Vanishing theorems and holomorphic forms

Vanishing theorems and holomorphic forms Mihnea Popa Northwestern AMS Meeting, Lansing March 14, 2015

Holomorphic one-forms and geometry X compact complex manifold, dim C X = n.

Holomorphic one-forms and geometry X compact complex manifold, dim C X = n. T X = holomorphic tangent bundle; Ω 1 X = T X = cotangent bundle.

Holomorphic one-forms and geometry X compact complex manifold, dim C X = n. T X = holomorphic tangent bundle; Ω 1 X = T X = cotangent bundle. Locally, in coordinates z 1,..., z n, a holomorphic 1-form is ω = n f i dz i, i=1 f i holomorphic. Global holomorphic 1-forms on X (the main objects in this talk) are the holomorphic sections of Ω 1 X : H 1,0 (X ) := Γ(X, Ω 1 X ). h 1,0 (X ) := dim C H 1,0 (X ) (a Hodge number).

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

Examples Non-trivial one-forms may or may not exist. Examples: 1) H 1,0 (P n ) = 0.

Examples Non-trivial one-forms may or may not exist. Examples: 1) H 1,0 (P n ) = 0. 2) X hypersurface in P n, n 3 = H 1,0 (X ) = 0.

Examples Non-trivial one-forms may or may not exist. Examples: 1) H 1,0 (P n ) = 0. 2) X hypersurface in P n, n 3 = H 1,0 (X ) = 0. 3) T = V /Λ compact complex torus = H 1,0 (T ) V.

Examples Non-trivial one-forms may or may not exist. Examples: 1) H 1,0 (P n ) = 0. 2) X hypersurface in P n, n 3 = H 1,0 (X ) = 0. 3) T = V /Λ compact complex torus = H 1,0 (T ) V. 4) C compact Riemann surface of genus g = h 1,0 (C) = g. If X is projective, or just compact Kähler, Hodge decomposition gives H 1 (X, C) H 1,0 (X ) H 1,0 (X ). In particular h 1,0 (X ) = b 1 (X )/2.

Holomorphic one-forms and geometry How can we use them geometrically? Examples:

Holomorphic one-forms and geometry How can we use them geometrically? Examples: There exist no nontrivial maps f : P n T, where T is a torus. Proof: f H 1,0 (T ) H 1,0 (P n ) = 0.

Holomorphic one-forms and geometry How can we use them geometrically? Examples: There exist no nontrivial maps f : P n T, where T is a torus. Proof: f H 1,0 (T ) H 1,0 (P n ) = 0. Recent, and much more subtle: there exist no submersions f : 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.

Holomorphic one-forms and geometry The condition h 1,0 (X ) 0 influences the global geometry of X.

Holomorphic one-forms and geometry The condition h 1,0 (X ) 0 influences the global geometry of X. There is an inclusion of H 1 (X, Z) as a lattice in H 1,0 (X ) given by γ ( ) γ

Holomorphic one-forms and geometry The condition h 1,0 (X ) 0 influences the global geometry of X. There is an inclusion of H 1 (X, Z) as a lattice in H 1,0 (X ) given by γ ( ) γ The Albanese torus of X is A = Alb(X ) := H 1,0 (X ) /H 1 (X, Z) = compact complex torus of dimension h 1,0 (X ); abelian variety (i.e. projective torus) if X is projective.

Up to fixing x 0 X, also have the Albanese map: X Alb(X ), x x x 0 ( ). 1 x x 0 2 Integrals well defined up to periods = elements of the lattice H 1 (X, Z) H 1,0 (X ).

Holomorphic one-forms and geometry Dual torus is Pic 0 (X ) := Â = H1,0 (X )/H 1 (X, Z), the Picard torus of X, i.e. the parameter space for line bundles L on X with c 1 (L) = 0 ( topologically trivial line bundles). Example: the Albanese variety of a Riemann surface C is its Jacobian, and the Albanese map is the famous Abel-Jacobi embedding C J(C). Pic 0 (C) = space of line bundles on C of degree 0 ( J(C)).

Interest in studying one-forms Why currently interesting? The invariant h 1,0 (X ) is crucial for classifying projective manifolds, or for bounding other numerical invariants. (Classically understood when dim X 2; but only recently in dimension 3.) Zeros of one-forms closely linked to the birational geometry of X. A bit of terminology: ω X = dim X Ω 1 X = canonical line bundle of X = bundle of top forms, locally of type ω = f dz 1... dz n. P m (X ) = dim C Γ(X, ω m X ) = m-th plurigenus of X.

Numerical applications: characterization of tori Example of classification result: T = torus = h 1,0 (T ) = dim T. Also, ω T T C trivial bundle = P m (T ) = 1, m 1. These are bimeromorphic invariants. Conversely: Theorem (Chen-Hacon, 01; conjecture of Kollár) If X is a projective manifold with P 1 (X ) = P 2 (X ) = 1 and h 1,0 (X ) = dim X, then X is birational to an abelian variety.

