## Duke Mathematical Journal

- Duke Math. J.
- Volume 168, Number 9 (2019), 1723-1736.

### Birational characterization of Abelian varieties and ordinary Abelian varieties in characteristic $p>0$

Christopher D. Hacon, Zsolt Patakfalvi, and Lei Zhang

#### Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$. We give a birational characterization of ordinary abelian varieties over $k$: a smooth projective variety $X$ is birational to an ordinary abelian variety if and only if ${\kappa}_{S}(X)=0$ and ${b}_{1}(X)=2dimX$. We also give a similar characterization of abelian varieties as well: a smooth projective variety $X$ is birational to an abelian variety if and only if $\kappa (X)=0$, and the Albanese morphism $a:X\to A$ is generically finite. Along the way, we also show that if ${\kappa}_{S}(X)=0$ (or if $\kappa (X)=0$ and $a$ is generically finite), then the Albanese morphism $a:X\to A$ is surjective and in particular $dimA\le dimX$.

#### Article information

**Source**

Duke Math. J., Volume 168, Number 9 (2019), 1723-1736.

**Dates**

Received: 28 August 2017

Revised: 9 January 2019

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1560326500

**Digital Object Identifier**

doi:10.1215/00127094-2019-0008

**Mathematical Reviews number (MathSciNet)**

MR3961214

**Subjects**

Primary: 14E99: None of the above, but in this section

Secondary: 14K05: Algebraic theory 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]

**Keywords**

birational geometry abelian varieties positive characteristic

#### Citation

Hacon, Christopher D.; Patakfalvi, Zsolt; Zhang, Lei. Birational characterization of Abelian varieties and ordinary Abelian varieties in characteristic $p\gt 0$. Duke Math. J. 168 (2019), no. 9, 1723--1736. doi:10.1215/00127094-2019-0008. https://projecteuclid.org/euclid.dmj/1560326500