Vanishing theorems and holomorphic forms

Mihnea Popa

Northwestern

AMS Meeting, LansingMarch 14, 2015

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 1

Holomorphic one-forms and geometry

X compact complex manifold, dimC X = n.

TX = holomorphic tangent bundle; Ω1X = T∨X = cotangent bundle.

Locally, in coordinates z1, . . . , zn, a holomorphic 1-form is

ω =n∑

i=1

fidzi , fi holomorphic.

Global holomorphic 1-forms on X (the main objects in this talk)are the holomorphic sections of Ω1

X :

H1,0(X ) := Γ(X ,Ω1X ).

h1,0(X ) := dimC H1,0(X ) (a Hodge number).

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 2

Holomorphic one-forms and geometry

X compact complex manifold, dimC X = n.

TX = holomorphic tangent bundle; Ω1X = T∨X = cotangent bundle.

Locally, in coordinates z1, . . . , zn, a holomorphic 1-form is

ω =n∑

i=1

fidzi , fi holomorphic.

Global holomorphic 1-forms on X (the main objects in this talk)are the holomorphic sections of Ω1

X :

H1,0(X ) := Γ(X ,Ω1X ).

h1,0(X ) := dimC H1,0(X ) (a Hodge number).

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 2

Holomorphic one-forms and geometry

X compact complex manifold, dimC X = n.

TX = holomorphic tangent bundle; Ω1X = T∨X = cotangent bundle.

Locally, in coordinates z1, . . . , zn, a holomorphic 1-form is

ω =n∑

i=1

fidzi , fi holomorphic.

Global holomorphic 1-forms on X (the main objects in this talk)are the holomorphic sections of Ω1

X :

H1,0(X ) := Γ(X ,Ω1X ).

h1,0(X ) := dimC H1,0(X ) (a Hodge number).

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 2

Holomorphic one-forms and geometry

X compact complex manifold, dimC X = n.

TX = holomorphic tangent bundle; Ω1X = T∨X = cotangent bundle.

Locally, in coordinates z1, . . . , zn, a holomorphic 1-form is

ω =n∑

i=1

fidzi , fi holomorphic.

Global holomorphic 1-forms on X (the main objects in this talk)are the holomorphic sections of Ω1

X :

H1,0(X ) := Γ(X ,Ω1X ).

h1,0(X ) := dimC H1,0(X ) (a Hodge number).

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 2

Examples

• Non-trivial one-forms may or may not exist. Examples:

1) H1,0(Pn) = 0.

2) X hypersurface in Pn, n ≥ 3 =⇒ H1,0(X ) = 0.

3) T = V /Λ compact complex torus =⇒ H1,0(T ) ' V ∨.

4) C compact Riemann surface of genus g =⇒ h1,0(C ) = g .

• If X is projective, or just compact Kahler, Hodge decompositiongives

H1(X ,C) ' H1,0(X )⊕ H1,0(X ).

In particular h1,0(X ) = b1(X )/2.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 3

Examples

• Non-trivial one-forms may or may not exist. Examples:

1) H1,0(Pn) = 0.

2) X hypersurface in Pn, n ≥ 3 =⇒ H1,0(X ) = 0.

3) T = V /Λ compact complex torus =⇒ H1,0(T ) ' V ∨.

4) C compact Riemann surface of genus g =⇒ h1,0(C ) = g .

• If X is projective, or just compact Kahler, Hodge decompositiongives

H1(X ,C) ' H1,0(X )⊕ H1,0(X ).

In particular h1,0(X ) = b1(X )/2.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 3

Examples

• Non-trivial one-forms may or may not exist. Examples:

1) H1,0(Pn) = 0.

2) X hypersurface in Pn, n ≥ 3 =⇒ H1,0(X ) = 0.

3) T = V /Λ compact complex torus =⇒ H1,0(T ) ' V ∨.

4) C compact Riemann surface of genus g =⇒ h1,0(C ) = g .

• If X is projective, or just compact Kahler, Hodge decompositiongives

H1(X ,C) ' H1,0(X )⊕ H1,0(X ).

In particular h1,0(X ) = b1(X )/2.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 3

Examples

• Non-trivial one-forms may or may not exist. Examples:

1) H1,0(Pn) = 0.

2) X hypersurface in Pn, n ≥ 3 =⇒ H1,0(X ) = 0.

3) T = V /Λ compact complex torus =⇒ H1,0(T ) ' V ∨.

4) C compact Riemann surface of genus g =⇒ h1,0(C ) = g .

• If X is projective, or just compact Kahler, Hodge decompositiongives

H1(X ,C) ' H1,0(X )⊕ H1,0(X ).

In particular h1,0(X ) = b1(X )/2.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 3

Examples

• Non-trivial one-forms may or may not exist. Examples:

1) H1,0(Pn) = 0.

2) X hypersurface in Pn, n ≥ 3 =⇒ H1,0(X ) = 0.

