# Vanishing theorems and holomorphic forms

Size: px
Start display at page:

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!

### FAKE PROJECTIVE SPACES AND FAKE TORI

FAKE PROJECTIVE SPACES AND FAKE TORI OLIVIER DEBARRE Abstract. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to P n is isomorphic to P n.

### DERIVED EQUIVALENCE AND NON-VANISHING LOCI

DERIVED EQUIVALENCE AND NON-VANISHING LOCI MIHNEA POPA To Joe Harris, with great admiration. 1. THE CONJECTURE AND ITS VARIANTS The purpose of this note is to propose and motivate a conjecture on the behavior

### HODGE MODULES ON COMPLEX TORI AND GENERIC VANISHING FOR COMPACT KÄHLER MANIFOLDS. A. Introduction

HODGE MODULES ON COMPLEX TORI AND GENERIC VANISHING FOR COMPACT KÄHLER MANIFOLDS GIUSEPPE PARESCHI, MIHNEA POPA, AND CHRISTIAN SCHNELL Abstract. We extend the results of generic vanishing theory to polarizable

### DERIVATIVE COMPLEX, BGG CORRESPONDENCE, AND NUMERICAL INEQUALITIES FOR COMPACT KÄHLER MANIFOLDS. Introduction

DERIVATIVE COMPLEX, BGG CORRESPONDENCE, AND NUMERICAL INEQUALITIES FOR COMPACT KÄHLER MANIFOLDS ROBERT LAZARSFELD AND MIHNEA POPA Introduction Given an irregular compact Kähler manifold X, one can form

### POSITIVITY FOR HODGE MODULES AND GEOMETRIC APPLICATIONS. Contents

POSITIVITY FOR HODGE MODULES AND GEOMETRIC APPLICATIONS MIHNEA POPA Abstract. This is a survey of vanishing and positivity theorems for Hodge modules, and their recent applications to birational and complex

### CANONICAL COHOMOLOGY AS AN EXTERIOR MODULE. To the memory of Eckart Viehweg INTRODUCTION

CANONICAL COHOMOLOGY AS AN EXTERIOR MODULE ROBERT LAZARSELD, MIHNEA POPA, AND CHRISTIAN SCHNELL To the memory of Eckart Viehweg INTRODUCTION Let X be a compact connected Kähler manifold of dimension d,

### arxiv: v1 [math.ag] 9 Mar 2011

INEQUALITIES FOR THE HODGE NUMBERS OF IRREGULAR COMPACT KÄHLER MANIFOLDS LUIGI LOMBARDI ariv:1103.1704v1 [math.ag] 9 Mar 2011 Abstract. Based on work of R. Lazarsfeld and M. Popa, we use the derivative

### The geometry of Landau-Ginzburg models

Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror

### Structure theorems for compact Kähler manifolds

Structure theorems for compact Kähler manifolds Jean-Pierre Demailly joint work with Frédéric Campana & Thomas Peternell Institut Fourier, Université de Grenoble I, France & Académie des Sciences de Paris

### ON VARIETIES OF MAXIMAL ALBANESE DIMENSION

ON VARIETIES OF MAXIMAL ALBANESE DIMENSION ZHI JIANG A smooth projective complex variety X has maximal Albanese dimension if its Albanese map X Alb(X) is generically finite onto its image. These varieties

### The geography of irregular surfaces

Università di Pisa Classical Algebraic Geometry today M.S.R.I., 1/26 1/30 2008 Summary Surfaces of general type 1 Surfaces of general type 2 3 and irrational pencils Surface = smooth projective complex

### arxiv: v3 [math.ag] 23 Jul 2012

DERIVED INVARIANTS OF IRREGULAR VARIETIES AND HOCHSCHILD HOMOLOG LUIGI LOMBARDI arxiv:1204.1332v3 [math.ag] 23 Jul 2012 Abstract. We study the behavior of cohomological support loci of the canonical bundle

### Contributors. Preface

Contents Contributors Preface v xv 1 Kähler Manifolds by E. Cattani 1 1.1 Complex Manifolds........................... 2 1.1.1 Definition and Examples.................... 2 1.1.2 Holomorphic Vector Bundles..................

