Projective normality of complete symmetric varieties

Abstract

We prove that in characteristic zero the multiplication of sections of line bundles generated by global sections on a complete symmetric variety $X=\overline{G/H}$ is a surjective map. As a consequence, the cone defined by a complete linear system over $X$ or over a closed $G$-stable subvariety of $X$ is normal. This gives an affirmative answer to a question raised by Faltings in [11]. A crucial point of the proof is a combinatorial property of root systems.