3) T = V /Λ compact complex torus =⇒ H1,0(T ) ' V ∨.

4) C compact Riemann surface of genus g =⇒ h1,0(C ) = g .

• If X is projective, or just compact Kahler, Hodge decompositiongives

H1(X ,C) ' H1,0(X )⊕ H1,0(X ).

In particular h1,0(X ) = b1(X )/2.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 3

Examples

• Non-trivial one-forms may or may not exist. Examples:

1) H1,0(Pn) = 0.

2) X hypersurface in Pn, n ≥ 3 =⇒ H1,0(X ) = 0.

3) T = V /Λ compact complex torus =⇒ H1,0(T ) ' V ∨.

4) C compact Riemann surface of genus g =⇒ h1,0(C ) = g .

• If X is projective, or just compact Kahler, Hodge decompositiongives

H1(X ,C) ' H1,0(X )⊕ H1,0(X ).

In particular h1,0(X ) = b1(X )/2.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 3

Holomorphic one-forms and geometry

How can we use them geometrically? Examples:

• There exist no nontrivial maps f : Pn → T , where T is a torus.

Proof: f ∗H1,0(T ) ⊂ H1,0(Pn) = 0.

• Recent, and much more subtle: there exist no submersionsf : X → T , where X is a variety of general type and T is a torus.

Later; will need a lot of the machinery discussed in the talk.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 4

Holomorphic one-forms and geometry

How can we use them geometrically? Examples:

• There exist no nontrivial maps f : Pn → T , where T is a torus.

Proof: f ∗H1,0(T ) ⊂ H1,0(Pn) = 0.

• Recent, and much more subtle: there exist no submersionsf : X → T , where X is a variety of general type and T is a torus.

Later; will need a lot of the machinery discussed in the talk.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 4

Holomorphic one-forms and geometry

How can we use them geometrically? Examples:

• There exist no nontrivial maps f : Pn → T , where T is a torus.

Proof: f ∗H1,0(T ) ⊂ H1,0(Pn) = 0.

• Recent, and much more subtle: there exist no submersionsf : X → T , where X is a variety of general type and T is a torus.

Later; will need a lot of the machinery discussed in the talk.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 4

Holomorphic one-forms and geometry

The condition h1,0(X ) 6= 0 influences the global geometry of X .

There is an inclusion of H1(X ,Z) as a lattice in H1,0(X )∨ given by

γ 7→∫γ

(·)

The Albanese torus of X is A = Alb(X ) := H1,0(X )∨/H1(X ,Z)=⇒ compact complex torus of dimension h1,0(X ); abelian variety(i.e. projective torus) if X is projective.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 5

Holomorphic one-forms and geometry

The condition h1,0(X ) 6= 0 influences the global geometry of X .

There is an inclusion of H1(X ,Z) as a lattice in H1,0(X )∨ given by

γ 7→∫γ

(·)

The Albanese torus of X is A = Alb(X ) := H1,0(X )∨/H1(X ,Z)=⇒ compact complex torus of dimension h1,0(X ); abelian variety(i.e. projective torus) if X is projective.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 5

Holomorphic one-forms and geometry

The condition h1,0(X ) 6= 0 influences the global geometry of X .

There is an inclusion of H1(X ,Z) as a lattice in H1,0(X )∨ given by

γ 7→∫γ

(·)

The Albanese torus of X is A = Alb(X ) := H1,0(X )∨/H1(X ,Z)=⇒ compact complex torus of dimension h1,0(X ); abelian variety(i.e. projective torus) if X is projective.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 5

Up to fixing x0 ∈ X , also have the Albanese map:

X −→ Alb(X ), x 7→∫ x

x0

(·).

x

x0

1

2

Integrals well defined up to “periods” = elements of the latticeH1(X ,Z) ⊂ H1,0(X )∨.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 6

Holomorphic one-forms and geometry

Dual torus is

Pic0(X ) := A = H1,0(X )/H1(X ,Z),

the Picard torus of X , i.e. the parameter space for line bundles Lon X with c1(L) = 0 (“topologically trivial” line bundles).

Example: the Albanese variety of a Riemann surface C is itsJacobian, and the Albanese map is the famous Abel-Jacobiembedding C → J(C ).

Pic0(C ) = space of line bundles on C of degree 0 (' J(C )).

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 7

Holomorphic one-forms and geometry

Dual torus is

Pic0(X ) := A = H1,0(X )/H1(X ,Z),

the Picard torus of X , i.e. the parameter space for line bundles Lon X with c1(L) = 0 (“topologically trivial” line bundles).

Example: the Albanese variety of a Riemann surface C is itsJacobian, and the Albanese map is the famous Abel-Jacobiembedding C → J(C ).

Pic0(C ) = space of line bundles on C of degree 0 (' J(C )).

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 7

Interest in studying one-forms

Why currently interesting?

