooth projective curve of genus ≥ 2.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?



C a smooth projective curve of genus ≥ 2.

C

f

B

X Y C= ×

CC

b bX Y C= ×

YbY C

b

C


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?


f : X = Y × C → B is an admissible family,and

C

f

B

X Y C= ×

CC

b bX Y C= ×

YbY C

b

C


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?


f : X = Y × C → B is an admissible family,and

any deformation of C gives a deformation of f .

C

f

B

X Y C= ×

CC

b bX Y C= ×

YbY C

b

C


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Rigidity

Therefore, (R) fails.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Rigidity


Question: Under what additional conditionsdoes (R) hold?


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Rigidity



K (2004), VIEHWEG-ZUO, (2004):Rigidity holds for strongly non-isotrivial families.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Rigidity



K (2004), VIEHWEG-ZUO, (2004):Rigidity holds for strongly1non-isotrivial families.

1unfortunately the margin is not wide enough to define

this term.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Current Work

Work in progress:


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Current Work

Work in progress:

• Geometric description of strongly non-isotrivialfamilies.


Sándor Kovács

Curves

Families


The Proof


Kodaira Dimension

Admissible Families

Weak Boundedness

Recent Results


Rigidity

Where Next?

Current Work

Work in progress:

• Geometric description of strongly non-isotrivialfamilies.

• Joint work with Stefan Kebekus (Köln):Families over two-dimensional bases.


Sándor Kovács

Curves

Families


The Proof

Generalizations

Acknowledgement

This presentation was made using thebeamertex LATEX macropackage of Till Tantau.http://latex-beamer.sourceforge.net

families of varieties of general type: the shafarevich ...kovacs/pdf/colloquium...families examples...

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