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Intersection Cohomology on Nonrational Polytopes[star]

Published online by Cambridge University Press:  04 December 2007

Paul Bressler  and
Valery A. Lunts
Paul Bressler
Affiliation:
Department of Mathematics, University of Arizona, 617 N. Santa Rita, Tucson, AZ 85721, U.S.A. e-mail: bressler@hedgehog.math.arizona.edu
Valery A. Lunts
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A. e-mail: vlunts@indiana.edu
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Abstract

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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.

Keywords

convex polytopesintersection cohomologytoric varieties
Type
Research Article
Information
Compositio Mathematica , Volume 135 , Issue 3 , February 2003 , pp. 245 - 278
Copyright
© 2003 Kluwer Academic Publishers