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Characteristic divisors on complex manifolds. (English) Zbl 0752.14008

In this paper, following Rokhlin’s definition of characteristic submanifolds, the authors introduce the notion of a characteristic divisor on a complex manifold. This means an effective divisor \(W\) in a complex (possibly non-compact) manifold \(M\), such that \(| W|\) is compact, and \(W\) is of the form \(W=2D-K\), where \(K\) is a divisor of \(\omega_ M\), the canonical sheaf, \(D\) is some divisor in \(M\) and \(| K|\), \(| D|\) are both compact. When \(n=\dim(M)\) is of the form \(n=4k+2\) the \(\bmod 2\)-index of the characteristic divisor \(W\) is defined as \(h(W,{\mathcal D})=\sum^{2k}_{i=0}h^{2i}(W,{\mathcal D}_{\mid W})\bmod 2\), where \({\mathcal D}\) is the bundle of \(D\). If \(M\) is compact of \(\mathbb{C}\)- dimension \(4k+2\) the authors prove that \(h(W,{\mathcal D})=\chi(M,{\mathcal D})\bmod 2\). This generalizes Rokhlin’s theorem to higher dimensions and to possibly singular and non-reduced, reducible divisors. When \(W\) is nonsingular this result is in fact a direct application of a result due to Atiyah.
Then the methods to prove the result above are applied to study desingularizations of isolated Gorenstein singularities with special reference to the surface case.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
32Q99 Complex manifolds
32S45 Modifications; resolution of singularities (complex-analytic aspects)
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