Subvarieties of moduli stacks - sites.math.washington.edukovacs/seattle... · Shafarevich’s Conjecture Generalizations Techniques and Ingredients Definitions and Notation Outline

Shafarevich’s ConjectureGeneralizations

Techniques and IngredientsDefinitions and Notation

Subvarieties of moduli stacks

Sándor Kovács

AMS Summer Institute on Algebraic GeometryUniversity of Washington, Seattle

August 4, 2005

Sándor Kovács Subvarieties of moduli stacks



Outline

1 Shafarevich’s Conjecture

2 Generalizations

3 Techniques and Ingredients




Section Outline

Definitions and Notation




NotationB – smooth projective curve of genus g,

∆ ⊆ B – finite set of points.

A family over B \∆, f : X → B \∆, is a surjectivemorphism with equidimensional, connected fibers.

For b ∈ B, the fiber f−1(b) will be denoted by Xb.

A family is smooth if Xb is smooth for ∀b ∈ B.

A family is projective if Xb is projective for ∀b ∈ B.




NotationB – smooth projective curve of genus g,

∆ ⊆ B – finite set of points.








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








































A family of curves

y2 − λx2 = 1

defines a family of conics.

Remark: all fibers for λ 6= 0 are isomorphic.




A family of curves

y2 − λx2 = 1defines a family of conics.





A family of curves

y2 − λx2 = 1defines a family of conics.





A family of curves of general type

y2 = x5 − 5λx + 4λ

defines a smooth family of curves parametrized by λ 6= 0, 1.




A family of curves of general type

y2 = x5 − 5λx + 4λ

defines a smooth family of curves parametrized by λ 6= 0, 1.




Isotrivial Families

A family f : X → B is isotrivial if there exists a finite set ∆ ⊂ Bsuch that Xb ' Xb′ for all b, b′ ∈ B \∆.

Example:y2 − λx2 =

For λ 6= 0, the fiber over λ is isomorphic to P1,and so this is an isotrivial family.

Warm-Up Question: Fix B and q ∈ Z. How many isotrivialfamilies of curves of genus q over B are there?




Isotrivial Families


Example:y2 − λx2 = 1






Isotrivial Families


Example:y2 − λx2 = z2






Isotrivial Families








Isotrivial Families








Non-isotrivial families

For the family defined by y2 = x5 − 5λx + 4λ one checkseasily that

Xλ1 6' Xλ2

for λ1 6= λ2.

Therefore this is a non-isotrivial family.




Non-isotrivial families

For the family defined by y2 = x5 − 5λx + 4λ one checkseasily that

Xλ1 6' Xλ2

for λ1 6= λ2.

Therefore this is a non-isotrivial family.



Techniques and Ingredients

StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method

Section Outline

1 Shafarevich’s ConjectureStatement and DefinitionsFirst Results and ConnectionsThe Arakelov-Parshin Method





Conjecture (SHAFAREVICH (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 \∆.

These will be called “admissible families”.

(II) If2g(B)− 2 + #∆ ≤ 0,

then there aren’t any admissible families.

Note:

2g(B)− 2 + #∆ ≤ 0 ⇔

g(B) = 0 & #∆ ≤ 2g(B) = 1 & ∆ = ∅






(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 & ∆ = ∅






(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 & ∆ = ∅







(II) If2g(B)− 2 + #∆ ≤ 0,


Note:

2g(B)− 2 + #∆ ≤ 0 ⇔

g(B) = 0 & #∆ ≤ 2g(B) = 1 & ∆ = ∅







(II) If2g(B)− 2 + #∆ ≤ 0,


Note:

2g(B)− 2 + #∆ ≤ 0 ⇔

g(B) = 0 & #∆ ≤ 2g(B) = 1 & ∆ = ∅





HyperbolicityA complex analytic space X is Brody hyperbolic if everyC → 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.

An algebraic variety X is algebraically Brody hyperbolic(or here and now simply hyperbolic) if• ∀ A1 \ 0 → X morphism is constant

, and

• ∀ A → X morphism is constant, for any abelian variety A.

Remark: There are other notions of hyperbolicity: incomplex analysis, “Kobayashi hyperbolicity”, and itsalgebraic counterpart introduced by Demailly.





