##
**Birational characterization of abelian varieties and ordinary abelian varieties in characteristic \(p>0\).**
*(English)*
Zbl 1436.14033

Largely using the techniques of the first two authors [Am. J. Math. 138, No. 4, 963–998 (2016; Zbl 1408.14073)], this paper gives criteria for a variety over an algebraically closed field of characteristic \(p>0\) to be birational to an abelian variety, or to an ordinary abelian variety.

In characteristic zero, a result of Y. Kawamata [Compos. Math. 43, 253–276 (1981; Zbl 0471.14022)] says that a projective variety \(X\) of dimension \(n\) is birational to an abelian variety if and only if \(b_2(X)=2n\) and \(\kappa(X)=0\). The corresponding result that is proved here has to take account of the action of Frobenius. Instead of \(\kappa(X)\), which measures the growth of the space of all sections of \(mK_X\), one should look at the growth of the the space of Frobenius-stable sections, as defined in [loc. cit.], given by the stable Kodaira dimension \(\kappa_S(X)\). One then has that \(X\) is birational to an ordinary abelian variety (i.e., one for which the Frobenius is injective on \(H^1(\mathcal{O}_A)\)) if and only if \(b_2(X)=2n\) and \(\kappa_S(X)=0\). The difficulties here mainly arise when the Albanese map \(a\colon X\to A\) is inseparable, or wildly ramified.

This criterion has the advantage of being expressed in terms of the Betti numbers, and thus being a cohomological condition, but the restriction to ordinary abelian varieties is not always convenient. The second result replaces the cohomological condition with a geometric one, but yields a characterisation of projective varieties \(X\) birational to an abelian variety \(A\), otherwise unrestricted. Now the condition is that \(\kappa(X)=0\) rather than a condition on \(\kappa_S(X)\), but the condition on the Betti numbers, is replaced by the condition that the Albanese map is generically finite. The authors explain the technical reason for the appearance of the generically finite condition but it is not clear whether a cohomological condition might suffice in this case also.

The route to the results is to prove that if \(\kappa_S(X)=0\), or if \(\kappa(X)=0\) and \(a\) is generically finite, then \(a\) is surjective. Then a base change argument allows one to conclude that \(a\) is birational.

In characteristic zero, a result of Y. Kawamata [Compos. Math. 43, 253–276 (1981; Zbl 0471.14022)] says that a projective variety \(X\) of dimension \(n\) is birational to an abelian variety if and only if \(b_2(X)=2n\) and \(\kappa(X)=0\). The corresponding result that is proved here has to take account of the action of Frobenius. Instead of \(\kappa(X)\), which measures the growth of the space of all sections of \(mK_X\), one should look at the growth of the the space of Frobenius-stable sections, as defined in [loc. cit.], given by the stable Kodaira dimension \(\kappa_S(X)\). One then has that \(X\) is birational to an ordinary abelian variety (i.e., one for which the Frobenius is injective on \(H^1(\mathcal{O}_A)\)) if and only if \(b_2(X)=2n\) and \(\kappa_S(X)=0\). The difficulties here mainly arise when the Albanese map \(a\colon X\to A\) is inseparable, or wildly ramified.

This criterion has the advantage of being expressed in terms of the Betti numbers, and thus being a cohomological condition, but the restriction to ordinary abelian varieties is not always convenient. The second result replaces the cohomological condition with a geometric one, but yields a characterisation of projective varieties \(X\) birational to an abelian variety \(A\), otherwise unrestricted. Now the condition is that \(\kappa(X)=0\) rather than a condition on \(\kappa_S(X)\), but the condition on the Betti numbers, is replaced by the condition that the Albanese map is generically finite. The authors explain the technical reason for the appearance of the generically finite condition but it is not clear whether a cohomological condition might suffice in this case also.

The route to the results is to prove that if \(\kappa_S(X)=0\), or if \(\kappa(X)=0\) and \(a\) is generically finite, then \(a\) is surjective. Then a base change argument allows one to conclude that \(a\) is birational.

Reviewer: G. K. Sankaran (Bath)

### MSC:

14E99 | Birational geometry |

14K05 | Algebraic theory of abelian varieties |

14K15 | Arithmetic ground fields for abelian varieties |

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