Titles and Abstracts
Hiraku Abe
Title: Hessenberg varieties I
Abstract: I will give an overview of recent topics on Hessenberg varieties. We will see that topology, combinatorics, and representation theory interact nicely on Hessenberg varieties. I will try to make this talk informal so that we can exchange ideas, tools, viewpoints, etc. This talk will continue to Tatsuya Horiguchi's talk.
Takuro Abe
Title: Solomon-Terao algebra of hyperplane arrangements
Abstract: Cohomology groups of regular nilpotent Hessenberg varieties are presented by using ideal arrangements and their logarithmic derivation modules by Horiguchi, Masuda, Murai, Sato and the speaker. This is a generalization of Borel's isomorphism between the coinvariant ring and the cohomology group of the flag varieties. Since logarithmic derivation modules can be defined for all hyperplane arrangements, we an obtain an Artinian ring from any arrangements which algebraically generalizes these cohomology groups and coinvariant algebras. We call it the Solomon-Terao algebra, and develop several general theory. This is a joint work with T. Maeno, S. Murai and Y. Numata.
Hajime Fujita
Title: On a metric on the moduli space of Delzant polytopes
Abstract: Pelayo-Pires-Ratiu-Sabatini constructed a metric on the set of Delzant polytopes. In 2-dimensional case they showed that the metric space is not complete and studied its completion. They also studied the moduli space of Delzant polytopes, the quotient space with respect to the integral affine transformation group.
Kaho Ohashi (Japan Women's university, master student) constructed a metric on the moduli space of 2-dimensional Delzant polytopes and showed that the metric topology is homeomorphic to the quotient topology of the moduli space. In this talk I will review her result and related problems.
Tatsuya Horiguchi
Title: Hessenberg varieties II
Abstract: Followed by Hiraku Abe's talk(the previous talk), I will introduce open problems for Hessenberg varieties. We hope to discuss these topics after our talks if you will be interested in Hessenberg varieties.
Taekgyu Hwang
Title: The Gromov width of generalized Bott manifolds
Abstract: The Gromov width is an invariant of a symplectic manifold defined as the maximal size of the embedded ball. I will explain the motivation and the methods of estimation that I know. I will talk about the computation for generalized Bott manifolds.
Yoshinobu Kamishima
Title: On locally homogeneous aspherical K\"ahler manifolds, aspherical Sasaki Manifolds
Abstract: PDF
We shall prove that a compact locally homogeneous aspherical K\"ahler manifold $M$ is holomorphically isometric to the quotient of the product $T^k_{\mathbb C}\times S_0\big /H_0$ by a discrete subgroup of ${\rm Isom}(\tilde M)$ where $S_0\big/H_0$ is a Hermitian symmetric manifold. In particular $M$ is a nonpositively curved manifold. We discuss the deformations space of K\"ahler structures on $M$. We also mention that every compact aspherical Sasaki manifold is an $S^1$-bundle over such $M$. This is a joint work with O. Baues.
Masaharu Kaneda
Title: Williamson's construction of torsion in the intersection cohomology of Schubert varieties
Abstract: A few years ago Geordie Williamson had shaken us in the modular representation theory by exhibiting large primes exceeding the long expected bound
for which Lusztig's formula for irreducible characters of reductive algebraic groups fails. Following his 2017 article in Journal of Algebra I will try to explain how he constructs such primes using equivariant cohomology on Demazure-Hansen-Bott-Samelson varieties.
Shintaro Kuroki
Title: Complexity one GKM manifolds with extended actions
Abstract : A complexity one GKM manifold is a (2n + 2)-dimensional GKM manifold with an n-dimensional torus action. In this talk, we introduce the progress work on a classification of complexity one GKM manifolds with (extended) non-abelian compact Lie group actions.
Hideya Kuwata
Title: Toric manifolds over an n-cube with one vertex cut
Abstract: PDF
A complete nonsingular toric variety (called a toric manifold) is over $P$ if its quotient by the compact torus is homeomorphic to $P$ as a manifold with corners. Bott manifolds are toric manifolds over an $n$-cube $I^n$ and blowing them up at a fixed point produces toric manifolds over $\mbox{vc}(I^n)$ an $n$-cube with one vertex cut. They are all projective. On the other hand, Oda's $3$-fold, the simplest non-projective toric manifold, is over $\mbox{vc}(I^3)$. In this paper, we classify toric manifolds over $\mbox{vc}(I^n)$ $(n\ge 3)$ as varieties and as smooth manifolds. It consequently turns out that there are many non-projective toric manifolds over $\mbox{vc}(I^n)$ but they are all diffeomorphic, and toric manifolds over $\mbox{vc}(I^n)$ in some class are determined by their cohomology rings as varieties. This is a joint work with Sho Hasui, Mikiya Masuda and Seonjeong Park.
Eunjeong Lee
Title: Flag Bott--Samelson Varieties and Generalized String Polytopes
Abstract: PDF
Let $G$ be a simply-connected semisimple algebraic group over $\mathbb{C}$ and let $B \subset G$ be a Borel subgroup. A Bott--Samelson variety $Z$ is an iterated sequence of $\mathbb{C} P^1$-bundles which has an action of $B$. It is known that Bott--Samelson varieties are closely intertwined with generalized Demazure modules, which are $B$-modules, and generalized string polytopes. In this talk, we introduce flag Bott--Samelson variety which is an extended notion of Bott--Samelson variety. We study $B$-representations related to flag Bott--Samelson varieties, and provide a connection between flag Bott--Samelson varieties and generalized string polytopes. This talk is based on an on-going project with Naoki Fujita and Dong Youp Suh.
Takahiro Nagaoka
Title: The universal Poisson deformation space of hypertoric varieties
Abstract: Hypertoric varieties are hyperkahler analogue of toric varieties and they can be defined as algebraic symplectic quotient of cotangent space of a vector space by algebraic torus action. Since they have holomorphic symplectic form, we can consider deformations of Poisson structure of them. From general theory of Poisson deformations of symplectic varieties, it is known that for each (conical) symplectic variety, there exsists the universal Poisson deformation space of it. In this talk, using holomorphic analogue of Duistermaat-Heckman theorem and some facts on Kirwan map of hypertoric varieties, I shall prove the Lawrence toric varietiey which includes the original hypertoric variety is exactly the universal Poisson deformation space of the hypertoric variety. This talk is based on my master thesis.
Jongbaek Song
Title: A rational CW structure on certain orbifolds
Abstract: The CW structure of certain spaces, such as effective orbifolds, can be too complicated for computational purposes. In this talk, we introduce the concept of ``rational CW complex'' structure on an orbifold, to detect torsion in its integral cohomology. This is a joint work with A. Bahri, D. Notbohm and S. Sarkar.
Takahiko Yoshida
Title: Adiabatic limits and geometric quantization of Lagrangian fibrations
Abstract: Abstract: In this talk, for a certain class of nonsingular Lagrangian fibrations, we show that Kahler quantization converges to the geometric quantization using the real polarization by the adiabatic limit. More precisely, for each Bohr-Sommerfeld point we construct a holomorphic section of the prequantum line bundle that satisfies the following properties
They forms a basis of the space of holomorphic sections of the prequantum line bundle.
The support of each section converges to the corresponding Bohr-Sommerfled fiber by the adiabatic limit.
Haozhi Zeng
Title: The orbifold fundamental group of torus orbifolds
Abstract: The orbifold fundamental group was introduced by Thurston. In this talk we discuss the orbifold fundamental group of torus orbifolds. This is joint work with Zhi Lu in progress.