Vanishing theorems and holomorphic forms

Transcription

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

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

3 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.

4 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.

5 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).

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

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

8 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.

9 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.

10 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.

11 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.

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

13 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.

14 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.

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

16 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

17 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.

18 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 ).

19 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).

20 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)).

21 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.

22 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.

23 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:

24 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.

25 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. Pareschi P. Schnell, 15: Same result when X is only compact Kähler (so X bimeromorphic to a compact complex torus).

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

27 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.

28 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 ).

29 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.

30 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. Consider E := H 1,0 (X ) = exterior algebra in H 1,0 (X ). Q X := n p=0 Hp,0 (X ) = the holomorphic cohomology algebra.

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

32 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.

33 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.

34 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.

35 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.

36 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.

37 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.

38 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. Theorem (Lazarsfeld P., 10) h p,0 (X ) function ( h 1,0 (X ) ).

39 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. Theorem (Lazarsfeld P., 10) h p,0 (X ) function ( h 1,0 (X ) ). If fibrations do exist, other semipositivity techniques of Kawamata, Kollár, Viehweg,..., apply as well; different story.

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

41 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.

42 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.

43 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.

44 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.

45 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.

46 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 ).

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

48 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.

49 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.

50 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. Kodaira-Nakano vanishing: If X is a projective manifold, and L is an ample line bundle on X, then H i (X, Ω j X L) = 0, i + j > dim X.

51 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.

52 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 ).

53 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.

54 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. So no nowhere vanishing one-forms if ω X ample!

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

56 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.

57 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.

58 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. Corollary. Surfaces of general type have no nowhere vanishing one-forms. Classical theorem of Castelnuovo: S is a surface of general type = χ(s) > 0.

59 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:

60 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).

61 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.

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

63 THANK YOU!