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ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
Generalizations
Families of varieties of generaltype: the Shafarevich conjecture
and related problems
Sándor Kovács
University of Washington
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
Generalizations
Motto
"To me,algebraic geometry
is algebra with a kick"–Solomon Lefschetz
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
Generalizations
Outline
1 Curves
2 Families
3 Shafarevich’s Conjecture
4 The Proof of Shafarevich’s Conjecture
5 Generalizations
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Outline
1 CurvesRiemann SurfacesGenusTopology, Arithmetic and Differential Geometry
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Curves
Smooth ComplexProjective Curve
=Compact Riemann
Surface
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Curves
Smooth ComplexProjective Curve
=Compact Riemann
Surface
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Outline
1 CurvesRiemann SurfacesGenusTopology, Arithmetic and Differential Geometry
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Genus
genus 0
genus 1
genus 2, ...
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Genus
genus 0
genus 1
genus 2, ...
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Genus
genus 0
genus 1
genus 2, ...
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Genus
genus 0
genus 1
genus 2, ...
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Genus
genus 0
genus 1
genus 2, ...
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Outline
1 CurvesRiemann SurfacesGenusTopology, Arithmetic and Differential Geometry
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Smooth Projective Curves
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Smooth Projective Curves
Smooth curves C come in three types:
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Smooth Projective Curves
Smooth curves C come in three types:
• C ' P1 (rational)
• C is a plane cubic (elliptic)
• C has genus ≥ 2 (general type)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Smooth Projective Curves
Smooth curves C come in three types:
• C ' P1 (rational)
• C is a plane cubic (elliptic)
• C has genus ≥ 2 (general type)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Smooth Projective Curves
Smooth curves C come in three types:
• C ' P1 (rational)
• C is a plane cubic (elliptic)
• C has genus ≥ 2 (general type)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Topology of Curves
Smooth curves C come in three types, and...
• C ' P1 (rational)
• C is a plane cubic (elliptic)
• C has genus ≥ 2 (general type)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Topology of Curves
...their fundamental groups come in three types:
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Topology of Curves
...their fundamental groups come in three types:
• C ' P1 π1 is trivial
• C is a plane cubic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Topology of Curves
...their fundamental groups come in three types:
• C ' P1 π1 is trivial
• C is a plane cubic π1 is abelian
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Topology of Curves
...their fundamental groups come in three types:
• C ' P1 π1 is trivial
• C is a plane cubic π1 is abelian
• C has genus ≥ 2 π1 is non-commutative
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Arithmetic of Curves
Smooth curves C come in three types, and...
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Arithmetic of Curves
...rational points on them come in three types:
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Arithmetic of Curves
...rational points on them come in three types:
• C ' P1 lots
• C is a plane cubic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Arithmetic of Curves
...rational points on them come in three types:
• C ' P1 non-finitely generated
• C is a plane cubic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Arithmetic of Curves
...rational points on them come in three types:
• C ' P1 non-finitely generated
• C is a plane cubic finitely generated
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Arithmetic of Curves
...rational points on them come in three types:
• C ' P1 non-finitely generated
• C is a plane cubic finitely generated
• C has genus ≥ 2 finite
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
Smooth curves C come in three types, and...
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1 positively curved
• C is a plane cubic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1 positively curved
• C is a plane cubic flat
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1 positively curved
• C is a plane cubic flat
• C has genus ≥ 2 negatively curved
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1 parabolic
• C is a plane cubic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1 parabolic
• C is a plane cubic elliptic
• C has genus ≥ 2
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1 parabolic
• C is a plane cubic elliptic
• C has genus ≥ 2 hyperbolic
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Outline
2 FamiliesExamples and DefinitionsIsotrivial and Non-isotrivial Families
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
X
y
π
A1λ
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
A2x ,y ⊇ Xλ = π−1(λ) ⊆ X
y
π
λ ∈ A1λ
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Notation
B – smooth projective curve of genus g,∆ ⊆ B – finite set of points.
A family over B \ ∆, f : X → B \ ∆, is asurjective (flat) map with equidimensional,connected fibers.
A family is smooth if Xb = f−1(b) is smooth for∀b ∈ B.
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Notation
B – smooth projective curve of genus g,∆ ⊆ B – finite set of points.