• The invariant h1,0(X ) is crucial for classifying projectivemanifolds, or for bounding other numerical invariants. (Classicallyunderstood when dimX ≤ 2; but only recently in dimension ≥ 3.)

• Zeros of one-forms closely linked to the birational geometry of X .

A bit of terminology:

• ωX = ∧dimXΩ1X = canonical line bundle of X = bundle of top

forms, locally of type ω = f · dz1 ∧ . . . ∧ dzn.

• Pm(X ) = dimC Γ(X , ω⊗mX ) = m-th plurigenus of X .

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 8

Interest in studying one-forms

Why currently interesting?

• The invariant h1,0(X ) is crucial for classifying projectivemanifolds, or for bounding other numerical invariants. (Classicallyunderstood when dimX ≤ 2; but only recently in dimension ≥ 3.)

• Zeros of one-forms closely linked to the birational geometry of X .

A bit of terminology:

• ωX = ∧dimXΩ1X = canonical line bundle of X = bundle of top

forms, locally of type ω = f · dz1 ∧ . . . ∧ dzn.

• Pm(X ) = dimC Γ(X , ω⊗mX ) = m-th plurigenus of X .

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 8

Numerical applications: characterization of tori

Example of classification result:

T = torus =⇒ h1,0(T ) = dimT .

Also, ωT ' T × C trivial bundle =⇒ Pm(T ) = 1, ∀ m ≥ 1.

These are bimeromorphic invariants. Conversely:

Theorem (Chen-Hacon, ’01; conjecture of Kollar)

If X is a projective manifold with P1(X ) = P2(X ) = 1 andh1,0(X ) = dimX , then X is birational to an abelian variety.

• Pareschi – P. – Schnell, ’15: Same result when X is only compactKahler (so X bimeromorphic to a compact complex torus).

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 9

Numerical applications: characterization of tori

Example of classification result:

T = torus =⇒ h1,0(T ) = dimT .

Also, ωT ' T × C trivial bundle =⇒ Pm(T ) = 1, ∀ m ≥ 1.

These are bimeromorphic invariants. Conversely:

Theorem (Chen-Hacon, ’01; conjecture of Kollar)

If X is a projective manifold with P1(X ) = P2(X ) = 1 andh1,0(X ) = dimX , then X is birational to an abelian variety.

• Pareschi – P. – Schnell, ’15: Same result when X is only compactKahler (so X bimeromorphic to a compact complex torus).

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 9

Numerical applications: characterization of tori

Example of classification result:

T = torus =⇒ h1,0(T ) = dimT .

Also, ωT ' T × C trivial bundle =⇒ Pm(T ) = 1, ∀ m ≥ 1.

These are bimeromorphic invariants. Conversely:

Theorem (Chen-Hacon, ’01; conjecture of Kollar)

If X is a projective manifold with P1(X ) = P2(X ) = 1 andh1,0(X ) = dimX , then X is birational to an abelian variety.

• Pareschi – P. – Schnell, ’15: Same result when X is only compactKahler (so X bimeromorphic to a compact complex torus).

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 9

Numerical applications: cup-product action

Examples of bounding invariants:

First, a little detour:

• ΩpX = ∧pΩ1

X = vector bundle of holomorphic p-forms.

• Hp,0(X ) = Γ(X ,ΩpX ) = global p-forms; dimension hp,0(X ).

• Any number of 1-forms acts on p-forms by cup-product:

q∧H1,0(X )⊗ Hp,0(X )→ Hp+q,0(X )

(ω1 ∧ · · · ∧ ωq, η) 7→ ω1 ∧ · · · ∧ ωq ∧ η.

Consider E :=∧•H1,0(X ) = exterior algebra in H1,0(X ).

QX :=⊕n

p=0Hp,0(X ) = the holomorphic cohomology algebra.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 10

Numerical applications: cup-product action

Examples of bounding invariants:

First, a little detour:

• ΩpX = ∧pΩ1

X = vector bundle of holomorphic p-forms.

• Hp,0(X ) = Γ(X ,ΩpX ) = global p-forms; dimension hp,0(X ).

• Any number of 1-forms acts on p-forms by cup-product:

q∧H1,0(X )⊗ Hp,0(X )→ Hp+q,0(X )

(ω1 ∧ · · · ∧ ωq, η) 7→ ω1 ∧ · · · ∧ ωq ∧ η.

Consider E :=∧•H1,0(X ) = exterior algebra in H1,0(X ).

QX :=⊕n

p=0Hp,0(X ) = the holomorphic cohomology algebra.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 10

Numerical applications: cup-product action

Examples of bounding invariants:

First, a little detour:

• ΩpX = ∧pΩ1

X = vector bundle of holomorphic p-forms.

• Hp,0(X ) = Γ(X ,ΩpX ) = global p-forms; dimension hp,0(X ).

