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Deformation of singularities and the homology of intersection spaces. (English) Zbl 1269.32017

A singular complex algebraic variety can be compared with a smooth one by resolution or deformation. Middle perversity intersection cohomology is stable under small resolutions, but not under deformations. The first author [Intersection spaces, spatial homology truncation, and string theory. Dordrecht: Springer (1997; Zbl 1219.55001)] introduced a new approach to intersection cohomology at the space level. For complex projective algebraic hypersurfaces with an isolated singularity it is shown that the cohomology of intersection spaces is stable under smooth deformations in all degrees except possibly the middle, and in the middle degree precisely when the monodromy action on the cohomology of the Milnor fiber is trivial. In many situations, the isomorphism is shown to be a ring homomorphism induced by a continuous map; note that the intersection space will in general not be algebraic. Then the rational cohomology of intersection spaces can be endowed with a mixed Hodge structure compatible with Deligne’s mixed Hodge structure on the ordinary cohomology of the singular hypersurface. The theory is illustrated by a number of examples, in particular of quintic threefolds.

MSC:

32S30 Deformations of complex singularities; vanishing cycles
55N33 Intersection homology and cohomology in algebraic topology
14J33 Mirror symmetry (algebro-geometric aspects)

Citations:

Zbl 1219.55001
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References:

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