Room 517, Meng Wah Complex, HKU
Professor Jungkai
Chen
On varieties with k = 0
Abstract
In algebraic geometry, the varieties
of Kodaira dimension k = 0 play an important role in many aspects. For example, elliptic
curves, abelian varieties, K3 surfaces, and Calabi-Yau varieties are varieties with k = 0. Moreover,
in classification theory, the natural Iitaka fibration constructed via pluricanonical
maps produces an algebraic fiber space with general fiber having k = 0.
Turning to the structure of varieties X with k(X) = 0, it's conjectured by Ueno that:
1. The Albenese map alb: X ® Alb(X) is surjective to the abelian variety Alb(X) with connected fibers,
2. The general fiber F of alb has k(F) = 0, and
3. After base change by a finite etale
morphism A' ® Alb(X), X'= X ´A' Alb(X) is birational to A' ´ F.
Kawamata has proved that (1) is
always true and the whole conjecture holds when a good minimal model exists.
The purpose of this talk is to present our recent progress toward this
conjecture. This is a joint work with Hacon.