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= 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.