Math 514. Complex Algebraic Geometry Instructor Syllabus

Main Text: Voisin, Hodge Theory and Complex Algebraic Geometry, I

Supplementary Texts:

• Griffiths & Harris, Principles of Algebraic Geometry
• Ballmann, Lectures on Kähler Manifolds, available on the author's webpage

Description: This course introduces the geometry of Kähler manifolds and the associated Hodge structure of cohomology. Note that one of the Clay Institute's Millenium Open Problems is to prove the Hodge conjecture: that every Hodge class of a non-singular projective variety over C is a rational linear combination of cohomology classes of algebraic cycles. Kähler manifolds lie at the intersection of complex geometry, Riemannian geometry, and symplectic geometry. Moreover, every smooth projective variety is a Kähler manifold. All of this structure is reflected in a rich theory of geometric and topological invariants. This course develops techniques from sheaf theory and linear elliptic theory to study the cohomology of Kähler manifolds. (click for pdf of syllabus)

Schedule

History and overview. Manifolds and fiber bundles. Complex manifolds, holomorphic vector bundles. Almost complex structures, theorems of Fröbenius and Newlander-Nirenberg. Dolbeault complex. Examples. (6 hours)
Sources: Voisin, Chapter 2; Chern: Complex manifolds without potential theory; Kazdan: Applications of Partial Differential Equations to Some Problems in Differential Geometry.

Hermitian and Kähler metrics. Characterization of Kähler metrics. (3 hours)
Sources: Voisin, Chapter 3; Ballmann, Chapters 1-4.

Sheaves. Derived functors. Sheaf cohomology. Interpretation of first cohomology group (Abelian and non-Abelian). Examples, including cohomology of line bundles on projective space. (6 hours)
Sources: Voisin, Chapter 4; Bredon: Sheaf Theory; Debremaeker: Non-Abelian cohomology.

Differential operators and adjoints. Hodge * and Hodge and Dolbeault Laplacians. Distributions and pseudodifferential operators on Euclidean space. The symbol calculus on a manifold. The parametrix of an elliptic operator. Elliptic regularity, Kodaira decomposition. Hodge cohomology. Poincaré Duality. (6 hours)
Sources: Voisin, Chapter 5; Albin: Linear Analysis on Manifolds lecture notes.

Hodge theory of Kähler manifolds: Kähler identities, Hodge decomposition. $\partial\overline{\partial}$-Lemma, Bott- Chern and Aeppli cohomologies. Lefschetz decomposition, Hard Lefschetz theorem. Intersection forms, Hodge index theorem. Applications. (6 hours)
Sources: Voisin, Chapter 6; Angella-Tomassini: On the $\partial\overline{\partial}$-Lemma and Bott-Chern cohomology; Ballmann, Chapter 5.

Chern forms of line bundles. Curvature of connections. Kähler identities for twisted differential forms. Hard Lefschetz and $\partial\overline{\partial}$-Lemma for flat bundles. Positive line bundles. Kodaira vanishing theorem. Applications. (3 hours)
Sources: Ballmann, Chapter 5.

Hodge structures and Hodge ltration. Fubini-Study metric and classification of line bundles on CP^n. Blow-ups. Kodaira's proof of Kodaira embedding. (4 hours)
Sources: Ballmann, Chapter 9; Wells: Differential analysis on complex manifolds.

Topological applications. Calculation of Hodge numbers of varieties in examples, including complete intersections. Examples of variation of Hodge structure. (6 hours)
Sources: Griffiths & Harris; J. Lewis, A Survey of the Hodge Conjecture, Chapter 9.

Leeway/Current Research. (3 hours)