Contractions, Fredholm operators and the cohomology of vector bundles on Hopf manifolds
Let \(f:(\mathbb{C}^n,0) \to (\mathbb{C}^n,0)\) be a contraction, that is an automorphism of \(\mathbb{C}^n\) fixing 0 and with the property that the linear map \(f'(0)\) has spectral radius less than one. The group generated by \(f\) acts freely and properly discontinuously on \(\mathbb{C}^n \backslash \{0\}\). The quotient space \(X\) is called a primary Hopf manifold, with projection \(p:\mathbb{C}^n \backslash \{0\}\to X\). Let \(A^p(E)\) denote the sheaf of holomorphic \(p\)-forms with coefficients in \(E\). The main result of the note is the following: Let \(X\) be a primary Hopf manifold of dimension greater or equal than 3 and let \(E\) be a holomorphic vector bundle on \(X\), with \(p^*E\) trivial. Then for every natural number \(p\), one has: 1) \(h^0(X,A^p(E)) = h^1(X,A^p (E))\), 2) \(h^j(X,A^p (E))=0\), for \(1<j<n-1\), 3) \(h^{n-1}(X,A^p(E)) = h^n(X,A^p(E))\). The proof uses an idea of Douady of computing the cohomology spaces via an adapted Čech complex, and on this complex, the fact that the index of a Fredholm operator between Fréchet spaces is invariant under compact perturbations.
scientific article; zbMATH DE number 937070
scientific article