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.


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


LiE; Macaulay2
Full Text: DOI arXiv


[1] D. Bayer and M. Stillman, Macaulay, version 3.1, 2000,
[2] N. Bourbaki, éléments de mathématique, fasc. 34: Groupes et algèbres de Lie, chapitres 4–6 , Actualités Sci. Indust. 1337 , Hermann, Paris, 1968.
[3] W. Bruns, Algebras defined by powers of determinantal ideals , J. Algebra 142 (1991), 150–163. · Zbl 0741.13005
[4] W. Bruns and A. Conca, Algebras of minors , J. Algebra 246 (2001), 311–330. · Zbl 1015.13004
[5] W. Bruns and U. Vetter, Determinantal Rings , Lecture Notes in Math. 1327 , Springer, Berlin, 1988. · Zbl 0673.13006
[6] R. Chirivì and A. Maffei, The ring of sections of a complete symmetric variety , J. Algebra 261 (2003), 310–326. · Zbl 1055.14052
[7] C. De Concini, ”Normality and non normality of certain semigroups and orbit closures” to appear in Algebraic Transformation Groups and Algebraic Varieties , Encyclopaedia Math. Sci. 132 , Springer, New York, 2004.
[8] C. De Concini, D. Eisenbud, and C. Procesi, Young diagrams and determinantal varieties , Invent. Math. 56 (1980), 129–165. · Zbl 0435.14015
[9] C. De Concini and C. Procesi, ”Complete symmetric varieties” in Invariant Theory (Montecatini, Italy, 1982) , Lecture Notes in Math. 996 , Springer, Berlin, 1983, 1–44. · Zbl 0581.14041
[10] C. De Concini and T. A. Springer, Compactification of symmetric varieties , Transform. Groups 4 (1999), 273–300. · Zbl 0966.14035
[11] G. Faltings, Explicit resolution of local singularities of moduli-spaces , J. Reine Angew. Math. 483 (1997), 183–196. · Zbl 0871.14012
[12] R. Hartshorne, Algebraic Geometry , Grad. Texts in Math. 52 , Springer, New York, 1977. · Zbl 0367.14001
[13] S. Helgason, A duality for symmetric spaces with applications to group representations , Adv. Math. 5 (1970), 1–154. · Zbl 0209.25403
[14] ——–, Differential Geometry, Lie Groups, and Symmetric Spaces , Pure Appl. Math. 80 , Academic Press, New York, 1978. · Zbl 0451.53038
[15] S. S. Kannan, Projective normality of the wonderful compactification of semisimple adjoint groups , Math. Z. 239 (2002), 673–682. · Zbl 0997.14012
[16] M. A. A. Van Leeuwen, A. M. Cohen, and B. Lisser, LiE, A package for Lie group computations, Computer Algebra Nederland, Amsterdam, 1992.
[17] J. R. Stembridge, The partial order of dominant weights , Adv. Math. 136 (1998), 340–364. · Zbl 0916.06001
[18] T. Vust, Opération de groupes réductifs dans un type de cônes presque homogènes , Bull. Math. Soc. France 102 (1974), 317–333. · Zbl 0332.22018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.