MAT 615 Topics in Algebraic Geometry

Instructor    Sorin Popescu   (office: Math 4-119, tel. 632-8358, e-mail sorin@math.sunysb.edu)

Course description

The course is somewhat a continuation of last semster's MAT 536 Algebra III: (Algebraic Geometry: An introduction) and aims to cover fundamental results and techniques in (higher dimensional) complex algebraic geometry. The language will be mainly geometric, and/or sometimes analytic and we will assume material covered in MAT 536, as well as MAT 539 (Algebraic Topology), MAT 542-543 (Complex Analysis). It should however be possible to fill in some of the gaps during the semester.

Textbook(s)

No particular text will be used exclusively for the course, but the plan is to review first certain sections in Griffiths' and Harris' textbook Principles of Algebraic Geometry, and then continue with an introduction to techniques and results in higher dimensional algebraic geometry.

Textbooks that you may find useful are the following: Basic Algebraic Geometry 1 & Basic Algebraic Geometry 2, I.R. Shafarevich, Springer 1995 Principles of Algebraic Geometry, Ph. Griffiths and J. Harris, John Wiley & Sons 1994 Algebraic Geometry, Robin Hartshorne, GTM 52, Springer 1997 Complex algebraic surfaces, A. Beauville, LMS Student Text Birational geometry of algebraic varieties, J. Kollar and S. Mori, Cambridge 2001 Higher-Dimensional Algebraic Geometry, O. Debarre, Spriger 2001 An Introduction to Mori's Program, K. Matsuki, Springer 2002

(The last two are not yet in the library, but the library is currently ordering them. They will be put on reserve.)

Topics

The following is a tentative list of what we will try to cover in class:

• Review of Complex manifolds and basic fundamental results in several complex variables. De Rham and Dolbeault Cohomology, Calculus on complex manifolds
• Review of sheaves and cohomology; de Rham and Dolbeault theorems
• The Hodge theorem (without proof) and applications. Kaehler manifolds and the Hodge decomposition
• Divisors, Line bundles, Chern classes, linear systems, ampleness criteria
• The Lefschetz Theorem on (1,1)-classes
• Kodaira vanishing and its generalizations
• Curves and their Jacobians; complex tori
• Birational geometry of surfaces. Minimal models of surfaces.
• The cone theorem. Introduction to the Minimal Model Program (Mori's Program)
• The Minimal Model Program in dimension two. (Surfaces revisited)
• Fano varieties. Calabi-Yau 3 folds. Other classes of examples.

Homework

I will assign problems in each lecture, ranging in difficulty from routine to more challenging.