• Any number of 1-forms acts on p-forms by cup-product:

q∧H1,0(X )⊗ Hp,0(X )→ Hp+q,0(X )

(ω1 ∧ · · · ∧ ωq, η) 7→ ω1 ∧ · · · ∧ ωq ∧ η.

Consider E :=∧•H1,0(X ) = exterior algebra in H1,0(X ).

QX :=⊕n

p=0Hp,0(X ) = the holomorphic cohomology algebra.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 10

Numerical applications: cup-product action

Examples of bounding invariants:

First, a little detour:

• ΩpX = ∧pΩ1

X = vector bundle of holomorphic p-forms.

• Hp,0(X ) = Γ(X ,ΩpX ) = global p-forms; dimension hp,0(X ).

• Any number of 1-forms acts on p-forms by cup-product:

q∧H1,0(X )⊗ Hp,0(X )→ Hp+q,0(X )

(ω1 ∧ · · · ∧ ωq, η) 7→ ω1 ∧ · · · ∧ ωq ∧ η.

Consider E :=∧•H1,0(X ) = exterior algebra in H1,0(X ).

QX :=⊕n

p=0Hp,0(X ) = the holomorphic cohomology algebra.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 10

Numerical applications: cup-product action

Examples of bounding invariants:

First, a little detour:

• ΩpX = ∧pΩ1

X = vector bundle of holomorphic p-forms.

• Hp,0(X ) = Γ(X ,ΩpX ) = global p-forms; dimension hp,0(X ).

• Any number of 1-forms acts on p-forms by cup-product:

q∧H1,0(X )⊗ Hp,0(X )→ Hp+q,0(X )

(ω1 ∧ · · · ∧ ωq, η) 7→ ω1 ∧ · · · ∧ ωq ∧ η.

Consider E :=∧•H1,0(X ) = exterior algebra in H1,0(X ).

QX :=⊕n

p=0Hp,0(X ) = the holomorphic cohomology algebra.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 10

Numerical applications: regularity

Rephrasing: QX is a graded module over E via cup-product.

Theorem (Lazarsfeld – P., ’10)

The Albanese map a : X → Alb(X ) has general fiber of dimensionk ⇐⇒ QX has Castelnuovo-Mumford regularity k over E .

Regularity = measure of the complexity of generators and relations.

Says that QX =⊕n

p=0Hp,0(X ) is generated in degrees at most

0, . . . , k , and the relations between the generators are constrained.

Input: Hodge theory.

Output: allows for applying commutative and homological algebramachinery (e.g. minimal free resolutions, Syzygy Theorem, BGGcorrespondence) to obtain new geometric information.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 11

Numerical applications: regularity

Rephrasing: QX is a graded module over E via cup-product.

Theorem (Lazarsfeld – P., ’10)

The Albanese map a : X → Alb(X ) has general fiber of dimensionk ⇐⇒ QX has Castelnuovo-Mumford regularity k over E .

Regularity = measure of the complexity of generators and relations.

Says that QX =⊕n

p=0Hp,0(X ) is generated in degrees at most

0, . . . , k , and the relations between the generators are constrained.

Input: Hodge theory.

Output: allows for applying commutative and homological algebramachinery (e.g. minimal free resolutions, Syzygy Theorem, BGGcorrespondence) to obtain new geometric information.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 11

Numerical applications: regularity

Rephrasing: QX is a graded module over E via cup-product.

Theorem (Lazarsfeld – P., ’10)

The Albanese map a : X → Alb(X ) has general fiber of dimensionk ⇐⇒ QX has Castelnuovo-Mumford regularity k over E .

Regularity = measure of the complexity of generators and relations.

Says that QX =⊕n

p=0Hp,0(X ) is generated in degrees at most

0, . . . , k , and the relations between the generators are constrained.

Input: Hodge theory.

Output: allows for applying commutative and homological algebramachinery (e.g. minimal free resolutions, Syzygy Theorem, BGGcorrespondence) to obtain new geometric information.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 11

Numerical applications: regularity

Rephrasing: QX is a graded module over E via cup-product.

Theorem (Lazarsfeld – P., ’10)

The Albanese map a : X → Alb(X ) has general fiber of dimensionk ⇐⇒ QX has Castelnuovo-Mumford regularity k over E .

Regularity = measure of the complexity of generators and relations.

Says that QX =⊕n

p=0Hp,0(X ) is generated in degrees at most

0, . . . , k , and the relations between the generators are constrained.

Input: Hodge theory.

Output: allows for applying commutative and homological algebramachinery (e.g. minimal free resolutions, Syzygy Theorem, BGGcorrespondence) to obtain new geometric information.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 11

Numerical applications: inequalities

Main application: Most interesting numerical invariants can bebounded below in terms of h1,0(X ). Besides hp,0(X ), recall:

• χ(X ) =∑n

p=0(−1)php,0(X ) = holomorphic Euler characteristic.