### Linear systems and Fano varieties: introduction

Linear systems and Fano varieties: introduction Caucher Birkar New advances in Fano manifolds, Cambridge, December 2017 References: [B-1] Anti-pluricanonical systems on Fano varieties. [B-2] Singularities

### GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES MIHNEA POPA AND CHRISTIAN SCHNELL Abstract. We extend the dimension and strong linearity results of generic vanishing theory to bundles of holomorphic forms

### Basic results on irregular varieties via Fourier Mukai methods

Current Developments in Algebraic Geometry MSRI Publications Volume 59, 2011 Basic results on irregular varieties via Fourier Mukai methods GIUSEPPE PARESCHI Recently Fourier Mukai methods have proved

### Hermitian vs. Riemannian Geometry

Hermitian vs. Riemannian Geometry Gabe Khan 1 1 Department of Mathematics The Ohio State University GSCAGT, May 2016 Outline of the talk Complex curves Background definitions What happens if the metric

### Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

### The Strominger Yau Zaslow conjecture

The Strominger Yau Zaslow conjecture Paul Hacking 10/16/09 1 Background 1.1 Kähler metrics Let X be a complex manifold of dimension n, and M the underlying smooth manifold with (integrable) almost complex

### K-stability and Kähler metrics, I

K-stability and Kähler metrics, I Gang Tian Beijing University and Princeton University Let M be a Kähler manifold. This means that M be a complex manifold together with a Kähler metric ω. In local coordinates

### GV-SHEAVES, FOURIER-MUKAI TRANSFORM, AND GENERIC VANISHING

GV-SHEAVES, FOURIER-MUKAI TRANSFORM, AND GENERIC VANISHING GIUSEPPE PARESCHI AND MIHNEA POPA Contents 1. Introduction 1 2. Fourier-Mukai preliminaries 5 3. GV-objects 7 4. Examples of GV -objects 12 5.

### A NEW FAMILY OF SYMPLECTIC FOURFOLDS

A NEW FAMILY OF SYMPLECTIC FOURFOLDS OLIVIER DEBARRE This is joint work with Claire Voisin. 1. Irreducible symplectic varieties It follows from work of Beauville and Bogomolov that any smooth complex compact

### Classifying complex surfaces and symplectic 4-manifolds

Classifying complex surfaces and symplectic 4-manifolds UT Austin, September 18, 2012 First Cut Seminar Basics Symplectic 4-manifolds Definition A symplectic 4-manifold (X, ω) is an oriented, smooth, 4-dimensional

### The structure of algebraic varieties

The structure of algebraic varieties János Kollár Princeton University ICM, August, 2014, Seoul with the assistance of Jennifer M. Johnson and Sándor J. Kovács (Written comments added for clarity that

### Homological mirror symmetry via families of Lagrangians

Homological mirror symmetry via families of Lagrangians String-Math 2018 Mohammed Abouzaid Columbia University June 17, 2018 Mirror symmetry Three facets of mirror symmetry: 1 Enumerative: GW invariants

### Hyperkähler geometry lecture 3

Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843

### Università degli Studi di Padova

Università degli Studi di Padova DIPARTIMENTO DI MATEMATICA "TULLIO LEVI-CIVITA" CORSO DI LAUREA MAGISTRALE IN MATEMATICA Generic Vanishing in Geometria Analitica e Algebrica Relatore: Prof. Mistretta

### CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents

CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS Contents 1. The Lefschetz hyperplane theorem 1 2. The Hodge decomposition 4 3. Hodge numbers in smooth families 6 4. Birationally

### Geometry of moduli spaces

Geometry of moduli spaces 20. November 2009 1 / 45 (1) Examples: C: compact Riemann surface C = P 1 (C) = C { } (Riemann sphere) E = C / Z + Zτ (torus, elliptic curve) 2 / 45 (2) Theorem (Riemann existence

### Introduction to algebraic surfaces Lecture Notes for the course at the University of Mainz

Introduction to algebraic surfaces Lecture Notes for the course at the University of Mainz Wintersemester 2009/2010 Arvid Perego (preliminary draft) October 30, 2009 2 Contents Introduction 5 1 Background

### RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY

RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY NICK MCCLEEREY 0. Complex Differential Forms Consider a complex manifold X n (of complex dimension n) 1, and consider its complexified tangent bundle T C

