Weights on cohomology, invariants of singularities, and dual complexes. (English) Zbl 1280.14005
The weight filtration (in the sense of mixed Hodge structure) for spaces related to singularities of complex algebraic varieties are studied. The weight-0 part reflects the combinatorics of resolution of singularities. Suppose that \(E\subset X\) is a simple normal crossing divisor in a smooth complete algebraic variety. It follows from the P. Deligne work on mixed Hodge structure [Publ. Math., Inst. Hautes Étud. Sci. 40, 5–57 (1971; Zbl 0219.14007)] that the cohomology of the dual complex \(\Sigma_E\) is isomorphic to \(W_0H^*(E)\) and it depends only on the complement \(X\setminus E\). Using weak factorization theorem [J. Włodarczyk, Invent. Math. 154, No. 2, 223–331 (2003; Zbl 1130.14014)] and [D. Abramovichet al., J. Am. Math. Soc. 15, No. 3, 531–572 (2002; Zbl 1032.14003)] it is shown that the homotopy type \(\Sigma_E\) is also determined by \(X\setminus E\) and moreover it depends only on the proper birational class of \(X\setminus E\). If \(E\) is the exceptional divisor of an isolated singularity resolution \(\pi:X\to Y\), then the homotopy type of \(\Sigma_E\) is an invariant of the singularity germ. For a rational singularity \(\Sigma_E\) is \(\mathbb{Q}\)-acyclic (but not necessarily contractible as conjectured in [D. A. Stepanov, Russ. Math. Surv. 61, No. 1, 181–183 (2006; Zbl 1134.14302)]; the counterexample was given by Payne). In the case when \(Y\) is a hypersurface rational singularity then \(\Sigma_E\) is contractible if \(\dim Y\geq 3\). If \(Y\) has Cohen-Macaulay singularities then \(\Sigma_E\) has the rational homotopy type of a wedge of spheres. Further nonisoleted singularities are studied. Again using weak factorization theorem it is shown that for \(y\in Y\) the dimensions of \(W_0H^i(\pi^{-1}(y))\) and \(H^1(\pi^{-1}(y))\) do not depend on a good resolution. Some estimates for remaining numbers \(\dim W_jH^i(\pi^{-1}(y))\) are obtained in general and for various classes of singularities. Similar estimates are obtained for the cohomology of the link \(H^*(Y,Y\setminus\{y\})\). The former results are achieved by applying the decomposition theorem of A. A. Beilinson, J. Bernstein and P. Deligne [Astérisque 100, 172 p. (1982; Zbl 0536.14011)].
Reviewer: Andrzej Weber (Warszawa)
MSC:
| 14F25 | Classical real and complex (co)homology in algebraic geometry |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
| 14F45 | Topological properties in algebraic geometry |
| 32S60 | Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) |
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