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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