The Serre duality theorem for holomorphic vector bundles over a strongly pseudo-convex manifold (Q2472524)

This paper in Cauchy--Riemann geometry presents a proof of an analogue of Serre duality for certain CR-manifolds. Theorem~1. Let \(M\) be a compact strongly pseudoconvex CR-manifold of real dimension \(2n-1\), \(E\) a holomorphic vector bundle over \(M\), \(E^*\) its dual, and \(H^{p,q}(M;E)\) the space of harmonic \((p,q)\)-forms on \(M\) with values in \(E\). Then there is a canonical isomorphism \(H^{p,q}(M;E)\cong H^{n-p,n-q-1}(M;E^*)\) for any \((p,q)\). The above isomorphism is induced by an analogue \(\#\) for vector valued forms of the Hodge star operator for ordinary forms. The proof sets up the necessary machinery to define \(\#\) and establish its relation with the Dolbeault operator \(\overline\partial\) and its dual to find that a form \(\#\psi\) is \(\overline\partial_{E^*}\)-closed (resp.\ coclosed) if and only if \(\psi\) is \(\overline\partial_E\)-coclosed (resp.\ closed). The proof also make use of the finiteness of the harmonic spaces and of the work of \textit{N. Tanaka} [`A differential geometric study on strongly pseudoconvex manifolds' (1975; Zbl 0331.53025)]. The paper is carefully written and pleasant to read.