### Rational Curves On K3 Surfaces

Rational Curves On K3 Surfaces Jun Li Department of Mathematics Stanford University Conference in honor of Peter Li Overview of the talk The problem: existence of rational curves on a K3 surface The conjecture:

### Algebraic Curves and Riemann Surfaces

Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex

### RESEARCH STATEMENT SOFIA TIRABASSI

RESEARCH STATEMENT SOFIA TIRABASSI I have very broad research interests, all of them focusing in the areas of algebraic geometry. More precisely, I use cohomological methods and tools coming from homological

### ON QUASI-ALBANESE MAPS

ON QUASI-ALBANESE MAPS OSAMU FUJINO Abstract. We discuss Iitaka s theory of quasi-albanese maps in details. We also give a detailed proof of Kawamata s theorem on the quasi-albanese maps for varieties

### Chern numbers and Hilbert Modular Varieties

Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point

### Intermediate Jacobians and Abel-Jacobi Maps

Intermediate Jacobians and Abel-Jacobi Maps Patrick Walls April 28, 2012 Introduction Let X be a smooth projective complex variety. Introduction Let X be a smooth projective complex variety. Intermediate

### Ω Ω /ω. To these, one wants to add a fourth condition that arises from physics, what is known as the anomaly cancellation, namely that

String theory and balanced metrics One of the main motivations for considering balanced metrics, in addition to the considerations already mentioned, has to do with the theory of what are known as heterotic

### arxiv: v1 [math.ag] 13 Mar 2019

THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show

### Cohomology jump loci of quasi-projective varieties

Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)

### arxiv: v1 [math.ag] 10 Jun 2016

THE EVENTUAL PARACANONICAL MAP OF A VARIETY OF MAXIMAL ALBANESE DIMENSION arxiv:1606.03301v1 [math.ag] 10 Jun 2016 MIGUEL ÁNGEL BARJA, RITA PARDINI AND LIDIA STOPPINO Abstract. Let X be a smooth complex

### LECTURES ON SINGULARITIES AND ADJOINT LINEAR SYSTEMS

LECTURES ON SINGULARITIES AND ADJOINT LINEAR SYSTEMS LAWRENCE EIN Abstract. 1. Singularities of Surfaces Let (X, o) be an isolated normal surfaces singularity. The basic philosophy is to replace the singularity

### Diagonal Subschemes and Vector Bundles

Pure and Applied Mathematics Quarterly Volume 4, Number 4 (Special Issue: In honor of Jean-Pierre Serre, Part 1 of 2 ) 1233 1278, 2008 Diagonal Subschemes and Vector Bundles Piotr Pragacz, Vasudevan Srinivas

### Chow Groups. Murre. June 28, 2010

Chow Groups Murre June 28, 2010 1 Murre 1 - Chow Groups Conventions: k is an algebraically closed field, X, Y,... are varieties over k, which are projetive (at worst, quasi-projective), irreducible and

### Hodge Theory of Maps

Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar

### GENERIC VANISHING AND THE GEOMETRY OF IRREGULAR VARIETIES IN POSITIVE CHARACTERISTIC

GENERIC VANISHING AND THE GEOMETRY OF IRREGULAR VARIETIES IN POSITIVE CHARACTERISTIC by Alan Marc Watson A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the

### DERIVED INVARIANCE OF THE NUMBER OF HOLOMORPHIC 1-FORMS AND VECTOR FIELDS 1. INTRODUCTION

DERIVED INVARIANCE OF THE NUMBER OF HOLOMORPHIC 1-FORMS AND VECTOR FIELDS MIHNEA POPA AND CHRISTIAN SCHNELL 1. INTRODUCTION Given a smooth projective variety X, we denote by DX) the bounded derived category

### The Calabi Conjecture

The Calabi Conjecture notes by Aleksander Doan These are notes to the talk given on 9th March 2012 at the Graduate Topology and Geometry Seminar at the University of Warsaw. They are based almost entirely

### KODAIRA-SAITO VANISHING AND APPLICATIONS

KODAIRA-SAITO VANISHING AND APPLICATIONS MIHNEA POPA Abstract. The first part of the paper contains a detailed proof of M. Saito s generalization of the Kodaira vanishing theorem, following the original

