Andrea Maffei

Projective normality of complete symmetric varieties

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. Introduction Let G be an adjoint semisimple algebraic group over an algebraically closed field of characteristic zero, and let G be its algebraic simply connected cover. Given an involutorial automorphism σ : G → G, denote by H the subgroup of fixed points of σ . A wonderful compactification X of the symmetric variety G/H has been constructed by De Concini and Procesi [9]. The main result of our paper can be stated as the following. THEOREM A If L and L ′ are line bundles generated by global sections on X , then the multiplication 0(X,L )⊗ 0(X,L )→ 0(X,L ⊗L ) is surjective. The projective normality of X follows by a standard argument. Hence we give an affirmative answer to a problem raised by Faltings in [11]. Our result has already been proved in [15] by Kannan in the special case of the compactification of a group, in which G = H×H and the involution exchanges the two copies of H , by a completely different method that does not apply to this situation. We stress that it is necessary to assume that the line bundles L and L ′ are generated by global sections, as the example after the proof of Theorem A in Section 3 shows. DUKE MATHEMATICAL JOURNAL Vol. 122, No. 1, c

The ring of sections of a complete symmetric variety

We study the ring of sections A(X) of a complete symmetric variety X, that is of the wonderful completion of G/H where G is an adjoint semisimple group and H is the fixed subgroup for an involutorial automorphism of G. We find generators for Pic (X), we generalize the PRV conjecture to complete symmetric varieties and construct a standard monomial theory for A(X) that is compatible with G orbit closures in X. This gives a degeneration result and the rational singularityness for A(X).  2003 Elsevier Science (USA). All rights reserved.

Normality and non-normality of group compactifications in simple projective spaces

If

Projective normality of model varieties and related results

G

Standard Monomial Theory for Wonderful Varieties

is a complex simply connected semisimple algebraic group and if

A Note on Normality of Cones Over Symmetric Varieties

\lambda

On exceptional completions of symmetric varieties

is a dominant weight, we consider the compactification

On normality of cones over symmetric varieties

X_\lambda

Equations Defining Symmetric Varieties and Affine Grassmannians

in the projectivisation of

A GENERALIZED STEINBERG SECTION AND BRANCHING RULES FOR QUANTUM GROUPS AT ROOTS OF 1

\End(V(\lambda))