Diagonal Subschemes and Vector Bundles
Diagonal Subschemes and Vector Bundles
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对角子方案和向量丛
DOI：
10.4310/pamq.2008.v4.n4.a11
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发表时间：
2006
影响因子：
0.7
通讯作者：
V. Pati
中科院分区：
文献类型：
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作者：
P. Pragacz;V. Srinivas;V. Pati
我们研究了一个对角嵌入在它的笛卡儿正方形中的光滑簇X是X×X上秩为Dim(X)的向量丛的一个截面的零点格式，我们称之为对角线性质(D)。已知它适用于所有旗形SLN/P。我们主要考虑真光滑簇的情形，以及光滑流形的类似问题(“拓扑情形”)。在真簇的情况下，我们的主要新观察结果是(D)与X上的上同调平凡线丛之间的关系，这是由Serre关于秩2向量丛和余维2子方案的经典论点的变体，结合Serre对偶得到的。在此基础上，我们得到了关于曲面的几个详细结果，以及一些关于高维的结果。对于光滑仿射簇，我们观察到对于代数闭域上的仿射代数群，对角线实际上是一个完全交；因此，使用平凡丛，(D)成立。我们猜想存在(D)不成立的光滑仿射复簇，这就引出了一个关于投影模的有趣问题。拓扑论中的论点有不同的风格，论点来自同伦理论、拓扑K-理论、指数理论等。对角问题有三种变体，取决于我们想要的矢丛的类型(任意的、定向的或复杂的)。对于一类同伦映射，我们得到了对角线性质作为扩张问题的同伦理论表述。在球面、奇维复射影二次超曲面和具有几乎复结构的偶维≤6的流形上也得到了详细的结果。
We study when a smooth variety X, embedded diagonally in its Cartesian square, is the zero scheme of a section of a vector bundle of rank dim(X) on X × X. We call this the diagonal property (D). It was known that it holds for all flag manifolds SLn/P . We consider mainly the cases of proper smooth varieties, and the analogous problems for smooth manifolds (“the topological case”). Our main new observation in the case of proper varieties is a relation between (D) and cohomologically trivial line bundles on X, obtained by a variation of Serre’s classic argument relating rank 2 vector bundles and codimension 2 subschemes, combined with Serre duality. Based on this, we have several detailed results on surfaces, and some results in higher dimensions. For smooth affine varieties, we observe that for an affine algebraic group over an algebraically closed field, the diagonal is in fact a complete intersection; thus (D) holds, using the trivial bundle. We conjecture the existence of smooth affine complex varieties for which (D) fails; this leads to an interesting question on projective modules. The arguments in the topological case have a different flavour, with arguments from homotopy theory, topological K-theory, index theory etc. There are 3 variants of the diagonal problem, depending on the type of vector bundle we want (arbitrary, oriented or complex). We obtain a homotopy theoretic reformulation of the diagonal property as an extension problem for a certain homotopy class of maps. We also have detailed results in several cases: spheres, odd dimensional complex projective quadric hypersurfaces, and manifolds of even dimension ≤ 6 with an almost complex structure.
芒福德假射影平面的算术结构
DOI：
--
发表时间：
--
期刊：
Math. Z. (掲載予定)(決定)
影响因子：
--
作者：
Fumiharu Kato;Hiroyuki Ochiai;Fumiharu Kato
通讯作者：
Fumiharu Kato