A family over B \ ∆, f : X → B \ ∆, is asurjective (flat) map with equidimensional,connected fibers.
A family is smooth if Xb = f−1(b) is smooth for∀b ∈ B.
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Notation
B – smooth projective curve of genus g,∆ ⊆ B – finite set of points.
A family over B \ ∆, f : X → B \ ∆, is asurjective (flat) map with equidimensional,connected fibers.
A family is smooth if Xb = f−1(b) is smooth for∀b ∈ B.
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Notation
B – smooth projective curve of genus g,∆ ⊆ B – finite set of points.
A family over B \ ∆, f : X → B \ ∆, is asurjective (flat) map with equidimensional,connected fibers.
A family is smooth if Xb = f−1(b) is smooth for∀b ∈ B.
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Outline
2 FamiliesExamples and DefinitionsIsotrivial and Non-isotrivial Families
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Isotrivial Family
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Isotrivial Family
All smooth fibers are isomorphic to P1.
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Isotrivial Family
All smooth fibers are isomorphic to P1.
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Non-isotrivial families
The family defined by y2 = x5 − 5λx + 4λ is notisotrivial, because
Xλ1 6' Xλ2
for λ1 6= λ2.
A family f : X → B is isotrivial if there exists afinite set ∆ ⊂ B such that Xb ' Xb′ for allb, b′ ∈ B \ ∆.
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Non-isotrivial families
The family defined by y2 = x5 − 5λx + 4λ is notisotrivial, because
Xλ1 6' Xλ2
for λ1 6= λ2.
A family f : X → B is isotrivial if there exists afinite set ∆ ⊂ B such that Xb ' Xb′ for allb, b′ ∈ B \ ∆.
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Non-isotrivial families
The family defined by y2 = x5 − 5λx + 4λ is notisotrivial, because
Xλ1 6' Xλ2
for λ1 6= λ2.
A family f : X → B is isotrivial if there exists afinite set ∆ ⊂ B such that Xb ' Xb′ for allb, b′ ∈ B \ ∆.
Warm-Up Question: Let q ∈ Z. How manyisotrivial families of curves of genus q over Bare there?
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Non-isotrivial families
The family defined by y2 = x5 − 5λx + 4λ is notisotrivial, because
Xλ1 6' Xλ2
for λ1 6= λ2.
A family f : X → B is isotrivial if there exists afinite set ∆ ⊂ B such that Xb ' Xb′ for allb, b′ ∈ B \ ∆.
Warm-Up Question: Let q ∈ Z. How manyisotrivial families of curves of genus q over Bare there? ∞
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Outline
3 Shafarevich’s ConjectureStatement and DefinitionsResults and Connections
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture (1962)
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture (1962)
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture (1962)
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.These will be called “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture (1962)
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.These will be called “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture (1962)
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.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 & ∆ = ∅
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic implies that
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic implies that
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic implies that
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic implies that
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic is equivalent to
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic is equivalent to
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
An algebraic variety X is algebraicallyhyperbolic if
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic is equivalent to
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
An algebraic variety X is algebraicallyhyperbolic if
• ∀ A1 \ {0} → X regular map is constant
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic is equivalent to
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
An algebraic variety X is algebraicallyhyperbolic if
• ∀ A1 \ {0} → X regular map is constant, and• ∀ A → X regular map is constant, for any
abelian variety (projective algebraic group) A.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity: Examples
B – smooth projective curve,∆ ⊆ B – finite subset.
• If g(B) ≥ 2, thenB \ ∆ is hyperbolic for arbitrary ∆.
• If g(B) = 1, thenB \ ∆ is hyperbolic ⇔ ∆ 6= ∅.
• If g(B) = 0, thenB \ ∆ is hyperbolic ⇔ #∆ ≥ 3.
B \ ∆ is hyperbolic ⇔ 2g(B) − 2 + #∆ > 0.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity: Examples
B – smooth projective curve,∆ ⊆ B – finite subset.
• If g(B) ≥ 2, thenB \ ∆ is hyperbolic for arbitrary ∆.
• If g(B) = 1, thenB \ ∆ is hyperbolic ⇔ ∆ 6= ∅.
• If g(B) = 0, thenB \ ∆ is hyperbolic ⇔ #∆ ≥ 3.
