奇异复空间与有理奇点上的$L^2$ -Serre对偶
$L^2$-Serre duality on singular complex spaces and rational singularities
摘要：
在本文中，我们设计了一个版本的拓扑$L^2$ -Serre对偶奇异复空间的任意奇点。利用这一对偶性，导出了奇异空间上$\bar{\partial}$ -方程的各种新的$L^2$ -消灭定理.证明了具有有理奇点的复空间在$L^2$ -意义下相对于$\bar{\partial}$ -方程表现得相当驯服.更精确地说：奇点是有理的当且仅当$L^2$ - $\bar{\partial}$ -复形在该点上是正合的。从而得到了结构层在有理奇点上的$L^2$ - $\bar{\partial}$ -分解.
英文摘要：
In the present paper, we devise a version of topological $L^2$-Serre duality for singular complex spaces with arbitrary singularities. This duality is used to deduce various new $L^2$-vanishing theorems for the $\bar{\partial}$-equation on singular spaces. It is shown that complex spaces with rational singularities behave quite tame with respect to the $\bar{\partial}$-equation in the $L^2$-sense. More precisely: a singular point is rational if and only if the $L^2$-$\bar{\partial}$-complex is exact in this point. So, we obtain an $L^2$-$\bar{\partial}$-resolution of the structure sheaf in rational singular points.
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