Intersection Cohomology on Nonrational Polytopes⋆

By P. Bressler. Compositio Mathematica, 2000

73 citations recorded by the source corpus.

AbstractWe 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 $$\mathcal{L}_\Phi$$ , 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 $$\mathcal{L}_\Phi$$ 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f13e3ddad536692267697c814b32a4c1595985fa

How researchers see this paper

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In Algebra and Number Theory

1. Ben Elias
2. T. Panov
3. M. A. Cataldo
4. L. Migliorini
5. Geordie Williamson
6. V. Buchstaber
7. Kalle Karu
8. M. Masuda

In Discrete Mathematics and Combinatorics

1. T. Panov
2. V. Buchstaber
3. Geordie Williamson
4. Ben Elias
5. L. Migliorini
6. M. A. Cataldo
7. Kalle Karu
8. M. Masuda

In Geometry and Topology

1. Ben Elias
2. Geordie Williamson
3. M. A. Cataldo
4. T. Panov
5. L. Migliorini
6. V. Buchstaber
7. Kalle Karu
8. M. Masuda

Citations in the corpus

• Hard Lefschetz theorem for nonrational polytopes (2001, methodology): “We start by recalling how E(∂σ) can be identified with the primitive cohomology of the “flattened boundary fan” [1,3].”
• The decomposition theorem, perverse sheaves and the topology of algebraic maps (2007, background): “Lunts have developed a theory of sheaves on the poset associated with the polytope P , or more generally to a fan, see [26].”
• The Hodge theory of Soergel bimodules (2012, background): “There are many formal similarities between the theory we develop here and the theory of intersection cohomology of nonrational polytopes, which was developed to prove Stanley’s conjecture on the unimodularity of the generalized h -vector [BL03], [Kar04], [BKBF07].”
• Torus Actions and their Applications in Topology and Combinatorics (2002, background): “Some progress in this direction has been achieved in [27], [134].”