HyperbolicityA complex analytic space X is Brody hyperbolic if everyC → 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.


, and







HyperbolicityA complex analytic space X is Brody hyperbolic if everyC → X holomorphic map is constant.X Brody hyperbolic implies that• ∀ C∗ → X holomorphic map is constant,

• ∀ T → X holomorphic map is constant for any complextorus T = Cn/Z2n.


, and







HyperbolicityA complex analytic space X is Brody hyperbolic if everyC → 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.


, and







HyperbolicityA complex analytic space X is Brody hyperbolic if everyC → X holomorphic map is constant.X Brody hyperbolic is equivalent to• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant for any complex

torus T = Cn/Z2n.


, and








torus T = Cn/Z2n.

An algebraic variety X is algebraically Brody hyperbolic(or here and now simply hyperbolic) if

• ∀ A1 \ 0 → X morphism is constant

, and








torus T = Cn/Z2n.


, and• ∀ A → X morphism is constant, for any abelian variety A.







torus T = Cn/Z2n.

An algebraic variety X is algebraically Brody hyperbolic(or here and now simply hyperbolic) if• ∀ A1 \ 0 → X morphism is constant, and• ∀ A → X morphism is constant, for any abelian variety A.







torus T = Cn/Z2n.

An algebraic variety X is algebraically Brody hyperbolic(or here and now simply hyperbolic) if• ∀ A1 \ 0 → X morphism is constant, and• ∀ A → X morphism is constant, for any abelian variety A.






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.













































Shafarevich’s Conjecture: Take TwoConjecture (SHAFAREVICH (1962))

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

(I) there exists only finitely many non-isotrivial smoothfamilies of curves of genus q over B \∆.“admissible families”.

(II) If2g(B)− 2 + #∆ ≤ 0,


(II∗) If there exist any admissible families, then B \∆ ishyperbolic.







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

“admissible families”.

(II) If2g(B)− 2 + #∆ ≤ 0,









(I) there exists only finitely many non-isotrivial smoothfamilies of curves of genus q over B \∆,“admissible families”.

(II) If2g(B)− 2 + #∆ ≤ 0,










(II) If2g(B)− 2 + #∆ ≤ 0,










(II) If2g(B)− 2 + #∆ ≤ 0,







Section Outline






History

Theorem (PARSHIN (1968))Shafarevich’s Conjecture holds for ∆ = ∅.

Theorem (ARAKELOV (1971))Shafarevich’s Conjecture holds in general.

Theorem (BEAUVILLE (1981))(II) holds.





History








History








Intermezzo: The number field case

Theorem (FALTINGS (1983))Let C be a smooth projective curve of genus ≥ 2defined over a number field F .Then C has only finitely many F-rational points.

Shafarevich’s Conjecture has a number field version as well.






Conjecture (MORDELL (1922))Let C be a smooth projective curve of genus ≥ 2defined over a number field F .Then C has only finitely many F -rational points.




















Conjecture (Geometric Mordell)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.

Theorem (MANIN (1963))The Geometric Mordell Conjecture holds.

Theorem (PARSHIN (1968))The Shafarevich Conjecture implies The Mordell Conjecture.

Proof“Parshin’s Covering Trick”





























Section Outline






Deformations of FamiliesA deformation of a family X → B (with fixed base) is afamily F : X → B × T such that there exists a t ∈ T that(X/B) ' (Xt/B × t), where Xt = F−1(B × t).





Deformation Type of FamiliesTwo families X1 → B and X2 → B have the samedeformation type if there exists a familyF : X → B × T , t1, t2 ∈ T such that(X1/B) ' (Xt1/B × t1) and (X2/B) ' (Xt2/B × t2).





The Arakelov-Parshin method

Conjecture (Reformulated Shafarevich)(B) Boundedness: There are only finitely many deformation

types of admissible families.

(R) Rigidity: Admissible families do not admit non-trivialdeformations.

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

Note:

• (B) and (R) together imply (I).

• (H) = (II∗) and hence is equivalent to (II).










Note:












Note:












Note:












Note:






Higher Dimensional FibersOther directionsHigher Dimensional Bases

Section Outline

2 GeneralizationsHigher Dimensional FibersOther directionsHigher Dimensional Bases





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 by the Hilbert polynomial.