### ON THE INTEGRAL HODGE CONJECTURE FOR REAL THREEFOLDS. 1. Motivation

ON THE INTEGRAL HODGE CONJECTURE FOR REAL THREEFOLDS OLIVIER WITTENBERG This is joint work with Olivier Benoist. 1.1. Work of Kollár. 1. Motivation Theorem 1.1 (Kollár). If X is a smooth projective (geometrically)

### A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY. We also have an isomorphism of holomorphic vector bundles

A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY LIVIU I. NICOLAESCU ABSTRACT. These are notes for a talk at a topology seminar at ND.. GENERAL FACTS In the sequel, for simplicity we denote the complex

### Constructing compact 8-manifolds with holonomy Spin(7)

Constructing compact 8-manifolds with holonomy Spin(7) Dominic Joyce, Oxford University Simons Collaboration meeting, Imperial College, June 2017. Based on Invent. math. 123 (1996), 507 552; J. Diff. Geom.

### HOLONOMIC D-MODULES ON ABELIAN VARIETIES

HOLONOMIC D-MODULES ON ABELIAN VARIETIES CHRISTIAN SCHNELL Abstract. We study the Fourier-Mukai transform for holonomic D-modules on complex abelian varieties. Among other things, we show that the cohomology

### Stable bundles with small c 2 over 2-dimensional complex tori

Stable bundles with small c 2 over 2-dimensional complex tori Matei Toma Universität Osnabrück, Fachbereich Mathematik/Informatik, 49069 Osnabrück, Germany and Institute of Mathematics of the Romanian

### Abelian varieties. Chapter Elliptic curves

Chapter 3 Abelian varieties 3.1 Elliptic curves An elliptic curve is a curve of genus one with a distinguished point 0. Topologically it is looks like a torus. A basic example is given as follows. A subgroup

### Kähler manifolds and variations of Hodge structures

Kähler manifolds and variations of Hodge structures October 21, 2013 1 Some amazing facts about Kähler manifolds The best source for this is Claire Voisin s wonderful book Hodge Theory and Complex Algebraic

### Algebraic geometry versus Kähler geometry

Algebraic geometry versus Kähler geometry Claire Voisin CNRS, Institut de mathématiques de Jussieu Contents 0 Introduction 1 1 Hodge theory 2 1.1 The Hodge decomposition............................. 2

### DERIVED CATEGORIES: LECTURE 4. References

DERIVED CATEGORIES: LECTURE 4 EVGENY SHINDER References [Muk] Shigeru Mukai, Fourier functor and its application to the moduli of bundles on an abelian variety, Algebraic geometry, Sendai, 1985, 515 550,

### Singularities of hypersurfaces and theta divisors

Singularities of hypersurfaces and theta divisors Gregor Bruns 09.06.2015 These notes are completely based on the book [Laz04] and the course notes [Laz09] and contain no original thought whatsoever by

### Math 797W Homework 4

Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

### Applications to the Beilinson-Bloch Conjecture

Applications to the Beilinson-Bloch Conjecture Green June 30, 2010 1 Green 1 - Applications to the Beilinson-Bloch Conjecture California is like Italy without the art. - Oscar Wilde Let X be a smooth projective

### F. LAYTIMI AND D.S. NAGARAJ

REMARKS ON RAMANUJAM-KAWAMATA-VIEHWEG VANISHING THEOREM arxiv:1702.04476v1 [math.ag] 15 Feb 2017 F. LAYTIMI AND D.S. NAGARAJ Abstract. In this article weproveageneralresult on anef vector bundle E on a

### HODGE NUMBERS OF COMPLETE INTERSECTIONS

HODGE NUMBERS OF COMPLETE INTERSECTIONS LIVIU I. NICOLAESCU 1. Holomorphic Euler characteristics Suppose X is a compact Kähler manifold of dimension n and E is a holomorphic vector bundle. For every p

### ON MAXIMAL ALBANESE DIMENSIONAL VARIETIES. Contents 1. Introduction 1 2. Preliminaries 1 3. Main results 3 References 6