B \ ∆ is hyperbolic ⇔ 2g(B) − 2 + #∆ > 0.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity: Examples
B – smooth projective curve,∆ ⊆ B – finite subset.
• If g(B) ≥ 2, thenB \ ∆ is hyperbolic for arbitrary ∆.
• If g(B) = 1, thenB \ ∆ is hyperbolic ⇔ ∆ 6= ∅.
• If g(B) = 0, thenB \ ∆ is hyperbolic ⇔ #∆ ≥ 3.
B \ ∆ is hyperbolic ⇔ 2g(B) − 2 + #∆ > 0.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity: Examples
B – smooth projective curve,∆ ⊆ B – finite subset.
• If g(B) ≥ 2, thenB \ ∆ is hyperbolic for arbitrary ∆.
• If g(B) = 1, thenB \ ∆ is hyperbolic ⇔ ∆ 6= ∅.
• If g(B) = 0, thenB \ ∆ is hyperbolic ⇔ #∆ ≥ 3.
B \ ∆ is hyperbolic ⇔ 2g(B) − 2 + #∆ > 0.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity: Examples
B – smooth projective curve,∆ ⊆ B – finite subset.
• If g(B) ≥ 2, thenB \ ∆ is hyperbolic for arbitrary ∆.
• If g(B) = 1, thenB \ ∆ is hyperbolic ⇔ ∆ 6= ∅.
• If g(B) = 0, thenB \ ∆ is hyperbolic ⇔ #∆ ≥ 3.
B \ ∆ is hyperbolic ⇔ 2g(B) − 2 + #∆ > 0.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture: Take Two
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q overB \ ∆. “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
(II∗) If there exists any admissible families, thenB \ ∆ is hyperbolic.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture: Take Two
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q overB \ ∆. “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
(II∗) If there exists any admissible families, thenB \ ∆ is hyperbolic.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture: Take Two
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q overB \ ∆, “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
(II∗) If there exists any admissible families, thenB \ ∆ is hyperbolic.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture: Take Two
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q overB \ ∆, “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
(II∗) If there exists any admissible families, thenB \ ∆ is hyperbolic.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture: Take Two
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q overB \ ∆, “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
(II∗) If there exists any admissible families, thenB \ ∆ is hyperbolic.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Outline
3 Shafarevich’s ConjectureStatement and DefinitionsResults and Connections
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Geometric Mordell Conjecture
Let f : X → B be a non-isotrivial family ofprojective curves of genus ≥ 2.Then there are only finitely many sections of f ,i.e., σ : B → X , such that f ◦ σ = idB.
This was proved by MANIN (1963).
And again by PARSHIN using
“Parshin’s Covering Trick” to prove that
The Shafarevich Conjectureimplies
The Mordell Conjecture.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Geometric Mordell Conjecture
Let f : X → B be a non-isotrivial family ofprojective curves of genus ≥ 2.Then there are only finitely many sections of f ,i.e., σ : B → X , such that f ◦ σ = idB.
This was proved by MANIN (1963).
And again by PARSHIN using
“Parshin’s Covering Trick” to prove that
The Shafarevich Conjectureimplies
The Mordell Conjecture.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Geometric Mordell Conjecture
Let f : X → B be a non-isotrivial family ofprojective curves of genus ≥ 2.Then there are only finitely many sections of f ,i.e., σ : B → X , such that f ◦ σ = idB.
This was proved by MANIN (1963).
And again by PARSHIN (1968). using
“Parshin’s Covering Trick” to prove that
The Shafarevich Conjectureimplies
The Mordell Conjecture.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Geometric Mordell Conjecture
Let f : X → B be a non-isotrivial family ofprojective curves of genus ≥ 2.Then there are only finitely many sections of f ,i.e., σ : B → X , such that f ◦ σ = idB.
This was proved by MANIN (1963).
And again by PARSHIN (1968) using
“Parshin’s Covering Trick” to prove that
The Shafarevich Conjectureimplies
The Mordell Conjecture.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Outline
4 The Proof of Shafarevich’s ConjectureDeformation TheoryThe Arakelov-Parshin method
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformations
A deformation of an algebraic variety X is afamily F : X → T such that there exists a t ∈ Tthat X ' Xt .
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformations
A deformation of an algebraic variety X is afamily F : X → T such that there exists a t ∈ Tthat X ' Xt .
