Log Homogeneous Varieties

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 TX(− log D) is generated by its global sections. Then the Albanese morphism α turns out to be 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 −KX −D is a spherical variety, and the irreducible components of all fibers of σ are equivariant compactifications of semiabelian 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.