The virtual number of \(D_ \infty\) points. II. (English) Zbl 0752.32011

de Jong, Johan; de Jong, Theo

Topology 29, No. 2, 185-188 (1990).

Let \(X\) be a divisor in a compact complex manifold \(Y\) of dimension \(n+1\). Suppose \(X\) has a one-dimensional singular locus \(\Sigma\), endowed with the reduced structure, such that \(X\) has only \(A_ \infty\) singularities on \(\Sigma-\{0\}\). Let \((X,\sigma)\) be a germ of the nonisolated singularity of \(X\) at \(\sigma\in \Sigma\). Then the virtual number \(VD_ \infty(X,\sigma)\) of \(D_ \infty\) points of the hypersurface germ \((X,\sigma)\) is defined [see Part I, the second author, ibid., 175-184 (1990; see the preceding review)]. The authors prove the following formula for the total virtual number of \(D_ \infty\) points of \(X:\sum_{\sigma\in\Sigma}VD_ \infty(X,\sigma)=\langle 2K_ Y+nX,\Sigma\rangle+4\chi(\Sigma,{\mathcal O}_ \Sigma)\), where \(K_ Y\) denotes the canonical divisor on \(Y\) and \(\langle , \rangle\) the intersection product on \(Y\).
As a good example the classical case of a surface in \(\mathbb{P}^ 3\) with only ordinary singularities is considered.

MSC:

32S20	Global theory of complex singularities; cohomological properties
32S10	Invariants of analytic local rings

Keywords:

nonisolated singularity; virtual number

Citations:

Zbl 0752.32010