Back to Departmental Colloquium: Spring 2001
Departmental Colloquium
Date: Monday, Feb 5, 2001
Time: 4:15PM
Location: JWB 335
Christopher Hacon
Riverside)
Title |
On numbers which classify algebraic varieties |
Abstract |
One useful way to organize our understanding of algebraic varieties, that is, solution sets of systems of polynomials, is by counting the number of linearly independent ( m -multiple-valued) top-degree holomorphic differential forms on the variety. These numbers, denoted as P_m, are called plurigenera of the variety. Another useful number is the number of linearly independent holomorphic one-forms, called the irregularity and denoted by q. It turns out that knowing even a few of these numbers completely determines the type of the variety we are dealing with, and so determines the important properties of all varieties X with the given numerical invariants P_m(X) and q(X). The focus usually is on varieties X with small P_m, since those with ``maximal P_m growth’’ are essentially unclassifiable. A turning point in modern classification theory is a theorem of Kawamata which roughly says that if all P_m ’s are <= 1 then X is a fibration over a q -dimensional torus. In this talk we will introduce and motivate the study of algebraic varieties via these numerical invariants, leading toward refinements and effective versions of Kawamata’s Theorem. Our goal will be the following results conjectured by Kollár: Theorem 1: Let X be smooth with P_2(X)=1. Then X maps surjectively to a complex projective torus of dimension q(X). (In particular dim( X ) >= q(X).) Theorem 2: Let X be smooth with P_2(X)=1 and dim( X )= q(X). Then X is birationally equivalent to a complex torus. (Both theorems are joint work with A. J. Chen) |
Algebraic Geometry