Log homogeneous varieties
By M. Brion. 2006
53 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/ca12e573818280328306461cdfc3084f7c459066
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In Computational Theory and Mathematics
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Citations in the corpus
- Some Basic Results on Actions of Nonaffine Algebraic Groups (2007, methodology): “Here we present a short proof of that result by analyzing rst the structure of G and then the Albanese morphism of X, while [Sa] proceeds by constructing the projection X ! Y via the study of the Ga -action on X. In characteristic zero, this projection is the Tits morphism that assigns to any point ”