## Duke Mathematical Journal

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

#### Abstract

Let $k$ be an algebraically closed field of characteristic $p\gt 0$. 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 $b_{1}(X)=2\dim X$. 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 $\dim A\leq \dim X$.

#### Article information

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

Dates
Revised: 9 January 2019
First available in Project Euclid: 12 June 2019

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

Digital Object Identifier
doi:10.1215/00127094-2019-0008

Mathematical Reviews number (MathSciNet)
MR3961214

#### 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

#### References

• [1] M. Blickle and K. Schwede, “$p^{-1}$-linear maps in algebra and geometry” in Commutative Algebra, Springer, New York, 2013, 123–205.
• [2] A. Chambert-Loir, Cohomologie cristalline: un survol. Exposition. Math. 16 (1998), 333–382
• [3] T. Ekedahl, “Foliations and inseparable morphisms” in Algebraic Geometry, Bowdoin, 1985 (Brunswick, ME, 1985), Proc. Sympos. Pure Math. 46, Amer. Math. Soc., Providence, 1987, 139–149.
• [4] T. Ekedahl, Canonical models of surfaces of general type in positive characteristic, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 97–144.
• [5] O. Gabber, “Notes on some $t$-structures” in Geometric Aspects of Dwork Theory, Vols. I, II, de Gruyter, Berlin, 2004, 711–734.
• [6] C. D. Hacon and Z. Patakfalvi, Generic vanishing in characteristic $p>0$ and the characterization of ordinary abelian varieties, Amer. J. Math. 138 (2016), no. 4, 963–998.
• [7] Y. Kawamata, Characterization of abelian varieties, Compos. Math. 43 (1981), no. 2, 253–276.
• [8] H. Lange and U. Stuhler, Vektorbundel auf Kurven und Darstellungen Fundamental-gruppe, Math Z. 156 (1977), 73–83.
• [9] C. Liedtke, “Algebraic surfaces in positive characteristic” in Birational Geometry, Rational Curves, and Arithmetic, Springer, Cham, 2013, 229–292.
• [10] B. Moonen and G. van der Geer, Abelian varieties, preprint, http://gerard.vdgeer.net/AV.pdf (accessed 22 May 2019).
• [11] S. Mukai, Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175.
• [12] J.-P. Serre, Quelques propriétés des variétés abéliennes en caractéristique $p$, Amer. J. Math. 80 (1958), 715–739.
• [13] L. Zhang, Abundance for non-uniruled 3-folds with non-trivial albanese maps in positive characteristics, J. Lond. Math. Soc. (2) 99 (2019), no. 2, 332–348.