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48. On deformations of complex analytic structures, III, Stability theorems for complex structures

2015; Princeton University Press; Linguagem: Inglês

10.1515/9781400869862-018

K. Kodaira, D. C. Spencer,

Nonlinear Waves and Solitons

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 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