Assumption: X does not admit “irregular fibrations”, i.e. roughlymappings f : X → Y with 0 < dimY < dimX and h1,0(Y ) 6= 0.

Theorem (Pareschi – P., ’09)

χ(X ) ≥ h1,0(X )− dimX .

When X is a surface, this is the celebrated Castelnuovo-deFranchis inequality from early 1900’s.

Theorem (Lazarsfeld – P., ’10)

hp,0(X ) ≥ function(h1,0(X )

).

If fibrations do exist, other semipositivity techniques of Kawamata,Kollar, Viehweg, ..., apply as well; different story.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 12

Numerical applications: inequalities

Main application: Most interesting numerical invariants can bebounded below in terms of h1,0(X ). Besides hp,0(X ), recall:

• χ(X ) =∑n

p=0(−1)php,0(X ) = holomorphic Euler characteristic.

Assumption: X does not admit “irregular fibrations”, i.e. roughlymappings f : X → Y with 0 < dimY < dimX and h1,0(Y ) 6= 0.

Theorem (Pareschi – P., ’09)

χ(X ) ≥ h1,0(X )− dimX .

When X is a surface, this is the celebrated Castelnuovo-deFranchis inequality from early 1900’s.

Theorem (Lazarsfeld – P., ’10)

hp,0(X ) ≥ function(h1,0(X )

).

If fibrations do exist, other semipositivity techniques of Kawamata,Kollar, Viehweg, ..., apply as well; different story.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 12

Numerical applications: inequalities

Main application: Most interesting numerical invariants can bebounded below in terms of h1,0(X ). Besides hp,0(X ), recall:

• χ(X ) =∑n

p=0(−1)php,0(X ) = holomorphic Euler characteristic.

Assumption: X does not admit “irregular fibrations”, i.e. roughlymappings f : X → Y with 0 < dimY < dimX and h1,0(Y ) 6= 0.

Theorem (Pareschi – P., ’09)

χ(X ) ≥ h1,0(X )− dimX .

When X is a surface, this is the celebrated Castelnuovo-deFranchis inequality from early 1900’s.

Theorem (Lazarsfeld – P., ’10)

hp,0(X ) ≥ function(h1,0(X )

).

If fibrations do exist, other semipositivity techniques of Kawamata,Kollar, Viehweg, ..., apply as well; different story.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 12

Numerical applications: inequalities

Main application: Most interesting numerical invariants can bebounded below in terms of h1,0(X ). Besides hp,0(X ), recall:

• χ(X ) =∑n

p=0(−1)php,0(X ) = holomorphic Euler characteristic.

Assumption: X does not admit “irregular fibrations”, i.e. roughlymappings f : X → Y with 0 < dimY < dimX and h1,0(Y ) 6= 0.

Theorem (Pareschi – P., ’09)

χ(X ) ≥ h1,0(X )− dimX .

When X is a surface, this is the celebrated Castelnuovo-deFranchis inequality from early 1900’s.

Theorem (Lazarsfeld – P., ’10)

hp,0(X ) ≥ function(h1,0(X )

).

If fibrations do exist, other semipositivity techniques of Kawamata,Kollar, Viehweg, ..., apply as well; different story.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 12

Numerical applications: inequalities

Main application: Most interesting numerical invariants can bebounded below in terms of h1,0(X ). Besides hp,0(X ), recall:

• χ(X ) =∑n

p=0(−1)php,0(X ) = holomorphic Euler characteristic.

Assumption: X does not admit “irregular fibrations”, i.e. roughlymappings f : X → Y with 0 < dimY < dimX and h1,0(Y ) 6= 0.

Theorem (Pareschi – P., ’09)

χ(X ) ≥ h1,0(X )− dimX .

When X is a surface, this is the celebrated Castelnuovo-deFranchis inequality from early 1900’s.

Theorem (Lazarsfeld – P., ’10)

hp,0(X ) ≥ function(h1,0(X )

).

If fibrations do exist, other semipositivity techniques of Kawamata,Kollar, Viehweg, ..., apply as well; different story.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 12

Zeros of holomorphic one-forms

Different direction, and main focus here: existence of zeros ofone-forms.

Main result. Conjecture of Hacon-Kovacs and Luo-Zhang (’05),partially due to Carrell as well:

Theorem (P. – Schnell, ’13)

If X is a projective manifold of general type, then everyholomorphic one-form on X vanishes at some point.

Example: X = C curve of genus g is of general type ⇐⇒ g ≥ 2⇐⇒ 2g − 2 > 0. Each non-zero one-form has 2g − 2 zeros.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 13

Zeros of holomorphic one-forms

Different direction, and main focus here: existence of zeros ofone-forms.

Main result. Conjecture of Hacon-Kovacs and Luo-Zhang (’05),partially due to Carrell as well:

Theorem (P. – Schnell, ’13)

If X is a projective manifold of general type, then everyholomorphic one-form on X vanishes at some point.

