Hodge Theory

Description: This is a Ph.D. level course at IMPA. The course is divided into four parts. In the first part we study the topology of smooth proyective complex varieties focusing on Picard-Lefschetz theory and applications of the Lefschetz hyperplane sections theorem. The second part is about the classical L2-Hodge theory on compact Kähler manifolds, harmonic forms, the Hodge decomposition theorem and the Hodge index theorem. The third part is about cohomology of coherent algebraic and analytic sheaves, we treat Stein varieties and the classical results due to Serre (FAC and GAGA). In the fourth part we go to the algebraic study of the Hodge filtration in terms of hypercohomology and spectral sequences, we treat Atiyah-Hodge theorem, logarithmic forms and Griffiths basis theorem.

References:

1.  Hodge Theory and Complex Algebraic Geometry, vol. 1 and 2, Claire Voisin.

2.  Géométrie Algébrique et Géométrie Analytique, Jean-Pierre Serre.

3.  A Course in Hodge Theory: With Emphasis on Multiple Integrals, Hossein Movasati.

4.  A Course in Hodge Theory: Periods of algebraic cycles, Hossein Movasati and Roberto Villaflor.

Exercises: List 1, List 2, List 3.

Period and time: August 05 to November 29, 2019. Monday 15h30-17h00 (room 232), Friday 17h00-18h30 (room 232).

• Lecture 1: Definition of singular homology, functoriality and homotopy invariance. Relative homology, long exact sequence of a pair. Relation between the first homology group and the fundamental group. Examples from algebraic geometry. Homology with coefficients in a group, cohomology with coefficients in a group, universal coefficient theorems. 

• Lecture 2: Künneth formula, compact support cohomology groups, Poincaré duality. Axioms for a homology theory, Eilenberg-Steenrod theorem, long exact sequence for triples, Mayer-Vietoris sequence. Leray-Thom-Gysin isomorphism, Leray-Thom-Gysin exact sequence associated to a pair, intersection map. 

• Lecture 3: Statement of Lefschetz main theorem (relative homology of the affine part of a projective variety relative to a hyperplane section). Consequences: Homology of affine varieties, properties of the intersection map, Lefschetz hyperplane section theorem, homology of complete intersections and affine comple intersections. Hard Lefschetz theorem, primitive homology, Lefschetz decomposition. 

• Lecture 4: Ehresman fibration theorem, strong deformation retracts. Relative homology of a one parameter family with isolated critical points relative to a smooth fiber, reduction to the local case, proof in the non-degenerated case. Lefschetz pencils, weak version of Lefschetz main theorem. 

• Lecture 5: Lefschetz thimbles, vanishing cycles. Proof of Lefschetz main theorem in the degenerate local case, Milnor number of a singularity. Application: Middle homology of a smooth hypersurface of the projective space. 

• Lecture 6: Complexified cotangent bundle, De Rham cohomology, differential (p,q)-forms, Dolbeault cohomology. Hermitian and Kähler manifolds, projective varieties are Kähler with the Fubini-Study metric. Poincaré-De Rham isomorphism, integral and rational classes, Kodaira embedding theorem. 

• Lecture 7: Volume form on a Hermitian manifold, induced metric on exterior powers of the complexified cotangent bundle. L2 product on the space of (p,q)-forms over a compact manifold, Hodge star operator, formal adjoint of anti-holomorphic differential, harmonic forms. Finiteness theorem for elliptic operators, Hodge theorem on harmonic forms, Kodaira-Serre duality, Poincaré duality. 

• Lecture 8: Kähler metrics are constant up to second order approximations. Lefschetz operator and its formal adjoint, commutators with differential operators, relation between Laplacians for compact Kähler manifolds. Cohomology groups of type (p,q), Hodge decomposition. 

• Lecture 9: Lefschetz operator commutes with the Laplacian, hard Lefschetz theorem. Hodge cycles and classes, fundamental class of an algebraic subvariety, algebraic cycles are Hodge cycles, Hodge conjecture. Statement of Lefschetz (1,1) theorem, Hodge conjecture is true for dimension at most 3, Hodge conjecture for hypersurfaces reduces to the middle homology of even dimensional hypersurfaces. 

