Intersection Cohomology. Simplicial Blow-up and Rational Homotopy

Abstract

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 Sullivan's differential forms over the field Q. In a second step, we use these forms to extend Sullivan's 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 (CDGA's) due to Hovey. We establish also the existence and unicity of a positively graded, minimal model of some perverse CDGA's, 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 PLpseudomanifold 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.

Mixed Hodge structures on the rational models of intersection spaces

“…The section 6 is completely devoted to the proof of the theorem 6.1 : suppose X to be a complex projective algebraic threefold with isolated singularities such that there exist a resolution of singularities with a smooth exceptional divisor, then if the links are simply connected the intersection spaces I p X are formal topological spaces for any perversity p. The proof being rather long and intricate, we made the choice of giving it its own section. This result goes well with the result of [7,Theorem E p.76] stating that any nodal hypersurface in CP 4 is intersection-formal.…”

“…(2) This convention also has to be compared at the level of algebraic models with [7], where a p-perverse rational model of a cone cL on a topological space L of dimension n is given by a truncation in degree p(n) of the rational model of L. In our case, a rational model of the intersection space I p cL is then given by a unital cotruncation in degree p(n) of the rational model of L.…”

Hodge theory for intersection space cohomology

“…Then the results of [7] show that under a mild torsion-freeness hypothesis on the homology of links, complex projective hypersurfaces X with only isolated singularities, whose monodromy operators on the cohomology of the associated Milnor fibers are trivial, are middle-perversity (m) intersection formal, since there is an algebra isomorphism from HI * m (X) to the ordinary cohomology algebra of a nearby smooth deformation, which is formal, being a Kähler manifold. This agrees nicely with the result of [10,Section 3.4], where it is shown that any nodal hypersurface in CP 4 is "totally" (i.e. with respect to an algebra that involves all perversities at once) intersection formal.…”

Singular decompositions of a cap product

“…We recall the basic definitions and properties we need, sending the reader to [7], [5], [1] or [2] for more details. Definition 1.1.…”

“…Cup product and intersection product. In [1], [2], we define from the local structure on the Euclidean simplices, a cup product in intersection cohomology, induced by a chain map ∪ : N k 1 TW,p 1 (X) ⊗ N k 2 TW,p 2 (X) → N k 1 +k 2 TW,p 1 +p 2 (X).…”