On the Infinite Dimensionality of the Middle $L^2$ Cohomology of Complex Domains

Abstract

In 1983, H. Donnelly and C. Fefferman [3] discovered a strikingly new phenomenon in complex analysis by establishing a vanishing theorem for the invariant L cohomology. According to their result, for any strictly pseudoconvex bounded domain D in a Stein manifold of dimension n, the L cohomology groups of D vanish except for that of the middle degree n. Their proof is based on a rather original estimate of H. Donnelly and F. Xavier [4] which may well be called the Hardy's inequality on manifolds. As for this new estimate, K. Takegoshi and the author [10] noticed that it is a direct consequence of Jacobi's identity, and applied it later to show an extension theorem for L holomorphic functions (cf. [9]). It has applications to the Hodge theory on singular complex spaces, too. (cf. [7] and [8]). As for the L cohomology in the middle degree, it was also shown in [3] that their (p, q) components are all infinite dimensional. Compared to the vanishing theorem, the basic reason for such infinite dimensionality seems to remain less transparent, although it is discussed under several different geometric situations (cf. [1], [6]). Therefore it might make sense to ask for a general geometric situation under which the infinite dimensionality is valid. The present paper is meant for that purpose. Let D be a domain in a connected complex manifold of dimension n. By definition, a regular boundary point of D is a point p e dD which admits a realvalued C°° function cp defined on a neighbourhood U B p such that dcp(p) ^ 0 and [/n D = {xeU; <p(x) < 0}. We call such <p a defining function of dD around p. A regular boundary point pedD shall be called non-degenerate if rank dde = n at p, for any defining function (p around p. Then our result is stated as follows.