This is the main page for **Math 392C (Complex Geometry)**, unique number **57175**.

I am Andy Neitzke; my office is RLM 9.134. My office hours are 2-3:30pm on Wednesday, or by appointment.

This course meets TuTh from 9:30-11:00am, in RLM 12.166.

The main text will be **Huybrechts, "Complex Geometry: An Introduction."** Other useful
resources are Griffiths and Harris, "Algebraic Geometry"; Voisin, "Hodge Theory and Complex Algebraic Geometry I"; Wells, "Differential Analysis
on Complex Manifolds"; Moroianu, "Lectures on Kahler geometry."

I will assign exercises; I strongly encourage you to do them. I will mention a few exercises during lecture, but also will post slightly more organized exercise sheets (below.)

**A very optimistic list of topics, along with some brief motivation,
appears here.** At most we can cover a couple of the more advanced
ones. Therefore *I would very much like feedback*
about which potential topics are of the most interest to you.

Here are some briefer comments. The first part of the course will be devoted to the basic technology of complex geometry. The main goals are the Hodge theory of Kahler manifolds and at least some of its consequences such as the Lefshetz theorems. Other phrases which will be explained along the way include "sheaf cohomology", "Dolbeault theorem", "holomorphic line bundle", and "divisor".

For the second part of the course we have somewhat more freedom, and what we do will depend somewhat on the tastes of the class. I hope to have time to explain the notion of variation of Hodge structure and the analogue of the Lefshetz theory for hyperkahler manifolds.

I will post my notes from the lectures below. The mapping between files and lectures is not necessarily 1-1. Notes will be updated to correct errors/omissions where they are helpfully pointed out or where I notice them later. I apologize for whatever errors may remain.

- 01.0: Introduction
- 02.0: Complex manifolds, holomorphic functions and holomorphic vector bundles
- 03.0: Linear algebra of complexified vector spaces
- 03.5: The holomorphic tangent bundle
- 03.8: Forms on almost complex manifolds
- 03.9: Holomorphic forms
- 04.0: Integrability of almost complex structures, and its analog for vector bundles
- 04.5: Holomorphic line bundles on the torus, first look
- 05.0: Hermitian structures on vector spaces
- 05.8: Hermitian metrics
- 06.0: Kahler metrics
- 06.3: Hodge theory for Riemannian manifolds
- 06.5: More Laplacians
- 07.0: Kahler identities
- 08.0: Hodge theory for Kahler manifolds
- 08.5: The Hopf surface
- 08.7: Holomorphic line bundles on the torus, revisited
- 09.0: Sheaves
- 09.1: Divisors and line bundles
- 09.3: Complexes of sheaves
- 09.5: Flabby, soft and fine sheaves
- 10.0: Sheaf cohomology
- 10.5: Calculating sheaf cohomology
- 10.7: Double complexes
- 10.8: Picard and Jacobian
- 11.0: Jacobians of compact Kahler manifolds
- 12.0: Connections in complex bundles
- 13.0: Curvature in complex bundles
- 14.0: Chern classes
- 14.5: Interpretations of c_1
- 15.0: Computations on projective space
- 16.0: Characteristic classes for K3
- 17.0: Serre duality
- 18.0: Kodaira vanishing
- 19.0: Hirzebruch-Riemann-Roch formula
- 20.0: Grothendieck lemma
- 20.5: Special holonomy
- 21.0: Calabi-Yau manifolds
- 22.0: Strominger-Yau-Zaslow picture
- 23.0: Hyperkahler manifolds

The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.