ON MAXIMAL ALBANESE DIMENSIONAL VARIETIES OSAMU FUJINO Abstract. We prove that any smooth projective variety with maximal Albanese dimension has a good minimal model. Contents 1. Introduction 1 2. Preliminaries

### APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY

APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY In this appendix we begin with a brief review of some basic facts about singular homology and cohomology. For details and proofs, we refer to [Mun84]. We then

### Geometry of the Calabi-Yau Moduli

Geometry of the Calabi-Yau Moduli Zhiqin Lu 2012 AMS Hawaii Meeting Department of Mathematics, UC Irvine, Irvine CA 92697 March 4, 2012 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 1/51

### ENOKI S INJECTIVITY THEOREM (PRIVATE NOTE) Contents 1. Preliminaries 1 2. Enoki s injectivity theorem 2 References 5

ENOKI S INJECTIVITY THEOREM (PRIVATE NOTE) OSAMU FUJINO Contents 1. Preliminaries 1 2. Enoki s injectivity theorem 2 References 5 1. Preliminaries Let us recall the basic notion of the complex geometry.

### Homological Mirror Symmetry and VGIT

Homological Mirror Symmetry and VGIT University of Vienna January 24, 2013 Attributions Based on joint work with M. Ballard (U. Wisconsin) and Ludmil Katzarkov (U. Miami and U. Vienna). Slides available

### ON A THEOREM OF CAMPANA AND PĂUN

ON A THEOREM OF CAMPANA AND PĂUN CHRISTIAN SCHNELL Abstract. Let X be a smooth projective variety over the complex numbers, and X a reduced divisor with normal crossings. We present a slightly simplified

### arxiv:alg-geom/ v1 29 Jul 1993

Hyperkähler embeddings and holomorphic symplectic geometry. Mikhail Verbitsky, verbit@math.harvard.edu arxiv:alg-geom/9307009v1 29 Jul 1993 0. ntroduction. n this paper we are studying complex analytic

### Mirror Symmetry: Introduction to the B Model

Mirror Symmetry: Introduction to the B Model Kyler Siegel February 23, 2014 1 Introduction Recall that mirror symmetry predicts the existence of pairs X, ˇX of Calabi-Yau manifolds whose Hodge diamonds

### LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES

LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.

### Topics in Geometry: Mirror Symmetry

MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:

### Fundamental groups, polylogarithms, and Diophantine

Fundamental groups, polylogarithms, and Diophantine geometry 1 X: smooth variety over Q. So X defined by equations with rational coefficients. Topology Arithmetic of X Geometry 3 Serious aspects of the

### Holomorphic symmetric differentials and a birational characterization of Abelian Varieties

Holomorphic symmetric differentials and a birational characterization of Abelian Varieties Ernesto C. Mistretta Abstract A generically generated vector bundle on a smooth projective variety yields a rational

### Geometry of the Moduli Space of Curves and Algebraic Manifolds

Geometry of the Moduli Space of Curves and Algebraic Manifolds Shing-Tung Yau Harvard University 60th Anniversary of the Institute of Mathematics Polish Academy of Sciences April 4, 2009 The first part

### Finite generation of pluricanonical rings

University of Utah January, 2007 Outline of the talk 1 Introduction Outline of the talk 1 Introduction 2 Outline of the talk 1 Introduction 2 3 The main Theorem Curves Surfaces Outline of the talk 1 Introduction

### Dedicated to Mel Hochster on his 65th birthday. 1. Introduction

GLOBAL DIVISION OF COHOMOLOGY CLASSES VIA INJECTIVITY LAWRENCE EIN AND MIHNEA POPA Dedicated to Mel Hochster on his 65th birthday 1. Introduction The aim of this note is to remark that the injectivity

### Topological Abel-Jacobi Mapping and Jacobi Inversion

Topological Abel-Jacobi Mapping and Jacobi Inversion by Xiaolei Zhao A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in the University

### KODAIRA DIMENSION OF LEFSCHETZ FIBRATIONS OVER TORI

KODAIRA DIMENSION OF LEFSCHETZ FIBRATIONS OVER TORI JOSEF G. DORFMEISTER Abstract. The Kodaira dimension for Lefschetz fibrations was defined in [1]. In this note we show that there exists no Lefschetz

