## 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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### References:

 [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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