Student Algebraic Geometry Seminar


Title: Some combinatorial torus-equivariant cohomology


Abstract: For a projective toric variety $X_\Sigma$ smooth or with only quotient singularities the intersection cohomology and ordinary cohomology coincide. In particular this is the case whenever $\Sigma$ is a simplicial fan. One can compute the intersection cohomology as a quotient of the torus-equivariant intersection cohomology. In the simplicial case, a theorem of Jurkiewicz and Danilov asserts that this is modelled by the Stanley-Reisner ring of $\Sigma$. Later, it was observed that, even more concretely, it is the algebra $A(\Sigma)$ of (continuous) conewise polynomial functions on $\Sigma$. The (non-equivariant) intersection cohomology is then the quotient of $A(\Sigma)$ by the global linear functions. This viewpoint has many uses. For one, it is quite computable from the combinatorial data: $A(\Sigma)$ is on the one hand a free module over the ("global") polynomial functions, and on the other generated as an algebra by characteristic functions of rays of $\Sigma$. For another, in 1993 Peter McMullen gave a combinatorial proof of the Hard Lefschetz theorem in this case (actually, he used weight spaces in the polytope algebra, another model of this algebra). In 2002-03 Barthel-Brasselet-Fieseler-Kaup and Bressler-Lunts (independently) constructed a combinatorial equivariant intersection cohomology sheaf on an arbitrary fan $\Sigma$, (not necessarily simplicial, or even rational). In 2004 Karu showed that when a complete $\Sigma$ has a strictly convex conewise linear function $\ell$ (which corresponds to an ample divisor in the case that $\Sigma$ is rational, i.e., when the toric variety $X_\Sigma$ exists), this $\ell$ is a Lefschetz operator, and the combinatorial intersection cohomology again satisfies the Hard Lefschetz theorem. (Among other things, this settled an outstanding conjecture of Stanley on the unimodality of the generalised h-vector associated to $\Sigma$.) Thus for any $X_\Sigma$ there is now a combinatorial proof of the deep geometric fact that the intersection cohomology satisfies the Hard Lefschetz theorem. We'll talk examples, results, and methods. There will be no derived categories; there will be convex geometry.