Duke Mathematical Journal

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

Christopher D. Hacon, Zsolt Patakfalvi, and Lei Zhang

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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 κS(X)=0 and b1(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 κ(X)=0, and the Albanese morphism a:XA is generically finite. Along the way, we also show that if κS(X)=0 (or if κ(X)=0 and a is generically finite), then the Albanese morphism a:XA is surjective and in particular dimAdimX.

Article information

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

Received: 28 August 2017
Revised: 9 January 2019
First available in Project Euclid: 12 June 2019

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Mathematical Reviews number (MathSciNet)

Primary: 14E99: None of the above, but in this section
Secondary: 14K05: Algebraic theory 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]

birational geometry abelian varieties positive characteristic


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

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