Recall: For curves, the Hilbert polynomial is determinedby the genus.
































Admissible FamiliesSetup: Fix B a smooth projective curve, ∆ ⊆ B a finitesubset, 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 Hilbert polynomial h.

“Canonically embedded” means that it is embedded bythe global sections of ωm

Xb. In particular, Xb is of general

type.





Admissible FamiliesSetup: Fix B a smooth projective curve, ∆ ⊆ B a finitesubset, 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,




type.





Admissible FamiliesSetup: Fix B a smooth projective curve, ∆ ⊆ B a finitesubset, 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 Hilbert polynomial h.



type.





Admissible FamiliesSetup: Fix B a smooth projective curve, ∆ ⊆ B a finitesubset, 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,




type.









type.









type.





Conjecture (Higher Dimensional Shafarevich)(B) Boundedness: There are only finitely many deformation








Applications of Hyperbolicity

Theorem (CATANESE-SCHNEIDER (1994))(H) can be used for giving upper bounds on the size ofautomorphisms groups of varieties of general type.

Theorem (SHOKUROV (1995))(H) can be used to prove quasi-projectivity of moduli spacesof varieties of general type.





Recent

1

Contributors to theHigher Dimensional Case

MIGLIORINI (1995)

K (1996), (1997), (2000), (2002), (2003)

ZHANG (1997)

BEDULEV AND VIEHWEG (2000)

VIEHWEG AND ZUO (2001), (2002), (2003)

1 most of the activity took place since Santa Cruz ’95





Recent

1

Contributors to theHigher Dimensional Case

MIGLIORINI (1995)

K (1996), (1997), (2000), (2002), (2003)

ZHANG (1997)


VIEHWEG AND ZUO (2001), (2002), (2003)






Recent1 Contributors to theHigher Dimensional Case

MIGLIORINI (1995)

K (1996), (1997), (2000), (2002), (2003)

ZHANG (1997)


VIEHWEG AND ZUO (2001), (2002), (2003)






Current KnowledgeTheorem (BEDULEV-VIEHWEG, K )

Hyperbolicity holds in all dimensions.

Boundedness holds in all dimensions in which Mh, themoduli space of canonically polarized varieties withHilbert polynomial h, admits a modular compactification.In particular boundedness holds for families of surfaces.

Theorem (VIEHWEG-ZUO, K )Both of these statements can be extended to include familiesof smooth minimal varieties of general type.





Current KnowledgeTheorem (BEDULEV-VIEHWEG, K )

Hyperbolicity holds in all dimensions.

Boundedness holds in all dimensions in which Mh, themoduli space of canonically polarized varieties withHilbert polynomial h, admits a modular compactification.In particular boundedness holds for families of surfaces.

Theorem (VIEHWEG-ZUO, K )Both of these statements can be extended to include familiesof smooth minimal varieties of general type.





But

, what about Rigidity?





But, what about Rigidity?





Rigidity – An Example





Rigidity – An ExampleY → B a non-isotrivial family of curves of genus ≥ 2, and

C a smooth projective curve of genus ≥ 2.

















Rigidity – An Examplef : X = Y × C → B is an admissible family, and

any deformation of C gives a deformation of f .





Rigidity – An Examplef : X = Y × C → B is an admissible family, and

any deformation of C gives a deformation of f .





RigidityTherefore, (R) fails.

Question:Under what additional conditions does (R) hold?

(Partial) Answer:

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

1unfortunately the margin is not wide enough to define this term.







(Partial) Answer:









(Partial) Answer:









(Partial) Answer:

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






Open ProblemGive a geometric description of strong non-isotriviality.

lt g that pl it w l





Open ProblemGive a geometric description of strong non-isotriviality,or at least give a geometric condition that implies it.





Section Outline






Uniform Boundedness

Currently only known for families of curves:

Caporaso (2002), (2003)

Heier (2004)





Uniform Boundedness

Currently only known for families of curves.

