ON DEFORMATIONS OF COMPLEX ANALYTIC STRUCTURES, III. STABILITY THEOREMS FOR COMPLEX STRUCTURES
ON DEFORMATIONS OF COMPLEX ANALYTIC STRUCTURES, III. STABILITY THEOREMS FOR COMPLEX STRUCTURES
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DOI：
10.2307/1969879
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发表时间：
1960
影响因子：
4.9
通讯作者：
K. Kodaira;D. Spencer
中科院分区：
文献类型：
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作者：
K. Kodaira;D. Spencer
本文是类似标题的系列文章中的第三篇，作者将椭圆型微分方程理论应用于复杂解析结构的变形问题，导出了第一篇论文第二节中未详细证明的基本定理，而第一篇和第二篇论文(参考文献[8])的许多结果都依赖于该基本定理。作者还证明了各种稳定性定理，其中一些已在第一篇文章中阐述过。虽然这是一个续篇，但本文并不假定有关于第一篇和第二篇论文的知识。本文分为两个部分。第一部分系统地讨论了紧可微流形上的强椭圆型偏微分方程组，它依赖于几个实参数。第一部分的主要结果总结在第一节的定理2-5中。关键命题1的证明由L.Nirenberg传达给作者。在本文的第二部分，由第一部分的结果得到了紧致复流形的可微族的稳定性定理。紧致复流形的可微族是一族复流形，其复结构可微地依赖于参数t在连通可微流形上的运动(见下文第四节定义1)。第四节和第五节中的稳定性定理涉及紧致复流形V的可微族，每个紧致复流形上都给出了一个复解析向量丛B(见下文第四节的定义2)。第四节的定理6是“上半连续原理”，它断言：系数在B的全纯截面芽的鞘中的V的上同调的维度是t的上半连续函数(第一篇论文的定理2.1，参考文献[8])。这一原理最早是由作者在他们的论文[7]中提出的，最近被证明具有类似于任意域上代数簇的上同调(见Chow和Igusa[1])。定理7是第一篇文献[8]的“基本定理”。在第五节中，引入了作用于dif43的线性空间上的拉普拉斯函数在Kodaira[6]意义下的特征函数的典范基
In this paper, the third in a series under similar title, the authors apply the theory of elliptic differential equations to questions concerning the deformation of complex analytic structures and derive the fundamental theorem stated without detailed proof in Section 2 of the first paper, on which many of the results in the first and the second papers (reference [8] of the Bibliography) depend. The authors prove also various stability theorems, some of which are stated in the first paper. A knowledge of the first and the second papers is not assumed in this paper though it is a sequel. This paper is divided into two parts. Part I contains a systematic treatment of strongly elliptic systems of partial differential equations on a compact differentiable manifold which depend on several real parameters. The principal results of Part I are summarized in Theorems 2-5 of Section 1. The proof of the crucial Proposition 1 was communicated to the authors by L. Nirenberg. In Part II of the the present paper, stability theorems for a differentiable family of compact complex manifolds are derived from the results of Part I. A differentiable family of compact complex manifolds is a family of complex manifolds V, whose complex structures depend differentiably on a parameter t moving on a connected differentiable manifold (see Definition 1 of Section 4 below). Stability theorems in Sections 4 and 5 are concerned with a differentiable family of compact complex manifolds V, on each of which a complex analytic vector bundle B, is given (see Definition 2 of Section 4 below). Theorem 6 of Section 4 is the "principle of upper semi-continuity" which asserts that the dimension of the cohomology of V, with coefficients in the sheaf of germs of holomorphic sections of B, is an upper semi-continuous function of t (Theorem 2.1 of the first paper, reference [8]). This principle, first stated by the authors in their paper [7], has recently been shown to have an analogue for cohomologies of algebraic varieties over arbitrary fields (see Chow and Igusa [1]). Theorem 7 is the "fundamental theorem" of the first paper of reference [8]. In Section 5 canonical bases of eigenfunctions, in the sense of Kodaira [6], are introduced for the Laplacian acting on the linear space of dif43