Se p 20 06 LOG HOMOGENEOUS VARIETIES MICHEL

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 α 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 σ associated with −K X − D is a spherical variety, and the irreducible components of all fibers of σ are quasiabelian varieties. Generalizing the Borel–Remmert structure theorem for homogeneous varieties, we show that the product morphism α × σ 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.