:

Caporaso (2002), (2003)

Heier (2004)





Uniform Boundedness


Caporaso (2002), (2003)

Heier (2004)





Uniform Boundedness


Caporaso (2002), (2003)

Heier (2004)





Varieties of Kodaira dimension 0FALTINGS (1983): (B) holds, (R) fails for abelianvarieties.BORCHERDS, KATZARKOV, PANTEV, SHEPHERD-BARRON

(1998), OGUISO (2003): (R) holds for families of K3surfaces with fixed Picard number.OGUISO, VIEHWEG (2001): (H) holds for elliptic surfaces.JORGENSON, TODOROV (2002): Various partial andrelated results for K3s and Enriques surfaces.LIU, TODOROV, YAU, ZUO (2003): Various partial andrelated results for Calabi-Yau manifolds.VIEHWEG, ZUO (2003): (R) holds for stronglynon-isotrivial families of varieties of Kodairadimension 0 plus a condition.





Varieties of Kodaira dimension ≥ 0FALTINGS (1983): (B) holds, (R) fails for abelianvarieties.BORCHERDS, KATZARKOV, PANTEV, SHEPHERD-BARRON

(1998), OGUISO (2003): (R) holds for families of K3surfaces with fixed Picard number.OGUISO, VIEHWEG (2001): (H) holds for elliptic surfaces.JORGENSON, TODOROV (2002): Various partial andrelated results for K3s and Enriques surfaces.LIU, TODOROV, YAU, ZUO (2003): Various partial andrelated results for Calabi-Yau manifolds.VIEHWEG, ZUO (2003): (R) holds for stronglynon-isotrivial families of varieties of Kodairadimension ≥ 0 plus a condition.





Section Outline






Higher dimensional basesBack to canonically polarized varieties (ωXb is ample).

Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.A family being non-isotrivial is no longer a goodassumption. Instead: maximal variation in moduli.In this case (B) is only known in a few special cases(Work of VIEHWEG, ZUO and K ).(R) would follow from the case when dim B = 1.(H) is replaced by

Conjecture (VIEHWEG (2001))If there exists an admissible family over B,then (B, ∆) is of log general type.






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

A family being non-isotrivial is no longer a goodassumption. Instead: maximal variation in moduli.In this case (B) is only known in a few special cases(Work of VIEHWEG, ZUO and K ).(R) would follow from the case when dim B = 1.(H) is replaced by







Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.A family being non-isotrivial is no longer a goodassumption.

Instead: maximal variation in moduli.In this case (B) is only known in a few special cases(Work of VIEHWEG, ZUO and K ).(R) would follow from the case when dim B = 1.(H) is replaced by







Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.A family being non-isotrivial is no longer a goodassumption. Instead: maximal variation in moduli.

In this case (B) is only known in a few special cases(Work of VIEHWEG, ZUO and K ).(R) would follow from the case when dim B = 1.(H) is replaced by







Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.A family being non-isotrivial is no longer a goodassumption. Instead: maximal variation in moduli.In this case (B) is only known in a few special cases(Work of VIEHWEG, ZUO and K ).(R) would follow from the case when dim B = 1.

(H) is replaced by



























Admissible Families

Let B be a smooth projective variety, and∆ ⊆ B a divisor with simple normal crossings.

f : X → B is an admissible family if

• Xb is a smooth projective canonicallypolarized variety ∀b ∈ B \∆,

• Var f = dim B.





Admissible Families




• Var f = dim B.





Admissible Families




• Var f = dim B.





Admissible Families




• Var f = dim B.





Conjecture (VIEHWEG (2001))Let f : X → B be an admissible family. Then κ(B, ∆) = dim B.

Theorem (ZUO (2000))Viehweg’s conjecture holds if the local Torelli theorem is truefor the general fiber of f .

Theorem (VIEHWEG-ZUO, K (2002))Viehweg’s conjecture holds for B = Pn.

Theorem (VIEHWEG-ZUO (2002))A similar result holds for smooth complete intersections of lowcodimension.





























ConjectureLet f : X → B be an admissible family, then

max(κ(B, ∆), 0) ≥ Var f .

Evidence:

Theorem (K (1997))Let B be birational to an abelian variety and ∆ = ∅.Then Var f = 0.

Theorem (VIEHWEG-ZUO (2002))Let B be such that TB(− log ∆) is weakly positive.Then Var f = 0.






max(κ(B, ∆), 0) ≥ Var f .

Evidence:








max(κ(B, ∆), 0) ≥ Var f .

