On the infinite dimensionality of the middle \(L^ 2\) cohomology of complex domains
The author is dealing with \(L^ 2\)-cohomology groups \(H^{p,q}_{(2)}(D)\) of a bounded domain D in a complex manifold. Let \(p\in \partial D\) be such that \(rank \partial {\bar \partial}e^{\phi}=n\) \(=\dim D\) (\(\phi\) \(=\) defining function for \(\partial D\) on a neighborhood U of p) and assume that \(C^{-1} ds^ 2<(-\phi)^{-a} ds^ 2_ U+(-\phi)^{-b} \partial \phi {\bar \partial}\phi <c ds^ 2\) on \(U\cap D\) where a, b, c are constants \(1\leq a\leq b\leq a+3,\) \(ds^ 2\) is a hermitian metric on D, \(ds^ 2_ U\) some hermitian metric on U. Then dim \(H^{p,q}_{(2)}(D)=+\infty\) when \(p+1=n\). In particular he rediscovers a theorem proved by Donnelly and Fefferman in the case D is strictly pseudoconvex in a Stein manifold.
scientific article; zbMATH DE number 4159074