Effective characterization of ordinary abelian varieties, and beyond
Jefferson Baudin
TL;DR
The work develops a positive characteristic analogue of Chen–Hacon's birational characterization of abelian varieties by linking cohomological invariants to Albanese geometry via Cartier/Frobenius techniques. Central to the approach are the positive-characteristic generic vanishing framework, $V$-modules and Fourier–Mukai methods, and Verschiebung-based base changes that force the Albanese pushforward to have rank one, yielding surjectivity with connected fibers and, under extra hypotheses, birationality to an abelian variety. These results simultaneously provide a purely positive-characteristic proof of Chen–Hacon's complex-analytic characterization and extend to a delta-version with divisor data, as well as a characteristic-free bound in the maximal Albanese dimension case. The findings illuminate how $S^0$-type invariants and $P_2$ controls on global sections govern the Albanese map and ordinal properties of the Albanese variety, with potential implications for understanding Kodaira dimensions and deformation behavior in positive characteristic.
Abstract
We prove that the Albanese morphism of any normal proper variety $X$ in positive characteristic satisfying $S^0(X, ω_X) \neq 0$ and $P_2(X) = 1$ is surjective with connected fibers, adn that $\mathrm{Alb}(X)$ is ordinary. We obtain from a variant of the above a purely positive characteristic proof of Chen and Hacon's effective birational characterization of complex abelian varieties.