Evidence:








max(κ(B, ∆), 0) ≥ Var f .

Evidence:







New Progress


max(κ(B, ∆), 0) ≥ Var f .

Theorem (KEBEKUS-K (2005))This conjecture holds if dim B ≤ 2.





New Progress


max(κ(B, ∆), 0) ≥ Var f .






New Progress


max(κ(B, ∆), 0) ≥ Var f .






New Progress


max(κ(B, ∆), 0) ≥ Var f .





Positivity of direct imagesIterated Kodaira-Spencer maps

Section Outline

3 Techniques and IngredientsPositivity of direct imagesIterated Kodaira-Spencer maps





Positivity of direct imagesLet f : X → B be an admissible family and for simplicity,assume that dim B = 1. Then

Theorem (Fujita, Kawamata, Kollár, Viehweg)f∗ωm

X/B is ample.

Corollarydet f∗ωm

X/B is an ample line bundle, in particular deg f∗ωmX/B > 0.

Corollary (?, K )Assume that f : X → B is smooth. Then ωX/B is ample.






Theorem (Fujita, Kawamata, KOLLÁR , Viehweg)f∗ωm

X/B is ample.










X/B is ample.










X/B is ample.








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

X/B isbounded in terms of B, ∆, h, m.

good moduli theory + (WB) ⇒ (B)

(WB) ⇒ (H)

0 <

deg f∗ωmX/B ≤ (positive constant)

· (2g − 2 + #∆)

⇒ (2g − 2 + #∆) > 0.

Hence Weak Boundedness is the key statement towardsproving (B) and (H).








(WB) ⇒ (H)

0 <


· (2g − 2 + #∆)

⇒ (2g − 2 + #∆) > 0.









effective form of (WB)

⇒ (H)

0 <

deg f∗ωmX/B ≤ (positive constant) · (2g − 2 + #∆)

⇒ (2g − 2 + #∆) > 0.









effective form of (WB) ⇒ (H)

0 < deg f∗ωmX/B ≤ (positive constant) · (2g − 2 + #∆)

⇒ (2g − 2 + #∆) > 0.









(K , 2002) (WB) ⇒ (H)

0 <


· (2g − 2 + #∆)

⇒ (2g − 2 + #∆) > 0.









(K , 2002) (WB) ⇒ (H)

0 <


· (2g − 2 + #∆)

⇒ (2g − 2 + #∆) > 0.






Vanishing TheoremsLet X be a smooth projective variety of dimension n over Cand L an ample line bundle on X .

Then

Theorem (Kodaira, Akizuki-Nakano)Hq(X , Ωp

X ⊗L ) = 0 for p + q > n.

Theorem (Esnault-Viehweg)Let D be a normal crossing divisor on X.Then Hq(X , Ωp

X (log D)⊗L ) = 0 for p + q > n.

Theorem (Navarro-Aznar et al., K )These admit reasonable generalizations to singular varieties.





Vanishing TheoremsLet X be a smooth projective variety of dimension n over Cand L an ample line bundle on X . Then


X ⊗L ) = 0 for p + q > n.


X (log D)⊗L ) = 0 for p + q > n.








X ⊗L ) = 0 for p + q > n.


X (log D)⊗L ) = 0 for p + q > n.








X ⊗L ) = 0 for p + q > n.


X (log D)⊗L ) = 0 for p + q > n.






Theorem (K )Let f : X → B be an admissible family, and L an ample linebundle on X. Assume that L ⊗ f ∗ωB(∆)−(n−1) is also ample.Then Hn(X , L ⊗ f ∗ωB(∆)) = 0.

ProofCompose the edge morphisms to get a diagonal map.

ApplicationLet f : X → B be an admissible family and suppose that (H)fails, i.e., 2g(C)− 2 + #∆ ≤ 0, and hence ωB(∆)−1 is nef.Then for any ample line bundle L , Hn(X , L ⊗ f ∗ωB(∆)) = 0.If ∆ = ∅, we are done, as this implies thatHn(X , ωX ) = Hn(X , ωX/B ⊗ f ∗ωB) = 0, a contradiction.














ApplicationLet f : X → B be an admissible family and suppose that (H)fails, i.e., 2g(C)− 2 + #∆ ≤ 0, and hence ωB(∆)−1 is nef.

