## Geometry Colloquium

Date, time & place Friday 10:00 - 11:30 126Room #126 (Graduate School of Math. Sci. Bldg.)

Seminar information archive

### 2017/05/08

16:00-17:00   Room #056 (Graduate School of Math. Sci. Bldg.)

### 2016/06/24

10:00-14:30   Room #118 (Graduate School of Math. Sci. Bldg.)

### 2016/06/03

13:00-14:30   Room #118 (Graduate School of Math. Sci. Bldg.)
Takeo Nishinou (Rikkyo University)
On a construction of holomorphic disks (Japanese)
[ Abstract ]
Recent study of algebraic and symplectic geometry revealed that holomorphic disks play an important role in several situations, deforming the classical geometry in some sense. In this talk we give a construction of holomorphic disks based on deformation theory, mainly on certain algebraic surfaces.

### 2016/06/03

15:00-16:30   Room #118 (Graduate School of Math. Sci. Bldg.)
Makoto Miura (KIAS)
Caldero's toric degenerations and mirror symmetry (Japanese)
[ Abstract ]
In this talk, we explain some basic facts on toric degenerations of Fano varieties. In particular, we focus on the toric degenerations of Schubert varieties proposed by Caldero, where we use the string parametrizations of Lusztig--Kashiwara's dual canonical basis. As an application, we introduce a conjectural mirror construction of a linear section Calabi--Yau 3-fold in an orthogonal Grassmannian OG(2,7). This talk is based on joint works with Daisuke Inoue and Atsushi Ito.

### 2016/05/27

10:00-11:30   Room #118 (Graduate School of Math. Sci. Bldg.)
Yohsuke Imagi (Kavli IPMU)
Compact Special Lagrangian T^2-conifolds (Japanese)
[ Abstract ]
Special Lagrangian submanifolds may be defined as volume-minimizing Lagrangian submanifolds of Calabi--Yau manifolds. Some interesting but difficult topics are (1) the SYZ conjecture, (2) counting compact special Lagrangian homology spheres, and (3) relation to Fukaya categories---all concerned with singularity of special Lagrangian submanifolds. I first recall some basic facts about these things and then talk about a simple class of singularity modelled on a certain T^2-cone.

### 2016/05/27

13:00-14:30   Room #118 (Graduate School of Math. Sci. Bldg.)
Yoshihiko Matsumoto (Osaka University)
Deformation of Einstein metrics and $L^2$ cohomology on strictly pseudoconvex domains (Japanese)
[ Abstract ]
Any bounded strictly pseudoconvex domain of a Stein manifold carries a complete Kähler-Einstein metric of negative scalar curvature, which is unique up to homothety, as shown by S. Y. Cheng and S. T. Yau. I will discuss the fact that this Cheng-Yau metric deforms into a family of Einstein metrics parametrized by partially integrable CR structures on the boundary under the assumption that the dimension is at least three. The necessary analysis on the linearized Einstein operator can be reduced to a vanishing result of the $L^2$ Dolbeault cohomology with values in the holomorphic tangent bundle.

### 2016/04/21

17:00-18:00   Room #123 (Graduate School of Math. Sci. Bldg.)
Shouhei Honda (Tohoku University)
Spectral convergence under bounded Ricci curvature (Japanese)
[ Abstract ]
For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are sharp generalizations of Cheeger-Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. These spectral convergence have two direct corollaries. One of them is to give new bounds on such eigenvalues, in terms of bounds on volume, diameter and the Ricci curvature. The other is that we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. On the other hand we also define measurable curvature tensors of the noncollapsed Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a uniform bound of Ricci curvature, which include Riemannian curvature tensor, the Ricci curvature, and the scalar curvature. As fundamental properties of our Ricci curvature, we show that the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover we prove a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology by using our scalar curvature. This talk is based on arXiv:1510.05349.

### 2015/12/04

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Yoshihiro Takeyama (Graduate School of Pure and Applied Sciences, University of Tsukuba)
The q-Boson system and a deformation of the affine Hecke algebra (Japanese)
[ Abstract ]
The q-Boson system due to Sasamoto and Wadati is a one-dimensional "integrable" stochastic particle model. Its Q-matrix is constructed in the framework of the quantum inverse scattering method and we obtain the eigenvectors by means of the algebraic Bethe ansatz method. Recently it is found that the q-Boson model can be derived also from a representation of a deformation of the affine Hecke algebra and its representation. In this formulation we get the eigenvectors of the transpose of the Q-matrix which were constructed by the technique called the coordinate Bethe ansatz. In this talk I review the above results and discuss the relationship between the two methods.

