Remarks on the Combinatorial Intersection Cohomology of Fans

Abstract

Abstract:We review the theory of combinatorial intersection cohomology of fans developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal properties but makes sense even for non-rational fans, which do not define a toric variety. As a result, a number of interesting results on the toric g and h polynomials have been extended from rational polytopes to general polytopes. We present explicit complexes computing the combinatorial IH in degrees one and two; the degree two complex gives the rigidity complex previously used by Kalai to study g 2 . We present several new results which follow from these methods, as well as previously unpublished proofs of Kalai that g k (P ) = 0 implies g k (P * ) = 0 and g k+1 (P ) = 0.For a d-dimensional convex polytope P , Stanley [St2] defined a polynomial invariant h(P, t) = d k=0 h k (P )t k of its face lattice, which is usually called the "generalized" or "toric" h-polynomial of P . It is "generalized" in that it extends a previous definition from simplicial polytopes to general polytopes, while the adjective "toric" refers to the fact that if P is a rational polytope (meaning that all its vertices have all coordinates in Q), then the coefficients of h(P, t) are intersection cohomology Betti numbers of an associated projective toric variety X P :(1) h k (P ) = dim R IH 2k (X P ; R).

The decomposition theorem, perverse sheaves and the topology of algebraic maps

“…The endomorphism algebra contains in particular the idempotents giving the projection of f * Q X [n] on the irreducible summand of the canonical decomposition (23). Since, again by semismallness, H BM 2n (X × Y X) is the top dimensional Borel Moore homology, it is generated by the irreducible components of X × Y X.…”

“…a d−1 ) ∈ N d to be the face vector of a simplicial polytope. Exploiting more fully the content of the hard Lefschetz theorem, it is possible to characterize completely the sequences in N d which occur as the face vectors of some simplicial polytope; see [23], Theorem 1.1.…”

“…Another example of application of methods of intersection cohomology to the combinatorics of polytopes is the solution, due to T. Braden and R. MacPherson of a conjecture of G. Kalai concerning the behavior of the g-polynomial of a face with respect to the g-polynomial of the whole polytope. See [24] and the survey [23].…”

“…Correspondingly, we have a decomposition of local systems L S = χ∈Irr(π 1 (S)) L χ ⊗ V χ S , and, for each term IC S (L S ) in (23), an isotypical decomposition…”

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“…The recent survey [23] contains many historical details, motivation, a discussion of open problems and recent results, and an extensive bibliography.…”