Example: X = C curve of genus g is of general type ⇐⇒ g ≥ 2⇐⇒ 2g − 2 > 0. Each non-zero one-form has 2g − 2 zeros.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 13

Zeros of holomorphic one-forms

Different direction, and main focus here: existence of zeros ofone-forms.

Main result. Conjecture of Hacon-Kovacs and Luo-Zhang (’05),partially due to Carrell as well:

Theorem (P. – Schnell, ’13)

If X is a projective manifold of general type, then everyholomorphic one-form on X vanishes at some point.

Example: X = C curve of genus g is of general type ⇐⇒ g ≥ 2⇐⇒ 2g − 2 > 0. Each non-zero one-form has 2g − 2 zeros.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 13

Varieties of general type

Examples of varieties of general type:

• X ⊂ Pn hypersurface of degree d is of general type ⇐⇒d ≥ n + 2.

• “Most” subvarieties of abelian varieties, and their covers.

• Varieties with ωX ample, i.e. c1(X ) < 0. Equivalently (Yau’stheorem), TX has a metric of constant negative Ricci curvature.

Varieties not of general type:

We understand them reasonably well, either as having nopluricanonical forms (like Pn), or as Calabi-Yau-type (ωX ≡ 0, e.g.tori, K3 surfaces), or as being fibered in such over lowerdimensional varieties. Prototype: elliptic surfaces f : S −→ Cfibered in elliptic curves over a curve C .

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 14

Varieties of general type

Examples of varieties of general type:

• X ⊂ Pn hypersurface of degree d is of general type ⇐⇒d ≥ n + 2.

• “Most” subvarieties of abelian varieties, and their covers.

• Varieties with ωX ample, i.e. c1(X ) < 0. Equivalently (Yau’stheorem), TX has a metric of constant negative Ricci curvature.

Varieties not of general type:

We understand them reasonably well, either as having nopluricanonical forms (like Pn), or as Calabi-Yau-type (ωX ≡ 0, e.g.tori, K3 surfaces), or as being fibered in such over lowerdimensional varieties. Prototype: elliptic surfaces f : S −→ Cfibered in elliptic curves over a curve C .

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 14

Zeros of holomorphic one-forms

Different direction, and main focus here: existence of zeros ofone-forms.

Main result. Conjecture of Hacon-Kovacs and Luo-Zhang (’05),partially due to Carrell as well:

Theorem (P. – Schnell, ’13)

If X is a smooth projective variety of general type, then everyholomorphic one-form on X vanishes at some point.

Typical application: I said that there are no submersions from avariety of general type to a torus. Reason:

All non-trivial forms on a torus are nowhere vanishing. But asubmersion f : X → T would then give nowhere vanishing forms:

0 6= f ∗H1,0(T ) ⊆ H1,0(X ).

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 15

Zeros of holomorphic one-forms

Different direction, and main focus here: existence of zeros ofone-forms.

Main result. Conjecture of Hacon-Kovacs and Luo-Zhang (’05),partially due to Carrell as well:

Theorem (P. – Schnell, ’13)

If X is a smooth projective variety of general type, then everyholomorphic one-form on X vanishes at some point.

Typical application: I said that there are no submersions from avariety of general type to a torus. Reason:

All non-trivial forms on a torus are nowhere vanishing. But asubmersion f : X → T would then give nowhere vanishing forms:

0 6= f ∗H1,0(T ) ⊆ H1,0(X ).

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 15

Techniques: vanishing theorems

How does one attack the results above? We’ve seen the prevalenceof homological algebra.

Another common theme: intimate relationship between 1-formsand vanishing theorems for cohomology groups of line bundles.

Most famous vanishing theorem relies on positivity: a line bundle Lon X is called positive (or ample) if∫

Vc1(L)dimV > 0, ∀ subvariety V ⊆ X .

Equivalently L has a hermitian metric with positive curvature form.

Kodaira-Nakano vanishing: If X is a projective manifold, and L isan ample line bundle on X , then

H i (X ,ΩjX ⊗ L) = 0, ∀ i + j > dimX .

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 16

Techniques: vanishing theorems

How does one attack the results above? We’ve seen the prevalenceof homological algebra.

Another common theme: intimate relationship between 1-formsand vanishing theorems for cohomology groups of line bundles.

Most famous vanishing theorem relies on positivity: a line bundle Lon X is called positive (or ample) if∫

Vc1(L)dimV > 0, ∀ subvariety V ⊆ X .

Equivalently L has a hermitian metric with positive curvature form.

Kodaira-Nakano vanishing: If X is a projective manifold, and L isan ample line bundle on X , then

H i (X ,ΩjX ⊗ L) = 0, ∀ i + j > dimX .

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 16

Techniques: vanishing theorems

How does one attack the results above? We’ve seen the prevalenceof homological algebra.

Another common theme: intimate relationship between 1-formsand vanishing theorems for cohomology groups of line bundles.

