On the Hausdorff property of some Dolbeault cohomology groups
On the Hausdorff property of some Dolbeault cohomology groups
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一些Dolbeault上同调群的Hausdorff性质
DOI：
10.1007/s00209-012-1111-z
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发表时间：
2012
影响因子：
0.8
通讯作者：
Mei
中科院分区：
文献类型：
--
作者：
C. Laurent;Mei
设X是一个复流形。研究Cauchy-Riemann方程的闭程性质无论从层理论的观点还是从偏微分方程的观点来看都是非常重要的。根据相关上同调，这意味着相应的上同调是豪斯多夫的，因此是分离的。关于复流形上的上同调的Hausdorff性质有许多已知的结果（特别参见[16-18]）。例如，众所周知，对于Cn中的有界伪凸域D，Frechet空间C∞p，q（D）中的Dolbeault上同调Hp，q（D）对所有q > 0为零。还已知L2上同调也为零。关于拓扑不具有Hausdorff性质的上同调，甚至对于Cn中的域也是如此（一个例子在[13]，第14节中给出）。本文研究了各种函数空间中Cauchy-Riemann复形的对偶性以及相应的上同调的Hausdorff性质。这种对偶性是经典的，如果
Let X be a complex manifold. The study of the closed-range property of the Cauchy–Riemann equations is of fundamental importance both from the sheaf theoretic point of view and the PDE point of view. In terms of the associated cohomology, it means that the corresponding cohomology is Hausdorff, hence separated. There are many known results for the Hausdorff property of such cohomologies in complex manifolds (see in particular [16–18]). For instance, it is well-known that for a bounded pseudoconvex domain D inCn , the Dolbeault cohomology H p,q(D) in the Frechet space C∞p,q(D) vanishes for all q > 0. It is also known that the L2 cohomology also vanishes. Much less is known about the cohomologies whose topology does not have the Hausdorff property, even for domains in Cn (an example is given in [13], section 14). In this paper, we study duality of the Cauchy–Riemann complex in various function spaces and the Hausdorff property of the corresponding cohomologies. Such duality is classical if