\(L^{2}\) Serre duality on domains in complex manifolds and applications (Q2839943)

The authors generalize the following Serre duality theorem:NEWLINENEWLINEIf each of the two operators NEWLINE\[NEWLINE\mathcal C_{p,q-1}^{\infty}(\Omega, E) \overset{\bar\partial_{E}} \rightarrow \mathcal C_{p,q}^{\infty}(\Omega, E)\overset{\bar\partial_{E}} \rightarrow \mathcal C_{p,q+1}^{\infty}(\Omega, E)NEWLINE\]NEWLINE has closed range with respect to the natural Fréchet topolgy, then the dual of the topological vector space \(H^{p,q}(\Omega,E)\) with respect to the quotient Fréchet topology can be identified with the space \(H^{n-p,n-q}_{\text{comp}}(\Omega,E^*)\) with the quotient topology in a canonical way. The converse is also true.NEWLINENEWLINEHere \(E\) is a holomorphic vector bundle over a complex manifold \(\Omega\) (not necessarily compact), \(E^*\) is it's dual. \(H^{p,q}(\Omega, E)\) and \(\mathcal C_{p,q}^{\infty}(\Omega, E)\) are the \((p,q)\)-th Dolbeault cohomology group for \(E\)-valued forms on \(\Omega\) and the space of smooth \(E\)-valued \( (p,q)\)-forms on \(\Omega\), repectively. The subscript \(_{\text{comp}}\) is to specify that one takes only forms with compact support, \(n\) is the dimension of \(\Omega\) and \(\bar\partial_{E}\) is the Cauchy-Riemann operator. The spaces of compactly supported forms are endowed with the natural inductive limit topology.NEWLINENEWLINEThe authors use \(L^2\)- methods developed by Kodaira in the setting of compact complex manifolds, extend them to the setting of non-compact Hermitian manifolds and obtain the following \(L^2\) analog of the Serre duality theorem:NEWLINENEWLINEThe following conditions \((1)\) and \((2)\) are equivalent.NEWLINENEWLINE\((1)\) Each of the two operators NEWLINE\[NEWLINEL_{p,q-1}^{2}(\Omega, E) \underrightarrow{\bar\partial_{E}}^{\dagger}L_{p,q}^{2}(\Omega, E)\;\underrightarrow{\bar\partial_{E}}^{\dagger} L_{p,q+1}^{2}(\Omega, E) NEWLINE\]NEWLINENEWLINENEWLINEhas closed range. NEWLINENEWLINENEWLINENEWLINE \((2)\) The map \(\star_{E}: L_{p,q}^{2}(\Omega, E)\to L_{n-p,n-q}^{2}(\Omega, E)\) induces a conjugate-linear isomorphism of Hilbert spaces NEWLINE\[NEWLINEH^{p,q}_{L^2}(\Omega, E)\cong H^{p,q}_{L^2,\text{comp}}(\Omega, E^{*}).NEWLINE\]NEWLINE In fact the latter space can be identified with the Hilbert space dual of the former.NEWLINENEWLINEHere \(\star_{E}\) is the Hodge star operator. And the \({}^{\dagger}\) notation is used to underline the fact that the domain is not the whole space on the left but just some linear subspace.NEWLINENEWLINEThere is a very good introduction chapter in this paper which helps a lot to understand the somewhat complicated notation and setting of the authors.NEWLINENEWLINEThe authors discuss some applications of their generalization and point out a misleading statement in the paper [\textit{G. M. Henkin} and \textit{A. Iordan}, Asian J. Math. 4, No. 4, 855--884 (2000; Zbl 0998.32021)] and correct it.