David Chataur

Intersection Cohomology. Simplicial Blow-up and Rational Homotopy

Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. We do it simplicially in the setting of a filtered version of face sets, also called simplicial sets without degeneracies, in the sense of C.P. Rourke and B.J. Sanderson. We define perverse local systems over filtered face sets and intersection cohomology with coefficients in a perverse local system. In particular, as announced above when X is a pseudomanifold, we get a perverse local system of cochains quasi-isomorphic to the intersection cochains of Goresky and MacPherson, over a field. We show also that these two complexes of cochains are quasi-isomorphic to a filtered version of Sullivans differential forms over the field Q. In a second step, we use these forms to extend Sullivans presentation of rational homotopy type to intersection cohomology. For that, we construct a functor from the category of filtered face sets to a category of perverse commutative differential graded Q-algebras (cdgas) due to Hovey. We establish also the existence and unicity of a positively graded, minimal model of some perverse cdgas, including the perverse forms over a filtered face set and their intersection cohomology. Finally, we prove the topological invariance of the minimal model of a PL-pseudomanifold whose regular part is connected, and this theory creates new topological invariants. This point of view brings a definition of formality in the intersection setting and examples are given. In particular, we show that any nodal hypersurface in CP(4), is intersection-formal.

Rational homotopy of complex projective varieties with normal isolated singularities

Abstract Let X be a complex projective variety of dimension n with only isolated normal singularities. In this paper, we prove, using mixed Hodge theory, that if the link of each singular point of X is ( n - 2 )

Fiberwise localization and the cube theorem

{(n-2)}

Blown-up intersection cohomology

-connected, then X is a formal topological space. This result applies to a large class of examples, such as normal surface singularities, varieties with ordinary multiple points, hypersurfaces with isolated singularities and, more generally, complete intersections with isolated singularities. We obtain analogous results for contractions of subvarieties.

Realizing Operadic Plus-constructions as Nullifications

In this paper we explain when it is possible to construct fiberwise localizations in model categories. For pointed spaces, the general idea is to decompose the total space of a fibration as a diagram over the category of simplices of the base and replace it by the localized diagram. This of course is not possible in an arbitrary category. We have thus to adapt another construction which heavily depends on Mathers cube theorem. Working with model categories in which the cube theorem holds, we propose a few equivalent conditions under which fiberwise nullifications exist. We show that these techniques apply to yield a fiberwise plus-construction for differential graded algebras over cofibrant operads.

Dualit\'e de Poincar\'e et Homologie d'intersection

In previous works, we have introduced the blown-up intersection cohomology and used it to extend Sullivans minimal models theory to the framework of pseudomanifolds, and to give a positive answer to a conjecture of M. Goresky and W. Pardon on Steenrod squares in intersection homology. In this paper, we establish the main properties of this cohomology. One of its major feature is the existence of cap and cup products for any filtered space and any commutative ring of coefficients, at the cochain level. Moreover, we show that each stratified map induces an homomorphism between the blown-up intersection cohomologies, compatible with the cup and cap products. We prove also its topological invariance in the case of a pseudomanifold with no codimension one strata. Finally, we compare it with the intersection cohomology studied by G. Friedman and J.E. McClure. A great part of our results involves general perversities, defined independently on each stratum, and a tame intersection homology adapted to large perversities.