### 2015/11/27

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Hisamoto Tomoyuki (Nagoya University)
The strong version of K-stability derived from the coercivity property of the K-energy (Japanese)
[ Abstract ]
This is a joint work with S. Boucksom and M. Jonsson. We introduced the notion of J-uniform K-stability in relation to the coercivity property of the K-energy. As a result, one can see that any Kähler-Einstein
manifolds with no non-zero holomorphic vector field is J-uniformly K-stable.

### 2015/11/13

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
MASAI, Hideyoshi (The University of Tokyo)
Closed random mapping tori are asymmetric
(Japanese)
[ Abstract ]
We consider random walks on the mapping class group of closed surfaces and mapping tori of such random mapping classes. It has been shown that such random mapping tori admit hyperbolic structure, and hence their symmetry groups are finite groups. In this talk we prove that the symmetry group of random mapping tori are trivial.

### 2015/10/23

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Nobuhiko Otoba (Keio University)
Metrics of constant scalar curvature on sphere bundles (Japanese)
[ Abstract ]
This talk is based on joint work with Jimmy Petean (CIMAT).
I'd like to talk about our attempt to study the Yamabe PDE on Riemannian twisted product manifolds, more precisely, the total spaces of Riemannian submersions with totally geodesic fibers. To demonstrate how the argument works,
I construct metrics of constant scalar curvature on unit sphere bundles for real vector bundles of the type $E \oplus L$,
the Whitney sum of a vector bundle $E$ and a line bundle $L$ with respective inner products, and then estimate the number of solutions to the corresponding Yamabe PDE.

### 2015/10/16

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Daisuke Yamakawa (Tokyo Institute of Technology)
Meromorphic connections on the Riemann sphere and quiver varieties (Japanese)
[ Abstract ]
I will show that some moduli spaces of meromorphic connections on the Riemann sphere are isomorphic to Nakajima's quiver varieties as complex symplectic manifolds (joint work with Kazuki Hiroe). This was conjectured by Boalch and generalizes Crawley-Boevey's result for logarithmic connections. Also I will mention Weyl group symmetries of isomonodromic deformations of meromorphic connections.

### 2015/10/02

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Takahashi Ryosuke (Nagoya University, Graduate School of Mathematics)
Asymptotic stability for K¥"ahler-Ricci solitons (Japanese)
[ Abstract ]
K¥"ahler-Ricci solitons arise from the geometric analysis, such as Hamilton’s Ricci flow, and have been studied extensively in recent years. It is expected that the existence of a canonical metric is closely related to some GIT stability of manifolds. For instance, Donaldson showed that any cscK polarized manifold with discrete automorphisms admits a sequence of balanced metrics and this sequence converges to the cscK metric. In this talk, we explain that the same result holds for K¥ahler-Ricci solitons. This generalizes a previous work of Berman-Witt Nystr¥"om, and is an analogous result on asymptotic relative Chow stability for extremal metrics obtained by Mabuchi.

### 2015/07/24

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Asuka Takatsu (Tokyo Metropolitan University)
High-dimensional metric-measure limit of Stiefel manifolds (Japanese)
[ Abstract ]
A metric measure space is the triple of a complete separable metric space with a Borel measure on this space. Gromov defined a concept of convergence of metric measure spaces by the convergence of the sets of 1-Lipschitz functions on the spaces. We study and specify the high-dimensional limit of Stiefel manifolds in the sense of this convergence; the limit is the infinite-dimensional Gaussian space, which is drastically different from the manifolds. This is a joint work with Takashi SHIOYA (Tohoku univ).

### 2015/07/17

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Takeo Nishinou (Rikkyo University)
Realization of tropical curves in complex tori (Japanese)
[ Abstract ]
Tropical curves are combinatorial object satisfying certain harmonicity condition. They reflect properties of holomorphic curves, and rather precise correspondence is known between tropical curves in real affine spaces and holomorphic curves in toric varieties. In this talk we extend this correspondence to the periodic case. Namely, we give a correspondence between periodic plane tropical curves and holomorphic curves in complex tori. This is a joint work with Tony Yue Yu.

### 2015/07/03

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Takayuki OKUDA (HIroshima University)
Proper actions of reductive groups on pseudo-Riemannian symmetric spaces and its compact dual. (日本語)
[ Abstract ]
Let G be a non-compact semisimple Lie group. We take a pair of symmetric pairs (G,H) and (G,L) such that the diagonal action of G on G/H \times G/L is proper. In this talk, we show that by taking the compact dual of triple (G,H,L)'', we obtain a compact symmetric space M = U/K and its reflective submanifolds S_1 and S_2 satisfying that the intersection of S_1 and gS_2 is discrete in M for any g in U. In particular, we give a classification of such triples (G,H,L).