Then for any ample line bundle L , Hn(X , L ⊗ f ∗ωB(∆)) = 0.If ∆ = ∅, we are done, as this implies thatHn(X , ωX ) = Hn(X , ωX/B ⊗ f ∗ωB) = 0, a contradiction.







ApplicationLet f : X → B be an admissible family and suppose that (H)fails, i.e., 2g(C)− 2 + #∆ ≤ 0, and hence ωB(∆)−1 is nef.Then for any ample line bundle L , Hn(X , L ⊗ f ∗ωB(∆)) = 0.

If ∆ = ∅, we are done, as this implies thatHn(X , ωX ) = Hn(X , ωX/B ⊗ f ∗ωB) = 0, a contradiction.












Negativity of KS kernels

ZUO

observed that via duality the positivity theorem on directimages implies negativity of subbundles of the kernel ofKodaira-Spencer maps.

VIEHWEG and ZUO

combined this with the ’compose the edge morphisms to geta diagonal map’ trick to“localize” the argument. The result isan argument that makes it easier to handle singular fibers.





Negativity of KS kernels

ZUO

observed that via duality the positivity theorem on directimages implies negativity of subbundles of the kernel ofKodaira-Spencer maps.

VIEHWEG and ZUO

combined this with the ’compose the edge morphisms to geta diagonal map’ trick to“localize” the argument. The result isan argument that makes it easier to handle singular fibers.





Section Outline

3 Techniques and IngredientsPositivity of direct imagesIterated Kodaira-Spencer maps





Strong non-isotriviality (and rigidity)Let f : X → B be a smooth admissible family, dim X = n + 1,dim B = 1, T m

X = ∧mTX and T mX/B = ∧mTX/B. Let 1 ≤ p ≤ n.

0 → T pX/B ⊗ f ∗T⊗(n−p)

B →→ T p

X ⊗ f ∗T⊗(n−p)B → T p−1

X/B ⊗ f ∗T⊗(n−p+1)B → 0.

This induces an edge map,

ρ(p)f : Rp−1f∗T

p−1X/B ⊗ T⊗(n−p+1)

B → Rpf∗TpX/B ⊗ T⊗(n−p)

B .

DefinitionLet ρf := ρ

(n)f ρ

(n−1)f · · · ρ

(1)f : T⊗n

B −→ Rnf∗T nX/B and

call f strongly non-isotrivial if ρf 6= 0.







0 → T pX/B ⊗ f ∗T⊗(n−p)

B →→ T p

X ⊗ f ∗T⊗(n−p)B → T p−1

X/B ⊗ f ∗T⊗(n−p+1)B → 0.



p−1X/B ⊗ T⊗(n−p+1)


B .


(n)f ρ

(n−1)f · · · ρ

(1)f : T⊗n









0 → T pX/B ⊗ f ∗T⊗(n−p)

B →→ T p

X ⊗ f ∗T⊗(n−p)B → T p−1

X/B ⊗ f ∗T⊗(n−p+1)B → 0.



p−1X/B ⊗ T⊗(n−p+1)


B .


(n)f ρ

(n−1)f · · · ρ

(1)f : T⊗n









0 → T pX/B ⊗ f ∗T⊗(n−p)

B →→ T p

X ⊗ f ∗T⊗(n−p)B → T p−1

X/B ⊗ f ∗T⊗(n−p+1)B → 0.



p−1X/B ⊗ T⊗(n−p+1)


B .


(n)f ρ

(n−1)f · · · ρ

(1)f : T⊗n







Geometric Meaning

Definitionρf : T⊗n

B −→ Rnf∗T nX/B and f is strongly non-isotrivial if ρf 6= 0.

The condition ρf 6= 0 is supposed to mean that the family isnon-isotrivial in every vertical direction.

ProblemMake this precise, i.e., give a geometric description of "strongnon-isotriviality".





Geometric Meaning









Geometric Meaning








Acknowledgement

This presentation was made using thebeamertex LATEX macropackage of Till Tantau.

http://latex-beamer.sourceforge.net


Documents

Subvarieties of moduli stacks - sites.math.washington.edukovacs/seattle... · Shafarevich’s Conjecture Generalizations Techniques and Ingredients Definitions and Notation Outline