Paul Bressler

Intersection Cohomology on Nonrational Polytopes

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

Homological mirror symmetry, deformation quantization and noncommutative geometry

\mathcal{L}_\Phi

Chern Character for Twisted Complexes

, 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

On quasi-classical limits of DQ-algebroids

\mathcal{L}_\Phi

Mirror symmetry and deformation quantization

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 Stanleys conjecture on the unimodality of the generalized h-vector of a convex polytope.

Hard Lefschetz Theorem and Hodge-Riemann Relations for Intersection Cohomology of Nonrational Polytopes

We give a natural obstruction theoretic interpretation to the first Pontryagin class in terms of Courant algebroids. As an application we calculate the class of the stack of algebras of chiral differential operators. In particular, we establish the existence and uniqueness of the chiral de Rham complex.