### Abelian Varieties and the Fourier Mukai transformations (Foschungsseminar 2005)

Abelian Varieties and the Fourier Mukai transformations (Foschungsseminar 2005) U. Bunke April 27, 2005 Contents 1 Abelian varieties 2 1.1 Basic definitions................................. 2 1.2 Examples

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

### arxiv: v2 [math.ag] 10 May 2017

ariv:1611.08768v2 [math.ag] 10 May 2017 ALGEBRAIC FIBER SPACES OVER ABELIAN VARIETIES: AROUND A RECENT THEOREM BY CAO AND PĂUN CHRISTOPHER HACON, MIHNEA POPA, AND CHRISTIAN SCHNELL To Lawrence Ein, on

### Hodge structures from differential equations

Hodge structures from differential equations Andrew Harder January 4, 2017 These are notes on a talk on the paper Hodge structures from differential equations. The goal is to discuss the method of computation

### L 2 extension theorem for sections defined on non reduced analytic subvarieties

L 2 extension theorem for sections defined on non reduced analytic subvarieties Jean-Pierre Demailly Institut Fourier, Université de Grenoble Alpes & Académie des Sciences de Paris Conference on Geometry

### HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

### Construction problems in algebraic geometry and the Schottky problem. Dissertation

Construction problems in algebraic geometry and the Schottky problem Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität

### As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing).

An Interlude on Curvature and Hermitian Yang Mills As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing). Suppose we wanted

### Poisson geometry of b-manifolds. Eva Miranda

Poisson geometry of b-manifolds Eva Miranda UPC-Barcelona Rio de Janeiro, July 26, 2010 Eva Miranda (UPC) Poisson 2010 July 26, 2010 1 / 45 Outline 1 Motivation 2 Classification of b-poisson manifolds

### A Gauss-Bonnet theorem for constructible sheaves on reductive groups

A Gauss-Bonnet theorem for constructible sheaves on reductive groups V. Kiritchenko 1 Introduction In this paper, we prove an analog of the Gauss-Bonnet formula for constructible sheaves on reductive groups.

### LIFTABILITY OF FROBENIUS

LIFTABILITY OF FROBENIUS (JOINT WITH J. WITASZEK AND M. ZDANOWICZ) Throughout the tal we fix an algebraically closed field of characteristic p > 0. 1. Liftability to(wards) characteristic zero It often

### arxiv:math/ v3 [math.ag] 1 Mar 2006

arxiv:math/0506132v3 [math.ag] 1 Mar 2006 A NOTE ON THE PROJECTIVE VARIETIES OF ALMOST GENERAL TYPE SHIGETAKA FUKUDA Abstract. A Q-Cartier divisor D on a projective variety M is almost nup, if (D, C) >

### Introduction (Lecture 1)

Introduction (Lecture 1) February 2, 2011 In this course, we will be concerned with variations on the following: Question 1. Let X be a CW complex. When does there exist a homotopy equivalence X M, where

### A NEW BOUND FOR THE EFFECTIVE MATSUSAKA BIG THEOREM

Houston Journal of Mathematics c 2002 University of Houston Volume 28, No 2, 2002 A NEW BOUND FOR THE EFFECTIVE MATSUSAKA BIG THEOREM YUM-TONG SIU Dedicated to Professor Shiing-shen Chern on his 90th Birthday

### An Introduction to Spectral Sequences

An Introduction to Spectral Sequences Matt Booth December 4, 2016 This is the second half of a joint talk with Tim Weelinck. Tim introduced the concept of spectral sequences, and did some informal computations,

### COMPLEX ALGEBRAIC SURFACES CLASS 4

COMPLEX ALGEBRAIC SURFACES CLASS 4 RAVI VAKIL CONTENTS 1. Serre duality and Riemann-Roch; back to curves 2 2. Applications of Riemann-Roch 2 2.1. Classification of genus 2 curves 3 2.2. A numerical criterion

### THE HODGE DECOMPOSITION

THE HODGE DECOMPOSITION KELLER VANDEBOGERT 1. The Musical Isomorphisms and induced metrics Given a smooth Riemannian manifold (X, g), T X will denote the tangent bundle; T X the cotangent bundle. The additional

### Fourier Mukai transforms II Orlov s criterion

Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and