Vanishing theorems for Dolbeault cohomology of log homogeneous varieties

We consider a complete nonsingular variety $X$ over \bC, having a normal crossing divisor $D$ such that the associated logarithmic tangent bundle is generated by its global sections. We show that $H^i\big(X, L^{-1} \otimes \Omega_X^j(\log D)\big) = 0$ for any nef line bundle $L$ on $X$ and all $i < j - c$, where $c$ is an explicit function of $(X,D,L)$. This implies e.g. the vanishing of $H^i(X, L \otimes \Omega_X^j)$ for $L$ ample and $i > j$, and gives back a vanishing theorem of Broer when $X$ is a flag variety

Epimorphic subgroups of algebraic groups

In this note, we show that the epimorphic subgroups of an algebraic group are exactly the pull-backs of the epimorphic subgroups of its affinization. We also obtain epimorphicity criteria for subgroups of affine algebraic groups, which generalize a result of Bien and Borel. Moreover, we extend the affinization theorem for algebraic groups to homogeneous spaces.Comment: Final version, accepted for publication at Mathematical Research Letter

Anti-affine algebraic groups

We say that an algebraic group $G$ over a field is anti-affine if every regular function on $G$ is constant. We obtain a classification of these groups, with applications to the structure of algebraic groups in positive characteristics, and to the construction of many counterexamples to Hilbert's fourteenth problem.Comment: Prior work of Carlos Sancho de Salas acknowledged, additional minor changes

Some basic results on actions of non-affine algebraic groups

We study actions of connected algebraic groups on normal algebraic varieties, and show how to reduce them to actions of affine subgroups.Comment: 20 pages ; references and final example adde