View
0
Download
0
Category
Preview:
Citation preview
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
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
Outline
1 Shafarevich’s Conjecture
2 Generalizations
3 Techniques and Ingredients
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
Section Outline
Definitions and Notation
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
A family of curves
y2 − λx2 = 1
defines a family of conics.
Remark: all fibers for λ 6= 0 are isomorphic.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
A family of curves
y2 − λx2 = 1defines a family of conics.
Remark: all fibers for λ 6= 0 are isomorphic.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
A family of curves
y2 − λx2 = 1defines a family of conics.
Remark: all fibers for λ 6= 0 are isomorphic.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
A family of curves of general type
y2 = x5 − 5λx + 4λ
defines a smooth family of curves parametrized by λ 6= 0, 1.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
A family of curves of general type
y2 = x5 − 5λx + 4λ
defines a smooth family of curves parametrized by λ 6= 0, 1.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
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?
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
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 = 1
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?
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
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 = z2
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?
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
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 = z2
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?
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
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 = z2
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?
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and IngredientsDefinitions and Notation
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
Section Outline
1 Shafarevich’s ConjectureStatement and DefinitionsFirst Results and ConnectionsThe Arakelov-Parshin Method
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst 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 & ∆ = ∅
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst 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 & ∆ = ∅
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst 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 & ∆ = ∅
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst 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 & ∆ = ∅
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst 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 & ∆ = ∅
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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.
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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.
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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.
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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.
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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.
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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.
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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 Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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 Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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 Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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 Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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 Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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,
then there aren’t any admissible families.
(II∗) If there exist any admissible families, then B \∆ ishyperbolic.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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,
then there aren’t any admissible families.
(II∗) If there exist any admissible families, then B \∆ ishyperbolic.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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,
then there aren’t any admissible families.
(II∗) If there exist any admissible families, then B \∆ ishyperbolic.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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,
then there aren’t any admissible families.
(II∗) If there exist any admissible families, then B \∆ ishyperbolic.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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,
then there aren’t any admissible families.
(II∗) If there exist any admissible families, then B \∆ ishyperbolic.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
Section Outline
1 Shafarevich’s ConjectureStatement and DefinitionsFirst Results and ConnectionsThe Arakelov-Parshin Method
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
History
Theorem (PARSHIN (1968))Shafarevich’s Conjecture holds for ∆ = ∅.
Theorem (ARAKELOV (1971))Shafarevich’s Conjecture holds in general.
Theorem (BEAUVILLE (1981))(II) holds.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
History
Theorem (PARSHIN (1968))Shafarevich’s Conjecture holds for ∆ = ∅.
Theorem (ARAKELOV (1971))Shafarevich’s Conjecture holds in general.
Theorem (BEAUVILLE (1981))(II) holds.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
History
Theorem (PARSHIN (1968))Shafarevich’s Conjecture holds for ∆ = ∅.
Theorem (ARAKELOV (1971))Shafarevich’s Conjecture holds in general.
Theorem (BEAUVILLE (1981))(II) holds.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
Intermezzo: The number field case
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.
Shafarevich’s Conjecture has a number field version as well.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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”
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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”
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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”
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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”
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
Section Outline
1 Shafarevich’s ConjectureStatement and DefinitionsFirst Results and ConnectionsThe Arakelov-Parshin Method
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
StatementsFirst Results and ConnectionsThe Arakelov-Parshin Method
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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Section Outline
2 GeneralizationsHigher Dimensional FibersOther directionsHigher Dimensional Bases
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Conjecture (Higher Dimensional 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Recent1 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
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
But
, what about Rigidity?
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
But, what about Rigidity?
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Rigidity – An Example
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Rigidity – An ExampleY → B a non-isotrivial family of curves of genus ≥ 2, and
C a smooth projective curve of genus ≥ 2.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Rigidity – An ExampleY → B a non-isotrivial family of curves of genus ≥ 2, and
C a smooth projective curve of genus ≥ 2.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Rigidity – An ExampleY → B a non-isotrivial family of curves of genus ≥ 2, and
C a smooth projective curve of genus ≥ 2.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Rigidity – An Examplef : X = Y × C → B is an admissible family, and
any deformation of C gives a deformation of f .
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Rigidity – An Examplef : X = Y × C → B is an admissible family, and
any deformation of C gives a deformation of f .
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
RigidityTherefore, (R) fails.
Question:Under what additional conditions does (R) hold?
(Partial) Answer:
Theorem (VIEHWEG-ZUO, K )Rigidity holds for strongly1non-isotrivial families.
1unfortunately the margin is not wide enough to define this term.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Open ProblemGive a geometric description of strong non-isotriviality.
lt g that pl it w l
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Open ProblemGive a geometric description of strong non-isotriviality,or at least give a geometric condition that implies it.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Section Outline
2 GeneralizationsHigher Dimensional FibersOther directionsHigher Dimensional Bases
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Uniform Boundedness
Currently only known for families of curves:
Caporaso (2002), (2003)
Heier (2004)
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Uniform Boundedness
Currently only known for families of curves.
:
Caporaso (2002), (2003)
Heier (2004)
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Uniform Boundedness
Currently only known for families of curves:
Caporaso (2002), (2003)
Heier (2004)
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Uniform Boundedness
Currently only known for families of curves:
Caporaso (2002), (2003)
Heier (2004)
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
Section Outline
2 GeneralizationsHigher Dimensional FibersOther directionsHigher Dimensional Bases
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
New Progress
ConjectureLet f : X → B be an admissible family, then
max(κ(B, ∆), 0) ≥ Var f .
Theorem (KEBEKUS-K (2005))This conjecture holds if dim B ≤ 2.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
New Progress
ConjectureLet f : X → B be an admissible family, then
max(κ(B, ∆), 0) ≥ Var f .
Theorem (KEBEKUS-K (2005))This conjecture holds if dim B ≤ 2.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
New Progress
ConjectureLet f : X → B be an admissible family, then
max(κ(B, ∆), 0) ≥ Var f .
Theorem (KEBEKUS-K (2005))This conjecture holds if dim B ≤ 2.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Higher Dimensional FibersOther directionsHigher Dimensional Bases
New Progress
ConjectureLet f : X → B be an admissible family, then
max(κ(B, ∆), 0) ≥ Var f .
Theorem (KEBEKUS-K (2005))This conjecture holds if dim B ≤ 2.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
Section Outline
3 Techniques and IngredientsPositivity of direct imagesIterated Kodaira-Spencer maps
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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)
effective form of (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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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)
effective form of (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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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)
(K , 2002) (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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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)
(K , 2002) (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).
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
Section Outline
3 Techniques and IngredientsPositivity of direct imagesIterated Kodaira-Spencer maps
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity 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.
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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".
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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".
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Positivity of direct imagesIterated Kodaira-Spencer maps
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".
Sándor Kovács Subvarieties of moduli stacks
Shafarevich’s ConjectureGeneralizations
Techniques and Ingredients
Acknowledgement
This presentation was made using thebeamertex LATEX macropackage of Till Tantau.
http://latex-beamer.sourceforge.net
Sándor Kovács Subvarieties of moduli stacks
Recommended