X
T
f
Xt
t
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformation Type
Two algebraic varieties X1 and X2 have thesame deformation type if there exists a familyF : X → T , t1, t2 ∈ T such that X1 ' Xt1 andX2 ' Xt2 .
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformation Type
Two algebraic varieties X1 and X2 have thesame deformation type if there exists a familyF : X → T , t1, t2 ∈ T such that X1 ' Xt1 andX2 ' Xt2 .
X
T
ft1
t2
Xt1
Xt2
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformations of Families
A deformation of a family X → B is a familyF : X → B × T such that there exists a t ∈ Tthat (X → B) ' (Xt → B × {t}), whereXt = F−1(B × {t}).
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformations of Families
A deformation of a family X → B is a familyF : X → B × T such that there exists a t ∈ Tthat (X → B) ' (Xt → B × {t}), whereXt = F−1(B × {t}).
F
X
T×B
Xt
{ }B t×
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformation Type of Families
Two families X1 → B and X2 → B have thesame deformation type if there exists a familyF : X → B × T , t1, t2 ∈ T such that(X1 → B) ' (Xt1 → B × {t}) and(X2 → B) ' (Xt2 → B × {t}).
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformation Type of Families
Two families X1 → B and X2 → B have thesame deformation type if there exists a familyF : X → B × T , t1, t2 ∈ T such that(X1 → B) ' (Xt1 → B × {t}) and(X2 → B) ' (Xt2 → B × {t}).
F
XXt
1
2{ }B t×
1{ }B t×
Xt2
T×B
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Outline
4 The Proof of Shafarevich’s ConjectureDeformation TheoryThe Arakelov-Parshin method
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
The Arakelov-Parshin method
ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
The Arakelov-Parshin method
ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:(B) Boundedness: There are only finitely many
deformation types of admissible families.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
The Arakelov-Parshin method
ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:(B) Boundedness: There are only finitely many
deformation types of admissible families.(R) Rigidity: Admissible families do not admit
non-trivial deformations.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
The Arakelov-Parshin method
ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:(B) Boundedness: There are only finitely many
deformation types of admissible families.(R) Rigidity: Admissible families do not admit
non-trivial deformations.
Note:• (B) and (R) together imply (I).
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
The Arakelov-Parshin method
ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:(B) Boundedness: There are only finitely many
deformation types of admissible families.(R) Rigidity: Admissible families do not admit
non-trivial deformations.(H) Hyperbolicity: If there exist admissible families,
then B \ ∆ is hyperbolic.
Note:• (B) and (R) together imply (I).
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
The Arakelov-Parshin method
ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:(B) Boundedness: There are only finitely many
deformation types of admissible families.(R) Rigidity: Admissible families do not admit
non-trivial deformations.(H) Hyperbolicity: If there exist admissible families,
then B \ ∆ is hyperbolic.
Note:• (B) and (R) together imply (I).• (H) = (II∗) and hence is equivalent to (II).
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional fibers
First order of business:
Generalize the setup.
Find a replacement for “genus” and “q ≥ 2”.
• Hilbert polynomial, h.• Kodaira dimension, κ = −∞, 0, 1, . . . , dim.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional fibers
First order of business:
Generalize the setup.
Find a replacement for “genus” and “q ≥ 2”.
• Hilbert polynomial, h.• Kodaira dimension, κ = −∞, 0, 1, . . . , dim.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional fibers
First order of business:
Generalize the setup.
Find a replacement for “genus” and “q ≥ 2”.
• Hilbert polynomial, h.• Kodaira dimension, κ = −∞, 0, 1, . . . , dim.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional fibers
First order of business:
Generalize the setup.
Find a replacement for “genus” and “q ≥ 2”.
• Hilbert polynomial, h.• Kodaira dimension, κ = −∞, 0, 1, . . . , dim.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional fibers
First order of business:
Generalize the setup.
Find a replacement for “genus” and “q ≥ 2”.
