L 2 SERRE DUALITY ON DOMAINS IN COMPLEX MANIFOLDS AND APPLICATIONS
L 2 SERRE DUALITY ON DOMAINS IN COMPLEX MANIFOLDS AND APPLICATIONS
复制标题
复杂流形域上的 L 2 SERRE 对偶性及其应用
DOI：
10.1090/s0002-9947-2012-05511-5
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发表时间：
2010
影响因子：
1.3
通讯作者：
Mei
中科院分区：
文献类型：
--
作者：
D. Chakrabarti;Mei
建立了复流形中域上涉及@-算子的Hilbert空间实现的对偶的L-2形式的Serre对偶.这种对偶性被用来研究具有规定支集的@-方程的解。给出了形式的@-闭扩张以及CR函数的Bochner-Hartogs型扩张的应用。Comp(；E�)与商拓扑，其中我们赋予紧支集形式的空间以自然归纳极限拓扑。事实上，(1)中的两个映射具有闭合范围的条件也是对偶定理成立的必要条件(见(9)；关于这种类型的进一步结果，也见(26，27，28)。)Serre的原始证明(33)是基于鞘理论和拓扑向量空间理论的。当是紧复流形时，Kodaira使用Hodge理论(见(23)或(7))给出了这一结果的不同方法。在这篇注记中，我们将Kodaira的方法推广到非紧的厄米特流形上，得到了L 2类似的Serre对偶。使用L 2方法的特殊情形已经在许多情况下出现过(例如，见(25)或(11，定理5.1.7)和(19，20)。)我们的处理旨在将这些结果流线化和系统化，重点是非紧流形，并指出它与L选择柯西-黎曼算子的2-实现，或者选择形式复形拉普拉斯的L 2-实现的边界条件密切相关。L的2-对偶可以用多种方式解释。在一个层面上，它是具有@-诺依曼边界条件的标准�-拉普拉斯算子和具有对偶(@-狄利克雷)边界条件的c-拉普拉斯算子之间的对偶。用另一种方法，可以将L 2中关于@-方程的解的结果转化为关于@c方程的解的陈述。这导致了@-柯西问题的解，即具有规定支持的@-方程的解。问题的核心在于…的存在
An L 2 version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the @-operator is established. This duality is used to study the solution of the @-equation with prescribed support. Applications are given to @-closed extension of forms, as well to Bochner-Hartogs type extension of CR functions. comp (;E � ) with the quotient topology, where we endow spaces of compactly supported forms with the natural inductive limit topology. In fact, condition that the two maps in (1) have closed range is also necessary for the duality theorem to hold (see (9); also see (26, 27, 28) for further results of this type.) Serre's original proof (33) is based on sheaf theory and the theory of topological vector spaces. A different approach to this result, in the case when is a compact complex manifold, was given by Kodaira using Hodge theory (see (23) or (7).) In this note we extend Kodaira's method to non-compact Hermitian manifolds to obtain an L 2 analog of the Serre duality. Special cases of Serre-duality using L 2 methods have appeared before in many contexts (see (25), or (11, Theorem 5.1.7) and (19, 20), for example.) Our treatment aims to streamline and systematize these results, with emphasis on non-compact manifolds, and point out its close relation with the choice of L 2 -realizations of the Cauchy-Riemann operator @, or alternatively, choice of boundary conditions for the L 2 -realizations of the formal complex Laplacian @E#E + #E@E. The L 2 -duality can be interpreted in many ways. At one level, it is a duality between the standard � - Laplacian with @-Neumann boundary conditions, and thec-Laplacian with dual ( "@-Dirichlet") boundary conditions. Using another approach, results regarding solution of the @-equation in L 2 can be converted to statements regarding the solution of the @c equation. This leads to a solution of the @-Cauchy problem, i.e., solution of the @-equation with prescribed support. At the heart of the matter lies the existence of