Projective normality of complete toroidal symmetric varieties

In [R. Chirivì, A. Maffei, Projective normality of complete symmetric varieties, Duke Math. J. 122 (1) (2004) 93–123] Chirivì and Maffei have proved that the multiplication of sections of any two line bundles generated by global sections on a wonderful symmetric variety is surjective. We prove two criterions that allow us to reduce the same problem on a (smooth) complete toroidal symmetric variety to the analogous problem on the corresponding complete toric variety (respectively on the corresponding open toric variety). We have also studied in details some family of complete toroidal symmetric varieties, in particular those of rank 2.