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Polyhedral resolutions of algebraic varieties. (English) Zbl 0602.14012

In this paper, a method of constructing smooth simplicial resolutions which are small is described. (These resolutions are used to compute the mixed Hodge structure of complex projective varieties.)
Let X be an algebraic variety, \(\Sigma\subset X\) the singular locus and \(\pi: \tilde X\to X\) a resolution of singularities. Then the abstract mapping cylinder \(C(\pi)=[\tilde X\leftarrow {\tilde \Sigma}\to^{\pi}\Sigma]\), \({\tilde \Sigma}=\pi^{-1}(\Sigma)\), is a simplicial space over the unit interval \(I^ 1=[0,1]\). Although the singular locus of \(X| I^ 1:=C(\pi)\) may not be empty, it must be of smaller dimension than the singular locus of \(X| I^ 0:=X\). Therefore the main idea is to go on by induction in order to construct a smooth simplicial resolution. Suppose \(X| I^ p\), \(p\geq 1\), has been constructed with \(X| \{0\}\times I^{p-1}\) smooth and \(AX| I^ p\) epimorphic, \((AX)_{\sigma}=X_{\sigma}\), \(\sigma \not\in \{0\}\times I^{p-1}\), and \((AX)_{\{0\}\times \sigma}=im(X_{I^ 1\times \sigma}\to X_{\{0\}\times \sigma})\). Then one can show that a birational morphism \(\pi: (^{\sim})| I^ p\to AX| I^ p,\) \((^{\sim})\) smooth and epimorphic, exists. Thus it is possible to construct an adequate resolution \(\pi_ p: \tilde X| I^ p\to X| I^ p\) such that \(X| I^{p+1}:=C(\pi_ p)\) satisfies the two conditions above. At least, this process stops after \(n=\dim (X)\) steps and one obtains a smooth simplicial resolution.
Reviewer: M.Heep

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14A10 Varieties and morphisms
55U10 Simplicial sets and complexes in algebraic topology
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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[1] James A. Carlson, Extensions of mixed Hodge structures, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn — Germantown, Md., 1980, pp. 107 – 127.
[2] Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5 – 57 (French). Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5 – 77 (French).
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