##### Department of Mathematics,

University of California San Diego

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### Math 208 - Algebraic Geometry

## Ziquan Zhuang

#### MIT

## Positivity of CM line bundle on the K-moduli space

##### Abstract:

Recently there has been a lot of work on the construction of the K-moduli space, i.e. a good moduli space parametrizing K-polystable Fano varieties. It is conjectured that this moduli space is projective and the polarization is given by a natural line bundle, the Chow-Mumford (CM) line bundle. In this talk, I will present a recent joint work with Chenyang Xu where we show the CM line bundle is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties, which conjecturally should be the entire K-moduli. As an application, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective.

Host: James McKernan

### January 17, 2020

### 3:00 PM

### AP&M 7321

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