The \(L_ 2-{\bar \partial}\)-cohomology of an algebraic surface
Let S be a complex projective surface. Denote by \(H_{(2)}^{p,q}(S- Sing S)\) the corresponding \(L_ 2-{\bar \partial}\)-cohomology groups (defined via the Riemannian metric induced by a projective imbedding). Let \(\chi_ 2(S)=\Sigma (-1)^ i \dim H_{(2)}^{0,i}(S-Sing S)\) be the \(L_ 2\)-arithmetic genus of S (whenever \(H_{(2)}^{0,i}\) are finite-dimensional over \({\mathbb{C}}).\) The author proves the following results: 1. If S is normal, then each \(H_{(2)}^{0,i}(S\)-Sing S) is finite-dimensional. 2. Let \(\rho\) : \(X\to S\) be a desingularization such that \(Z=\rho^{-1}(Sing S)\) is a divisor with normal crossings. Then \(\chi\) (\({\mathcal O}_ X(Z-| Z|))\) is a birational invariant of S, \[ \chi ({\mathcal O}_ X)\geq \chi ({\mathcal O}_ X(Z-| Z|))\geq \chi_ 2(S), \] and the first inequality is in general strict. In such a way the author gives (in dimension 2) contra-examples to an older conjecture of MacPherson (stating that \(\chi\) (\({\mathcal O}_ X)=\chi_ 2(S)\), S projective variety and X a desingularization of it). Comments and results about revised conjectures are also given.
scientific article; zbMATH DE number 4116938