Projective normality of complete symmetric varieties
Projective normality of complete symmetric varieties
复制标题
完全对称簇的投影正态性
DOI：
10.1215/s0012-7094-04-12213-4
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发表时间：
2002
影响因子：
2.5
通讯作者：
A. Maffei
中科院分区：
文献类型：
--
作者：
R. Chirivì;A. Maffei
本文证明了在特征零点下，由完备对称簇X = G/H上的整体截面生成的线丛截面的乘积是满射映射。因此，由X上的完备线性系统或X的闭G-稳定子簇定义的锥是正规的。这对Faltings在[11]中提出的一个问题给出了肯定的回答。证明的一个关键点是根系的组合性质。设G是特征为零的代数闭域上的伴随半单代数群，G是其代数单连通覆盖。给定一个对合自同构σ：G → G，用H表示σ的不动点子群. De Concini和Procesi [9]构造了对称簇G/H的一个奇妙的紧化X。我们论文的主要结果可以表述如下。定理A如果L和L ′是由X上的整体截面生成的线丛，则乘法0（X，L）<$0（X，L）→ 0（X，L <$L）是满射的。X的投射正规性由一个标准的论证遵循。因此，我们对Faltings在[11]中提出的一个问题给出了肯定的回答.我们的结果已经在[15]中由Kannan在群的紧化的特殊情况下证明了，其中G = H×H，对合交换了H的两个副本，通过一种完全不同的方法，不适用于这种情况。我们强调，必须假定线丛L和L ′是由整体截面生成的，如第3节定理A证明后的例子所示。杜克数学杂志，第122卷，第1期，c © 2004年，2002年10月4日接收。2003年3月17日收到修订。2000年数学学科分类。小学14 M17;中学14 L30。
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 © 2004 Received 4 October 2002. Revision received 17 March 2003. 2000 Mathematics Subject Classification. Primary 14M17; Secondary 14L30.