• Hilbert polynomial, h.• Kodaira dimension, κ = −∞, 0, 1, . . . , dim.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Plane curves:Kodaira
type degree genus dimension
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Plane curves:Kodaira
type degree genus dimension
P1 deg = 1, 2 g = 0 κ = −∞
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Plane curves:Kodaira
type degree genus dimension
P1 deg = 1, 2 g = 0 κ = −∞
elliptic deg = 3 g = 1 κ = 0
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Plane curves:Kodaira
type degree genus dimension
P1 deg = 1, 2 g = 0 κ = −∞
elliptic deg = 3 g = 1 κ = 0
general deg ≥ 4 g ≥ 2 κ = 1type
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
deg < n + 1 κ = −∞
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
deg < n + 1 κ = −∞
deg = n + 1 κ = 0
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
deg < n + 1 κ = −∞
deg = n + 1 κ = 0
deg > n + 1 κ = dim
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
Fano deg < n + 1 κ = −∞
deg = n + 1 κ = 0
deg > n + 1 κ = dim
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
Fano deg < n + 1 κ = −∞
Calabi-Yau deg = n + 1 κ = 0
deg > n + 1 κ = dim
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
Fano deg < n + 1 κ = −∞
Calabi-Yau deg = n + 1 κ = 0
general type deg > n + 1 κ = dim
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
Fano deg < n + 1 κ = −∞
Calabi-Yau deg = n + 1 κ = 0
general type deg > n + 1 κ = dim
Products:
dim(X × Y ) = dim X + dim Y
κ(X × Y ) = κ(X ) + κ(Y )
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Varieties of General Type
X is of general type if κ(X ) = dim X .
Recall: A curve C is of general type iff g(C) ≥ 2.
Genus is replaced with the Hilbert polynomial.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Varieties of General Type
X is of general type if κ(X ) = dim X .
Recall: A curve C is of general type iff g(C) ≥ 2.
Genus is replaced with the Hilbert polynomial.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Varieties of General Type
X is of general type if κ(X ) = dim X .
Recall: A curve C is of general type iff g(C) ≥ 2.
Genus is replaced with the Hilbert polynomial.
For curves, the Hilbert polynomial is determinedby the genus, so this is a natural generalization.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Varieties of General Type
X is of general type if κ(X ) = dim X .
Recall: A curve C is of general type iff g(C) ≥ 2.
Genus is replaced with the Hilbert polynomial.
For curves, the Hilbert polynomial is determinedby the genus, so this is a natural generalization.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
The Shafarevich Conjecture
Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if
• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective
variety, canonically embedded with Hilbertpolynomial h.
“Canonically embedded” means that it isembedded by the global sections of ωm
Xb.
In particular, Xb is of general type.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
The Shafarevich Conjecture
Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if
• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective
variety, canonically embedded with Hilbertpolynomial h.
“Canonically embedded” means that it isembedded by the global sections of ωm
Xb.
In particular, Xb is of general type.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
The Shafarevich Conjecture
Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if
• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective
variety, canonically embedded with Hilbertpolynomial h.
“Canonically embedded” means that it isembedded by the global sections of ωm
Xb.
In particular, Xb is of general type.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
The Shafarevich Conjecture
Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if
• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective
variety, canonically embedded with Hilbertpolynomial h.
“Canonically embedded” means that it isembedded by the global sections of ωm
Xb.
In particular, Xb is of general type.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
The Shafarevich Conjecture
Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if
• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective
variety, canonically embedded with Hilbertpolynomial h.
“Canonically embedded” means that it isembedded by the global sections of ωm
Xb.
In particular, Xb is of general type.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
The Shafarevich Conjecture
Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if
• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective
variety, canonically embedded with Hilbertpolynomial h.
“Canonically embedded” means that it isembedded by the global sections of ωm
Xb.
In particular, Xb is of general type.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Shafarevich Conjecture
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Shafarevich Conjecture
(B) Boundedness: There are only finitely manydeformation types of admissible families.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Shafarevich Conjecture
(B) Boundedness: There are only finitely manydeformation types of admissible families.
(R) Rigidity: Admissible families do not admitnon-trivial deformations.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Shafarevich Conjecture
(B) Boundedness: There are only finitely manydeformation types of admissible families.
(R) Rigidity: Admissible families do not admitnon-trivial deformations.
(H) Hyperbolicity: If there exist admissible families,then B \ ∆ is hyperbolic.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Shafarevich Conjecture
(B) Boundedness: There are only finitely manydeformation types of admissible families.
(R) Rigidity: Admissible families do not admitnon-trivial deformations.
(H) Hyperbolicity: If there exist admissible families,then B \ ∆ is hyperbolic.
