Himeji seminar on geometry of torus actions 2023
Titles and Abstracts
Laura Escobar (Washington University in St.Louis)
Title: Determining the complexity of Kazhdan--Lusztig varieties
Abstract:
The defining ideals of Kazhdan--Lusztig varieties are generated by certain minors of a matrix, which are chosen by a combinatorial rule. The structure of these varieties can be understood from the combinatorics of permutations. Each Kazhdan--Lusztig variety has a natural torus action from which one can construct a polytope. The complexity of this torus action can be computed from the dimension of the polytope and, in some sense, indicates how close the geometry of the variety is to the combinatorics of the associated polytope. In joint work with Maria Donten-Bury and Irem Portakal we address the problem of classifying which Kazhdan--Lusztig varieties have a given complexity utilizing the rich combinatorics of these varieties.
Hiroaki Ishida (Kagoshima University)
Title: Canonical foliations of complex manifolds
Abstract:
For a compact complex manifold with a torus action, we introduce a holomorphic foliation associated with the action and complex structure. In this talk, I will explain that the foliation plays an important role in transverse Kaehler geometry.
Shintaro Kuroki (Okayama University of Science)
Title: Classification of 6 and 8-dimensional complexity one GKM manifolds with Sp(2)-extended actions
Abstract:
A complexity one GKM manifold M is a (2n+2)-dimensional manifolds with n-dimensional torus T-action whose 0 and 1-dimensional orbits have the structure of a graph.
We are interested in the question of when the T-action extends to the compact connected Lie group G-action, where the torus T is a maximal torus of G.
If the extended G-action is transitive, then M splits into a product of a homogeneous, complexity one GKM manifold and some homogeneous, complexity zero GKM manifolds (also called a torus manifold), i.e., even-dimensional spheres or complex projective spaces.
There are essentially six types of homogeneous, complexity one GKM manifolds (see RIMS Kokyuroku 1922, 135--146, 2014).
One of the groups acting on them is the rank 2 symplectic group Sp(2).
In this talk, I will introduce the classification of 6 and 8-dimensional complexity one GKM manifolds with Sp(2)-extended actions.
Takashi Sato (Osaka Central Advanced Mathematical Institute)
Title: Regular semisimple Hessenberg varieties and the modular law
Abstract:
In this talk, I introduce the proof of the modular law of regular semisimple Hessenberg varieties shown by Kiem and Lee.
The proof is elementary and then the algorithm of Abreu and Nigro gives an elementary proof of Brosnan-Chow theorem, which connects regular semisimple Hessenberg varieties with chromatic symmetric functions.
Grigory Solomadin (Okayama University of Science)
Title: Sheaves on posets, distributivity and cohomology of orbit spaces for moment-angle complexes
Abstract:
Consider the moment-angle complex (MAC, for short) corresponding to any simplicial complex K on m vertices. Let H be any closed subgroup in the naturally acting compact torus T^m on it. The cohomology groups (with integer or rational coefficients) of the respective quotient are known in a number of various cases (depending on K and H). (Including toric manifolds in sense of Davis-Januszkiewicz, for instance.) The general Eilenberg-Moore SS method is given by computing the Tor-groups of the Stanley-Reisner algebra for K; however, the resulting answer is implicit. At present, the general cohomology computation problem seems to remain open. In the first part of the talk, cohomology of sheaves over posets will be defined and related to such task. Relevant properties (distributivity, duality for K Cohen-Macaulay, etc.) and computations (including M. Franz's example with torsion in cohomology) will be given. Over rational numbers, this approach is related to finite arrangements of linear subspaces in a vector space. In the second part of the talk, three increasingly restrictive versions of distributivity for the collection of H and coordinate subgroups T^I, I\in K, will be introduced and explained. The weakest of these (introduced by Limonchenko and Solomadin) implies that the equivariant cohomology of MAC quotient is the limit of a certain diagram for abelian groups. Medium condition is equivalent to flasqueness of the stabilizer (co)sheaf for the MAC quotient. The strongest is the distributivity condition for coordinate subgroups (corresponding to simplices in K) and H.