Most famous vanishing theorem relies on positivity: a line bundle Lon X is called positive (or ample) if∫

Vc1(L)dimV > 0, ∀ subvariety V ⊆ X .

Equivalently L has a hermitian metric with positive curvature form.

Kodaira-Nakano vanishing: If X is a projective manifold, and L isan ample line bundle on X , then

H i (X ,ΩjX ⊗ L) = 0, ∀ i + j > dimX .

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 16

Techniques: vanishing theorems

How does one attack the results above? We’ve seen the prevalenceof homological algebra.

Another common theme: intimate relationship between 1-formsand vanishing theorems for cohomology groups of line bundles.

Most famous vanishing theorem relies on positivity: a line bundle Lon X is called positive (or ample) if∫

Vc1(L)dimV > 0, ∀ subvariety V ⊆ X .

Equivalently L has a hermitian metric with positive curvature form.

Kodaira-Nakano vanishing: If X is a projective manifold, and L isan ample line bundle on X , then

H i (X ,ΩjX ⊗ L) = 0, ∀ i + j > dimX .

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 16

Koszul complex and vanishing

How do one-forms and vanishing theorems come together?Example:

For ω ∈ H1,0(X ), d = · ∧ ω gives a Koszul complex:

K• : 0 −→ OX∧ω−→ Ω1

X∧ω−→ Ω2

X∧ω−→ · · · ∧ω−→ Ωn

X −→ 0.

Commutative algebra: Z (ω) = ∅ =⇒ K• exact. Assume this.

Twist with ωX , and pass to cohomology; relevant groups are:

H i (X ,ΩjX ⊗ ωX ).

Assume now ωX ampleNakano=⇒ H i (X ,Ωj

X ⊗ ωX ) = 0 for i + j > n.

Chasing diagram gives 0 = Hn(X , ωX ) = dual of space ofholomorphic functions on X (= constants), contradiction.

So no nowhere vanishing one-forms if ωX ample!

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 17

Koszul complex and vanishing

How do one-forms and vanishing theorems come together?Example:

For ω ∈ H1,0(X ), d = · ∧ ω gives a Koszul complex:

K• : 0 −→ OX∧ω−→ Ω1

X∧ω−→ Ω2

X∧ω−→ · · · ∧ω−→ Ωn

X −→ 0.

Commutative algebra: Z (ω) = ∅ =⇒ K• exact. Assume this.

Twist with ωX , and pass to cohomology; relevant groups are:

H i (X ,ΩjX ⊗ ωX ).

Assume now ωX ampleNakano=⇒ H i (X ,Ωj

X ⊗ ωX ) = 0 for i + j > n.

Chasing diagram gives 0 = Hn(X , ωX ) = dual of space ofholomorphic functions on X (= constants), contradiction.

So no nowhere vanishing one-forms if ωX ample!

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 17

Koszul complex and vanishing

How do one-forms and vanishing theorems come together?Example:

For ω ∈ H1,0(X ), d = · ∧ ω gives a Koszul complex:

K• : 0 −→ OX∧ω−→ Ω1

X∧ω−→ Ω2

X∧ω−→ · · · ∧ω−→ Ωn

X −→ 0.

Commutative algebra: Z (ω) = ∅ =⇒ K• exact. Assume this.

Twist with ωX , and pass to cohomology; relevant groups are:

H i (X ,ΩjX ⊗ ωX ).

Assume now ωX ampleNakano=⇒ H i (X ,Ωj

X ⊗ ωX ) = 0 for i + j > n.

Chasing diagram gives 0 = Hn(X , ωX ) = dual of space ofholomorphic functions on X (= constants), contradiction.

So no nowhere vanishing one-forms if ωX ample!

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 17

Koszul complex and vanishing

How do one-forms and vanishing theorems come together?Example:

For ω ∈ H1,0(X ), d = · ∧ ω gives a Koszul complex:

K• : 0 −→ OX∧ω−→ Ω1

X∧ω−→ Ω2

X∧ω−→ · · · ∧ω−→ Ωn

X −→ 0.

Commutative algebra: Z (ω) = ∅ =⇒ K• exact. Assume this.

Twist with ωX , and pass to cohomology; relevant groups are:

H i (X ,ΩjX ⊗ ωX ).

Assume now ωX ampleNakano=⇒ H i (X ,Ωj

X ⊗ ωX ) = 0 for i + j > n.

Chasing diagram gives 0 = Hn(X , ωX ) = dual of space ofholomorphic functions on X (= constants), contradiction.

So no nowhere vanishing one-forms if ωX ample!

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 17

Generic Vanishing Theorems

General case much more complicated, but still relying on vanishing.Fundamental tool: generic vanishing theorems.

Another special example:

Theorem (Green-Lazarsfeld, ’87)

If X has a nowhere vanishing holomorphic one-form, then

H i (X , ωX ⊗ L) = 0, ∀ i ≥ 0, ∀ L ∈ Pic0(X ) general.

