Log homogeneous varieties

TL;DR

This paper studies log homogeneous varieties, showing their structure involves spherical and toric varieties, and reduces their classification to automorphism groups of spherical varieties under certain conditions.

Contribution

It generalizes the Borel-Remmert theorem to log homogeneous varieties and links their structure to automorphism groups of spherical varieties.

Findings

The Albanese morphism is a fibration with spherical fibers.

Irreducible components of the boundary divisor are nonsingular.

The product of Albanese and sigma morphisms is surjective with toric fibers.

Abstract

Given a complete nonsingular algebraic variety $X$ and a divisor $D$ with normal crossings, we say that $X$ is log homogeneous with boundary $D$ if the logarithmic tangent bundle $T_X(- \log D)$ is generated by its global sections. We then show that the Albanese morphism $\alpha$ is a fibration with fibers being spherical (in particular, rational) varieties. It follows that all irreducible components of $D$ are nonsingular, and any partial intersection of them is irreducible. Also, the image of $X$ under the morphism $\sigma$ associated with $- K_X - D$ is a spherical variety, and the irreducible components of all fibers of $\sigma$ are quasiabelian varieties. Generalizing the Borel-Remmert structure theorem for homogeneous varieties, we show that the product morphism $\alpha \times \sigma$ is surjective, and the irreducible components of its fibers are toric varieties. We reduce the classification of log homogeneous varieties to a problem concerning automorphism groups of spherical varieties, that we solve under an additional assumption.