• Lecture 10: Primitive k-forms, the star of a primitive form, Lefschetz decomposition in cohomology. Polarization and its compatibility with Lefschetz and Hodge decompositions, Hodge index theorem. Hodge filtration, intermediate Jacobians, the first Jacobian is an Abelian variety. 

• Lecture 11: Abelian categories with enough injectives, the derived object of a complex is well defined, simple complex associated to a double complex resolution, every complex admits a derived object, the derived objects are defined in the derived category. 

• Lecture 12: Derived long exact sequence associated to a short exact sequence of complexes. Divisible groups are injective, the category of sheaves of abelian groups have enough injectives, sheaf cohomology. Cech cohomology for coverings, Cech resolution of a sheaf, Leray theorem, long exact sequence in Cech cohomology, flabby sheaves are Cech acyclic, Cech cohomology as limit of refinements is isomorphic to sheaf cohomology. 

• Lecture 13: Fine sheaves are acyclic, De Rham theorem, Dolbeault theorem. Definition of analytic varieties, coherent analytic sheaves, Oka coherence, Cartan theorem, coherent sheaves form an abelian category, Oka theorem. Stein varieties, closed subvarieties of a Stein variety are Stein, the extended polydisc is a Stein variety.

• Lecture 14: Holomorphically convex varieties with separating holomorphic functions are Stein, intersection of Stein varieties is Stein, every analytic variety admits a good Stein covering. Frechet spaces, compact operators, Schwartz theorem. Frechet structure on the sections of a coherent sheaf induced by the uniform convergence on compact subsets, Cartan-Serre finiteness theorem. 

• Lecture 15: Definition of complex algebraic varieties and morphisms, definition of algebraic quasi-coherent and coherent sheaves, Serre's theorem on the cohomology of affine varieties (quasi-coherent sheaves are acyclic). Projective varieties, twisted sheaves, Serre's theorem about coherent algebraic sheaves over projective varieties (are quotients of finite direct sums of twisted sheaves). Cohomology of the projective space with coefficients in the twisted sheaves, Serre's theorem on the cohomology of projective varieties with coefficients in a coherent sheaf (finiteness and vanishing after a high enough twisting). 

• Lecture 16: Analytification functor, holomorphic functions are flat over regular functions, first GAGA theorem (correspondence of sheaf cohomology groups). Sheaf of morphisms between coherent sheaves is coherent, second GAGA theorem (every analytic morphism comes from an algebraic morphism). Third GAGA theorem (every analytic coherent sheaf comes from an algebraic coherent sheaf), Chow theorem. 

• Lecture 17: Definition of hypercohomology as derived functor and some properties. Hypercohomology relative to a open covering, long exact sequence in hypercohomology with respect to a covering and Leray theorem in hypercohomology. Naive filtration in hypercohomology, spectral sequence, criterion for degeneration at Er. 

• Lecture 18: Degeneration at E1 of the spectral sequence associated to the naive filtration. Analytic De Rham cohomology as hypercohomology with coefficients in the analytic De Rham complex, correspondence of the naive filtration with the Hodge filtration in the projective case. Meromorphic forms and meromorphic forms with logarithmic poles, local description of meromorphic forms with logarithmic poles, Atiyah-Hodge and Deligne theorems. 

• Lecture 19: Algebraic differential forms, algebraic De Rham cohomology, algebraic Hodge filtration for smooth projective varieties. Poincaré residue sequence, algebraic De Rham cohomology of affine varieties with logarithmic forms, algebraic Hodge filtration for affine varieties. Carlson-Griffiths lemma (operator reducing the order of the pole). 

• Lecture 20: Euler's short exact sequence, Bott formula and corollary. First Griffiths theorem (about generators of the Hodge filtration), second Griffiths theorem (about the kernel of the residue map). Carlson-Griffiths theorem (about the explicit values of the residue map on Griffiths basis).