##### Department of Mathematics,

University of California San Diego

****************************

### Advancement to Candidacy

## Iacopo Brivio

#### UCSD

## Deformation invariance of plurigenera in analytic and algebraic geometry

##### Abstract:

In the classification theory of higher dimensional algebraic varieties a central role is played by the canonical divisor $K_X$ and its multiples. A famous theorem of Y. T. Siu states that if ${\pi:X\longrightarrow}$ T is a smooth projective family of varieties, then the plurigenera of the fibres $\lbrace h^0(X_t, mK_{X_t})\rbrace_{m\geq 0}$ are constant in t. Despite being an algebraic problem, Siu's proof employs methods which are essentially analytic in nature. After giving an overview of the techniques involved, we outline a path to a possible algebraic proof, based on a reduction to the general type case via the Iitaka fibration.

Advisor: James McKernan

### May 22, 2017

### 11:00 AM

### AP&M B412

****************************