Numerical applications: cup-product action Examples of bounding invariants: First, a little detour:

Numerical applications: cup-product action Examples of bounding invariants: First, a little detour: Ω p X = p Ω 1 X = vector bundle of holomorphic p-forms.

Numerical applications: cup-product action Examples of bounding invariants: First, a little detour: Ω p X = p Ω 1 X = vector bundle of holomorphic p-forms. H p,0 (X ) = Γ(X, Ω p X ) = global p-forms; dimension hp,0 (X ). Any number of 1-forms acts on p-forms by cup-product: q H 1,0 (X ) H p,0 (X ) H p+q,0 (X ) (ω 1 ω q, η) ω 1 ω q η.

Numerical applications: regularity Rephrasing: Q X is a graded module over E via cup-product.

Numerical applications: regularity Rephrasing: Q X is a graded module over E via cup-product. Theorem (Lazarsfeld P., 10) The Albanese map a : X Alb(X ) has general fiber of dimension k Q X has Castelnuovo-Mumford regularity k over E. Regularity = measure of the complexity of generators and relations. Says that Q X = n p=0 Hp,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 algebra machinery (e.g. minimal free resolutions, Syzygy Theorem, BGG correspondence) to obtain new geometric information.

Numerical applications: inequalities Main application: Most interesting numerical invariants can be bounded below in terms of h 1,0 (X ). Besides h p,0 (X ), recall: χ(x ) = n p=0 ( 1)p h p,0 (X ) = holomorphic Euler characteristic. Assumption: X does not admit irregular fibrations, i.e. roughly mappings f : X Y with 0 < dim Y < dim X and h 1,0 (Y ) 0. Theorem (Pareschi P., 09) χ(x ) h 1,0 (X ) dim X. When X is a surface, this is the celebrated Castelnuovo-de Franchis inequality from early 1900 s.

Zeros of holomorphic one-forms Different direction, and main focus here: existence of zeros of one-forms.

Zeros of holomorphic one-forms Different direction, and main focus here: existence of zeros of one-forms. Main result. Conjecture of Hacon-Kovács and Luo-Zhang ( 05), partially due to Carrell as well: Theorem (P. Schnell, 13) If X is a projective manifold of general type, then every holomorphic 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.

Varieties of general type Examples of varieties of general type: X P n hypersurface of degree d is of general type d n + 2. Most subvarieties of abelian varieties, and their covers. Varieties with ω X ample, i.e. c 1 (X ) < 0. Equivalently (Yau s theorem), T X has a metric of constant negative Ricci curvature. Varieties not of general type: We understand them reasonably well, either as having no pluricanonical forms (like P n ), or as Calabi-Yau-type (ω X 0, e.g. tori, K3 surfaces), or as being fibered in such over lower dimensional varieties. Prototype: elliptic surfaces f : S C fibered in elliptic curves over a curve C.

Zeros of holomorphic one-forms Different direction, and main focus here: existence of zeros of one-forms. Main result. Conjecture of Hacon-Kovács 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 every holomorphic one-form on X vanishes at some point. Typical application: I said that there are no submersions from a variety of general type to a torus. Reason: All non-trivial forms on a torus are nowhere vanishing. But a submersion f : X T would then give nowhere vanishing forms: 0 f H 1,0 (T ) H 1,0 (X ).

Techniques: vanishing theorems How does one attack the results above? We ve seen the prevalence of homological algebra.

Techniques: vanishing theorems How does one attack the results above? We ve seen the prevalence of homological algebra. Another common theme: intimate relationship between 1-forms and vanishing theorems for cohomology groups of line bundles. Most famous vanishing theorem relies on positivity: a line bundle L on X is called positive (or ample) if c 1 (L) dim V > 0, subvariety V X. V Equivalently L has a hermitian metric with positive curvature form.

Koszul complex and vanishing How do one-forms and vanishing theorems come together? Example: For ω H 1,0 (X ), d = ω gives a Koszul complex: K : 0 O X ω Ω 1 X ω Ω 2 X ω ω Ω n X 0.

Koszul complex and vanishing How do one-forms and vanishing theorems come together? Example: For ω H 1,0 (X ), d = ω gives a Koszul complex: K : 0 O X ω Ω 1 X ω Ω 2 X ω Commutative algebra: Z(ω) = = K exact. ω Ω n X 0. Assume this. Twist with ω X, and pass to cohomology; relevant groups are: H i (X, Ω j X ω X ). Assume now ω X ample Nakano = H i (X, Ω j X ω X ) = 0 for i + j > n. Chasing diagram gives 0 = H n (X, ω X ) = dual of space of holomorphic functions on X (= constants), contradiction.

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

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, ω X L) = 0, i 0, L Pic 0 (X ) general. Corollary. X has a nowhere vanishing one-form = χ(x ) = 0.

Ideas in higher dimension In dimension 3, no simple numerical obstruction to being of general type! Instead, we use more sophisticated generic vanishing statements, based on two modern developments:

Ideas in higher dimension In dimension 3, no simple numerical obstruction to being of general type! Instead, we use more sophisticated generic vanishing statements, 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.

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