Corollary. X has a nowhere vanishing one-form =⇒ χ(X ) = 0.

Corollary. Surfaces of general type have no nowhere vanishingone-forms.

Classical theorem of Castelnuovo: S is a surface of general type=⇒ χ(S) > 0.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 18

Generic Vanishing Theorems

General case much more complicated, but still relying on vanishing.Fundamental tool: generic vanishing theorems.

Another special example:

Theorem (Green-Lazarsfeld, ’87)

If X has a nowhere vanishing holomorphic one-form, then

H i (X , ωX ⊗ L) = 0, ∀ i ≥ 0, ∀ L ∈ Pic0(X ) general.

Corollary. X has a nowhere vanishing one-form =⇒ χ(X ) = 0.

Corollary. Surfaces of general type have no nowhere vanishingone-forms.

Classical theorem of Castelnuovo: S is a surface of general type=⇒ χ(S) > 0.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 18

Generic Vanishing Theorems

General case much more complicated, but still relying on vanishing.Fundamental tool: generic vanishing theorems.

Another special example:

Theorem (Green-Lazarsfeld, ’87)

If X has a nowhere vanishing holomorphic one-form, then

H i (X , ωX ⊗ L) = 0, ∀ i ≥ 0, ∀ L ∈ Pic0(X ) general.

Corollary. X has a nowhere vanishing one-form =⇒ χ(X ) = 0.

Corollary. Surfaces of general type have no nowhere vanishingone-forms.

Classical theorem of Castelnuovo: S is a surface of general type=⇒ χ(S) > 0.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 18

Generic Vanishing Theorems

General case much more complicated, but still relying on vanishing.Fundamental tool: generic vanishing theorems.

Another special example:

Theorem (Green-Lazarsfeld, ’87)

If X has a nowhere vanishing holomorphic one-form, then

H i (X , ωX ⊗ L) = 0, ∀ i ≥ 0, ∀ L ∈ Pic0(X ) general.

Corollary. X has a nowhere vanishing one-form =⇒ χ(X ) = 0.

Corollary. Surfaces of general type have no nowhere vanishingone-forms.

Classical theorem of Castelnuovo: S is a surface of general type=⇒ χ(S) > 0.

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 18

Ideas in higher dimension

In dimension ≥ 3, no simple numerical obstruction to being ofgeneral type! Instead, we use more sophisticated generic vanishingstatements, based on two modern developments:

• Derived category approach to generic vanishing (Hacon, ’03).

• Extension to mixed Hodge modules (P. – Schnell, ’11).

Key concepts that are used:

• Derived categories of coherent sheaves, Fourier-Mukai transform.

• Variations of Hodge structure, Hodge filtration.

• Filtered regular holonomic D-modules, mixed Hodge modules.

• Decomposition Theorem.

Q: Powerful tools, but is there a more geometric approach?

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 19

Ideas in higher dimension

In dimension ≥ 3, no simple numerical obstruction to being ofgeneral type! Instead, we use more sophisticated generic vanishingstatements, based on two modern developments:

• Derived category approach to generic vanishing (Hacon, ’03).

• Extension to mixed Hodge modules (P. – Schnell, ’11).

Key concepts that are used:

• Derived categories of coherent sheaves, Fourier-Mukai transform.

• Variations of Hodge structure, Hodge filtration.

• Filtered regular holonomic D-modules, mixed Hodge modules.

• Decomposition Theorem.

Q: Powerful tools, but is there a more geometric approach?

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 19

Ideas in higher dimension

In dimension ≥ 3, no simple numerical obstruction to being ofgeneral type! Instead, we use more sophisticated generic vanishingstatements, based on two modern developments:

• Derived category approach to generic vanishing (Hacon, ’03).

• Extension to mixed Hodge modules (P. – Schnell, ’11).

Key concepts that are used:

• Derived categories of coherent sheaves, Fourier-Mukai transform.

• Variations of Hodge structure, Hodge filtration.

• Filtered regular holonomic D-modules, mixed Hodge modules.

• Decomposition Theorem.

Q: Powerful tools, but is there a more geometric approach?

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 19

Ideas in higher dimension

In dimension ≥ 3, no simple numerical obstruction to being ofgeneral type! Instead, we use more sophisticated generic vanishingstatements, based on two modern developments:

• Derived category approach to generic vanishing (Hacon, ’03).

• Extension to mixed Hodge modules (P. – Schnell, ’11).

Key concepts that are used:

• Derived categories of coherent sheaves, Fourier-Mukai transform.

• Variations of Hodge structure, Hodge filtration.

• Filtered regular holonomic D-modules, mixed Hodge modules.

• Decomposition Theorem.

Q: Powerful tools, but is there a more geometric approach?

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 19

THANK YOU!

Mihnea Popa (Northwestern) One-forms AMS Meeting, Lansing March 14, 2015 20