An Analytic Grothendieck Riemann Roch Theorem
An Analytic Grothendieck Riemann Roch Theorem
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DOI：
10.1016/j.aim.2016.02.031
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发表时间：
2014-04
期刊：
影响因子：
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通讯作者：
R. Douglas;Xiang Tang;Guoliang Yu
中科院分区：
文献类型：
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作者：
R. Douglas;Xiang Tang;Guoliang Yu
根据Baum，Douglas和Taylor的相对K-同调理论，我们将Boutet de Monvel Toeplitz指数定理推广到具有孤立奇点的复流形上。我们应用这个指标定理来研究Arveson-Douglas猜想。设Bm是Cm中的单位球，I是多项式代数C[z1，⋯，zm]中的理想。证明了当零簇Zi是只有孤立奇点的完备交空间且与单位球面S 2m−1横交时，L a2(Bm)中C[z1，⋯，zm]在I的闭包上的表示以及相应的商空间Qi实质上是正规的.进一步，我们证明了Qi上Toeplitz算子的一个指标定理，证明了商空间Qi上的C[z1，⋯，zm]的表示给出了边界ZI的基本类∩S 2m−1。在附录中，我们和王凯证明了如果f∈L a2(Bm)在Zi∩Bm上消失，则f包含在L a2(Bm)中的理想i的闭包内。
We extend the Boutet de Monvel Toeplitz index theorem to complex manifolds with isolated singularities following the relative K-homology theory of Baum, Douglas, and Taylor for manifolds with boundary. We apply this index theorem to study the Arveson–Douglas conjecture. Let B m be the unit ball in C m, and I an ideal in the polynomial algebra C [z 1,⋯, z m]. We prove that when the zero variety Z I is a complete intersection space with only isolated singularities and intersects with the unit sphere S 2 m− 1 transversely, the representations of C [z 1,⋯, z m] on the closure of I in L a 2 (B m) and also the corresponding quotient space Q I are essentially normal. Furthermore, we prove an index theorem for Toeplitz operators on Q I by showing that the representation of C [z 1,⋯, z m] on the quotient space Q I gives the fundamental class of the boundary Z I∩ S 2 m− 1. In the appendix, we prove with Kai Wang that if f∈ L a 2 (B m) vanishes on Z I∩ B m, then f is contained inside the closure of the ideal I in L a 2 (B m).