## Algebraic Geometry Day

**Speaker(s): ** Chenyang Xu (BICMR)

**Time: **March 21, 2013

**Venue: ** BICMR, 甲乙丙楼 82J04

### Algebraic Geometry Day

Time: March 21, 2013

Address: BICMR, 甲乙丙楼 82J04

Organizer: Chenyang Xu

### Schedule

9:30-10:30

Baohua Fu: Uniqueness of equivariant compactifications of C^n by a Fano manifold of Picard number one

coffee break

11:00-noon

Brion: Structure of algebraic monoids

Lunch break

14:30-15:30

Cascini: Topological bounds in birational geometry

Coffee Break

16:00-17:00

Looijenga: Meromorphic automorphic forms in algebraic geometry

17:30 banquet

### Program

Michael Brion

**Title**: Structure of algebraic monoids

**Abstract**: An algebraic monoid is an algebraic variety M equipped with an associative composition law which has a neutral element. Then the invertible elements form an algebraic group G, open in M; we may view M as a partial compactification of G, equivariant with respect to the action of G x G by left and right multiplication. In this talk, we will present a characterization of monoids among such compactifications, and derive some consequences on the structure of algebraic monoids.

Paolo Cascini

**Title**: Topological bounds in birational geometry.

**Abstract**: Many birational invariants of a smooth projective three-fold, such as the number and the singularities of its minimal models, are related with the topology of the underlying manifold. Using these methods, I will discuss some recent progress towards a question by Hirzebruch on the Chern numbers of a smooth projective threefold.

Baohua Fu

**Title**: Uniqueness of equivariant compactifications of C^n by a Fano manifold of Picard number one

**Abstract**: Let X be an n-dimensional Fano manifold of Picard number 1. We study how many different ways X can compactify the complex vector group C^n equivariantly. Hassett and Tschinkel showed that when X = P^n with n \geq 2, there are many distinct ways that X can be realized as equivariant compactifications of C^n. Our result says that projective space is an exception: among Fano manifolds of Picard number 1 with smooth VMRT, projective space is the only one compactifying C^n equivariantly in more than one ways. This answers questions raised by Hassett-Tschinkel and Arzhantsev-Sharoyko. This is a joint work with Jun-Muk Hwang.

Eduard Looijenga

**Title**: Meromorphic automorphic forms in algebraic geometry

**Abstract**: TBA