Analytic cohomology in Fréchet spaces
Analytic cohomology in Fréchet spaces
复制标题
Fréchet 空间中的解析上同调
DOI：
10.4310/cag.2003.v11.n1.a2
复制
发表时间：
2003
影响因子：
0.7
通讯作者：
L. Lempert
中科院分区：
文献类型：
--
作者：
L. Lempert
五十多年来，束上同调群的计算一直是复杂分析和几何的中心问题。似乎在无限维设置中，束上同调是由 Douady 在 [Do] 中首次研究的。在本文中，我们解决了当 q > 1 并且 fi 是复 Frechet 空间 X 中的伪凸开集时，上同调群 H(Q,, O) 是否消失的问题。这里 O 表示 X 中全纯函数的胚芽束，如果 Q n Z 对于所有有限维子空间 Z C X 都是伪凸，则 O 称为伪凸；特别是 X 本身是伪凸的。到目前为止，已经很清楚，余学消失对所讨论的空间 X 的几何形状很敏感。在 Dineen 提出第一个非零例子之后，Meise 和 Vogt 制定了 H(X, O) — 0 成立的必要条件，请参阅 [D1,MV]。所有 Banach 空间以及快速递减序列的空间 s（及其子空间）都满足这种条件，即主导范数（DN）的存在。另一方面，我们在[LI]中证明，如果 ft 在无条件基的 Banach 空间 X 中是伪凸且 q > 1，则 ll{$\,0) = 0；例如，X 可以是空间 P，1 < p < oo，或 L[0,1]，1 < p < oo。在这里我们将证明，在非规范弗雷谢空间中也可以证明上同调消失，特别是在 5 中。空间 5 在弗雷谢空间中值得特别关注，因为几何中经常出现的函数空间与其同构；它还具有一定的普遍性。研究部分得到美国国家科学基金会 (NSF) 资助的支持。从技术上讲，[MV] 处理 Dolbeault 而不是层上同调群，但那里的推理也表明 H(X, O) = 0 意味着 (DN)。
The computation of sheaf cohomology groups has been a central problem of complex analysis and geometry for over fifty years now. It appears that in an infinite dimensional setting sheaf cohomologies were first investigated by Douady in [Do]. In this paper we address the question whether the cohomology groups H(Q,, O) vanish when q > 1 and fi is a pseudoconvex open set in a complex Frechet space X. Here O denotes the sheaf of germs of holomorphic functions in X and O is called pseudoconvex if Q n Z is pseudoconvex for all finite dimensional subspaces Z C X; in particular X itself is pseudoconvex. By now it has become clear that coholomogy vanishing is sensitive to the geometry of the space X in question. After the first examples of nonvanishing by Dineen, Meise and Vogt formulated a necessary condition for H(X, O) — 0 to hold, see [D1,MV]. This condition, the existence of a dominant norm, or (DN), is met by all Banach spaces and also by the space s of rapidly decreasing sequences (and its subspaces). On the other hand we proved in [LI] that ll{$\,0) = 0 if ft is pseudoconvex in a Banach space X with unconditional basis and q > 1; for example X could be the space P, 1 < p < oo, or L[0,1], 1 < p < oo. Here we shall show that it is possible to prove cohomology vanishing in nonnormable Frechet spaces as well, in particular in 5. The space 5 deserves special attention among Frechet spaces as function spaces that frequently occur in geometry are isomorphic to it; it also has certain universality properties. Research partially supported by an NSF grant. technically, [MV] deals with Dolbeault rather than sheaf cohomology groups, but the reasoning there also shows that H(X, O) = 0 implies (DN).