(WB) Weak Boundedness:If f : X → B is an admissible family, thendeg f∗ωm
X/B is bounded in terms of B, ∆, h, m.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Weak Boundedness
Moduli Theory + (WB) ⇒ (B)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Weak Boundedness
Moduli Theory + (WB) ⇒ (B)
(WB) ⇒ (H)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Weak Boundedness
Moduli Theory + (WB) ⇒ (B)
(K , 2002) (WB) ⇒ (H)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Weak Boundedness
Moduli Theory + (WB) ⇒ (B)
(K , 2002) (WB) ⇒ (H)
Hence Weak Boundedness is the key statementtowards proving (B) and (H).
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Weak Boundedness
Moduli Theory + (WB) ⇒ (B)
(K , 2002) (WB) ⇒ (H)
Hence Weak Boundedness is the key statementtowards proving (B) and (H).
(WB) Weak Boundedness:If f : X → B is an admissible family, thendeg f∗ωm
X/B is bounded in terms of B, ∆, h, m.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History
“modern” = last 25 years
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History
“modern” ≈ last 10 years
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
BEAUVILLE (1981):(H) holds for dim(X/B) = 1.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
BEAUVILLE (1981):(H) holds for dim(X/B) = 1.
CATANESE-SCHNEIDER (1994):(H) can be used for giving upper bounds on thesize of automorphisms groups of varieties ofgeneral type.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
BEAUVILLE (1981):(H) holds for dim(X/B) = 1.
CATANESE-SCHNEIDER (1994):(H) can be used for giving upper bounds on thesize of automorphisms groups of varieties ofgeneral type.
SHOKUROV (1995):(H) can be used to prove quasi-projectivity ofmoduli spaces of varieties of general type.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
MIGLIORINI (1995):(H) holds for dim(X/B) = 2 and g(B) = 1.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
MIGLIORINI (1995):(H) holds for dim(X/B) = 2 and g(B) = 1.
K (1996):(H) holds for dim(X/B) arbitrary and g(B) = 1.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
MIGLIORINI (1995):(H) holds for dim(X/B) = 2 and g(B) = 1.
K (1996):(H) holds for dim(X/B) arbitrary and g(B) = 1.
K (1997):(H) holds for dim(X/B) = 2.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
MIGLIORINI (1995):(H) holds for dim(X/B) = 2 and g(B) = 1.
K (1996):(H) holds for dim(X/B) arbitrary and g(B) = 1.
K (1997):(H) holds for dim(X/B) = 2.
K (2000):(H) holds for dim(X/B) arbitrary.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
MIGLIORINI (1995):(H) holds for dim(X/B) = 2 and g(B) = 1.
K (1996):(H) holds for dim(X/B) arbitrary and g(B) = 1.
K (1997):(H) holds for dim(X/B) = 2.
K (2000):(H) holds for dim(X/B) arbitrary.
VIEHWEG AND ZUO (2002):Brody hyperbolicity holds as well.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History:(Weak) Boundedness
BEDULEV AND VIEHWEG (2000):
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History:(Weak) Boundedness
BEDULEV AND VIEHWEG (2000):• (B) holds for dim(X/B) = 2, and
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History:(Weak) Boundedness
BEDULEV AND VIEHWEG (2000):• (B) holds for dim(X/B) = 2, and• (WB) holds for dim(X/B) arbitrary.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History:(Weak) Boundedness
BEDULEV AND VIEHWEG (2000):• (B) holds for dim(X/B) = 2, and• (WB) holds for dim(X/B) arbitrary.
As a byproduct of their proof they also obtainedthat (H) holds in these cases.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History:(Weak) Boundedness
BEDULEV AND VIEHWEG (2000):• (B) holds for dim(X/B) = 2, and• (WB) holds for dim(X/B) arbitrary.
As a byproduct of their proof they also obtainedthat (H) holds in these cases.
K (2002), VIEHWEG AND ZUO (2002):(WB) holds under more general assumptions.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History:(Weak) Boundedness
BEDULEV AND VIEHWEG (2000):• (B) holds for dim(X/B) = 2, and• (WB) holds for dim(X/B) arbitrary.
As a byproduct of their proof they also obtainedthat (H) holds in these cases.
K (2002), VIEHWEG AND ZUO (2002):(WB) holds under more general assumptions.
