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

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

Zbl 0752.32010
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