## 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

### Citations:

Zbl 0581.14041; Zbl 0997.14012

LiE; Macaulay2
Full Text:

### References:

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