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Projective normality of complete symmetric varieties. (English) Zbl 1064.14058

Let \(\sigma\) denote an involutorial automorphism on a semisimple algebraic group \(G\) of adjoint type over a field of characteristic \(0\). If \(H\) denotes the fixed points for \(\sigma\) then C. De Concini and C. Procesi [in: Invariant theory, Lect. Notes Math. 996, 1–44 (1983; Zbl 0581.14041)] have constructed the wonderful compactification \(X\) of \(G/H\).
In the present paper the authors prove that for any two globally generated line bundles on \(X\) the natural cup product map \(H^0(X, L_1)\otimes H^0(X, L_2) \rightarrow H^0(X, L_1 \otimes L_2)\) is surjective. A standard argument then gives that \(X\) is projectively normal. The authors’ proof of the the main result involves a close examination of so called low triples of dominant weights.
The theorem is not valid in prime characteristic. A counter example due to de Concini (\(\mathrm{PSL}(6)\) in characteristic \(3\)) is recalled in the paper. When \(G = H \times H\) and \(\sigma\) is the flip operator then the result was proved earlier by S. Kannan [Math. Z. 339, 673–682 (2002; Zbl 0997.14012)] by quite different arguments.

MSC:

14M17 Homogeneous spaces and generalizations
20G05 Representation theory for linear algebraic groups

Software:

LiE; Macaulay2
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References:

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