K (2002):(WB) implies (H).
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
More History
...more related results by
FALTINGS (1983)
ZHANG (1997)
OGUISO AND VIEHWEG (2001)
..and more.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
More History
...more related results by
FALTINGS (1983)
ZHANG (1997)
OGUISO AND VIEHWEG (2001)
..and more.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
More History
...more related results by
FALTINGS (1983)
ZHANG (1997)
OGUISO AND VIEHWEG (2001)
..and more.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
More History
...more related results by
FALTINGS (1983)
ZHANG (1997)
OGUISO AND VIEHWEG (2001)
..and more.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
More History
...more related results by
FALTINGS (1983)
ZHANG (1997)
OGUISO AND VIEHWEG (2001)
..and more.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
Instead of ∆ being a finite subset,
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.
A family being non-isotrivial is no longer a goodassumption.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.
A family being non-isotrivial is no longer a goodassumption. It has to be replaced by a familyhaving maximal variation in moduli.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.
A family being non-isotrivial is no longer a goodassumption. It has to be replaced by a familyhaving maximal variation in moduli.
Viehweg’s Conjecture (2001): If there exists anadmissible family with maximal variation inmoduli, then (B, ∆) is of log general type.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.
A family being non-isotrivial is no longer a goodassumption. It has to be replaced by a familyhaving maximal variation in moduli.
Viehweg’s Conjecture (2001): If there exists anadmissible family with maximal variation inmoduli, then (B, ∆) is of log general type.That is, the log-Kodaira dimension of B ismaximal: κ(B, ∆) = dim B.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Viehweg’s Conjecture (2001): If there exists anadmissible family with maximal variation inmoduli, then (B, ∆) is of log general type.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Viehweg’s Conjecture (2001): If there exists anadmissible family with maximal variation inmoduli, then (B, ∆) is of log general type.
K (2003), VIEHWEG-ZUO, (2003):Viehweg’s conjecture holds for B = Pn.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Viehweg’s Conjecture (2001): If there exists anadmissible family with maximal variation inmoduli, then (B, ∆) is of log general type.
K (2003), VIEHWEG-ZUO, (2003):Viehweg’s conjecture holds for B = Pn.
K (1997):If f : X → B is admissible and B is an abelianvariety, then ∆ 6= ∅.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity
But, what about Rigidity?
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity – An Example
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity – An Example
Let Y → B be an arbitrary non-isotrivial familyof curves of genus ≥ 2, and
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity – An Example
Let Y → B be an arbitrary non-isotrivial familyof curves of genus ≥ 2, and
C a smooth projective curve of genus ≥ 2.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity – An Example
Let Y → B be an arbitrary non-isotrivial familyof curves of genus ≥ 2, and
C a smooth projective curve of genus ≥ 2.
C
f
B
X Y C= ×
CC
b bX Y C= ×
YbY C
b
C
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity – An Example
f : X = Y × C → B is an admissible family,and
C
f
B
X Y C= ×
CC
b bX Y C= ×
YbY C
b
C
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity – An Example
f : X = Y × C → B is an admissible family,and
any deformation of C gives a deformation of f .
C
f
B
X Y C= ×
CC
b bX Y C= ×
YbY C
b
C
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity
Therefore, (R) fails.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity
Therefore, (R) fails.
Question: Under what additional conditionsdoes (R) hold?
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity
Therefore, (R) fails.
Question: Under what additional conditionsdoes (R) hold?
K (2004), VIEHWEG-ZUO, (2004):Rigidity holds for strongly non-isotrivial families.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity
Therefore, (R) fails.
Question: Under what additional conditionsdoes (R) hold?
K (2004), VIEHWEG-ZUO, (2004):Rigidity holds for strongly1non-isotrivial families.
1unfortunately the margin is not wide enough to define
this term.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Current Work
Work in progress:
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Current Work
Work in progress:
• Geometric description of strongly non-isotrivialfamilies.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Current Work
Work in progress:
• Geometric description of strongly non-isotrivialfamilies.
• Joint work with Stefan Kebekus (Köln):Families over two-dimensional bases.
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
Generalizations
Acknowledgement
This presentation was made using thebeamertex LATEX macropackage of Till Tantau.http://latex-beamer.sourceforge.net
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