Compositio Mathematica

Intersection Cohomology on Nonrational Polytopes[star, open]

Paul Bressler a1 and Valery A. Lunts a2
a1 Department of Mathematics, University of Arizona, 617 N. Santa Rita, Tucson, AZ 85721, U.S.A. e-mail:
a2 Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A. e-mail:

Article author query
bressler p   [Google Scholar] 
lunts va   [Google Scholar] 


We consider a fan as a ringed space (with finitely many points). We develop the corresponding sheaf theory and functors, such as direct image Rπ* (π is a subdivision of a fan), Verdier duality, etc. The distinguished sheaf ${\cal L}_\Phi$, called the minimal sheaf plays the role of an equivariant intersection cohomology complex on the corresponding toric variety (which exists if Φ is rational). Using ${\cal L}_\Phi$ we define the intersection cohomology space IH(Φ). It is conjectured that a strictly convex piecewise linear function on Φ acts as a Lefschetz operator on IH(Φ). We show that this conjecture implies Stanley's conjecture on the unimodality of the generalized h-vector of a convex polytope.

Key Words: convex polytopes; intersection cohomology; toric varieties.