Protective normality of complete symmetric varieties

none2We 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.Chirivi' Rocco; Maffei AndreaChirivi', Rocco; Maffei, Andre