Leonid Monin: Chain posets, monotone path polytopes and Chow polynomials

Abstract: Given a \mathbb{C}^*-action on a projective variety with finitely many fixed points, one can define a poset structure on the set of (connected components of) fixed points by considering the chains of orbits between them. The goal of my talk is to relate the combinatorics of this chain poset to the geometry of the action. In particular, I will discuss when the chain poset admits a natural grading and a natural kernel in the sense of Kazhdan–Lusztig–Stanley theory. In such cases, we obtain a geometric interpretation of the Chow function of the poset in terms of the Chow quotient of the action. I will primarily focus on the case of toric varieties, where a \mathbb{C}^*-action corresponds to a linear functional on a polytope, fixed points correspond to vertices, and the associated Chow quotient is the monotone path polytope.   Based on a joint work with Mateusz Michalek and Botong Wang.