### 2015/06/12

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Kota Hattori (Keio University)
The nonuniqueness of tangent cone at infinity of Ricci-flat manifolds (Japanese)
[ Abstract ]
For a complete Riemannian manifold (M,g), the Gromov-Hausdorff limit of (M, r^2g) as r to 0 is called the tangent cone at infinity. By the Gromov's Compactness Theorem, there exists tangent cone at infinity for every complete Riemannian manifolds with nonnegative Ricci curvatures. Moreover, if it is Ricci-flat, with Euclidean volume growth and having at least one tangent cone at infinity with a smooth cross section, then it is uniquely determined by the result of Colding and Minicozzi. In this talk I will explain that the assumption of the volume growth is essential for their uniqueness theorem.

### 2015/06/05

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Yoshihiko Shinomiya (Shizuoka University)
Veech groups of Veech surfaces and periodic points
(日本語)
[ Abstract ]
Flat surfaces are surfaces with singular Euclidean structures.　The Veech group of a flat surface is the group consisting of all matrices inducing affine mappings of the flat surface. In this talk, we give relations between some geometrical values of flat surfaces and the signatures of Veech groups as Fuchsian groups. As an application of these relations, we estimate the numbers of periodic points of certain flat surfaces.

### 2015/05/08

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Masashi Ishida (Osaka University)
On Perelman type functionals for the Ricci Yang-Mills flow (Japanese)
[ Abstract ]
In his works on the Ricci flow, Perelman introduced two functionals with monotonicity
formulas under the Ricci flow. The monotonicity formulas have many remarkable geometric applications. On the other hand, around 2007, Jeffrey Streets and Andrea Young independently and simultaneously introduced a new geometric flow which is called the Ricci Yang-Mills flow. The new flow can be regarded as the Ricci flow coupled with the Yang-Mills
heat flow. In this talk, we will introduce new functionals with monotonicity formulas under the Ricci Yang-Mills flow and discuss its applications.

### 2015/04/24

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Kento Fujita (Kyoto University)
On K-stability and the volume functions of Q-Fano varieties (JAPANESE)
[ Abstract ]
For Fano manifolds X, it is known that X admits K\"ahler-Einstein metrics if and only if the polarized pair
(X, -K_X) is K-polystable. In this talk, I will introduce a new effective stability named "divisorial stability" for Fano manifolds, which is weaker than K-stability and stronger than slope stability along divisors. I will show that:
1. We can easily test divisorial stability via the volume functions.
2. There is a relationship between divisorial stability and the structure property of Okounkov bodies of anti-canonical divisors.
3. For toric Fano manifolds, the existence of K\"ahler-Einstein metrics is equivalent to divisorial semistability.

### 2015/04/17

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Masaki TSUKAMOTO (Kyoto University)
Mean dimension of the dynamical system of Brody curves (日本語)
[ Abstract ]
Mean dimension is a topological invariant of dynamical systems with infinite dimension and infinite entropy. Brody curves are Lipschitz entire holomorphic curves, and they form an infinite dimensional dynamical system. Gromov started the problem of estimating its mean dimension in 1999. We solve this problem by proving the exact mean dimension formula. Our formula expresses the mean dimension by the energy density of Brody curves. A key novel ingredient is an information theoretic approach to mean dimension introduced by Lindenstrauss and Weiss.

### 2015/01/16

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Takeo Nishinou (Rikkyo University)
Degeneration and curves on K3 surfaces (Japanese)
[ Abstract ]
There is a well-known conjecture which states that all projective K3 surfaces contain infinitely many rational curves. By calculating obstructions in deformation theory through degeneration, we give a new approach to this problem. In particular, we show that there is a Zariski open subset in the moduli space of quartic K3 surfaces whose members fulfil the conjecture.

### 2014/12/19

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Yuichi KABAYA (Kyoto University)
Exotic components in linear slices of quasi-Fuchsian groups
[ Abstract ]
The linear slice of quasi-Fuchsian punctured torus groups is defined by fixing the length of some simple closed curve to be a fixed positive real number. It is known that the linear slice is a union of disks, and it has one `standard' component containing Fuchsian groups. Komori-Yamashita proved that there exist non-standard components if the length is sufficiently large. In this talk, I give another proof based on the theory of complex projective structures. If time permits, I will talk about a refined statement and a generalization to other surfaces.

### 2014/12/04

17:00-18:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Ryushi Goto (Osaka University)
Deformations and the moduli spaces of generalized complex manifolds (JAPANESE)

### 2014/11/14

10:00-11:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Manabu Akaho (Tokyo Metropolitan University)
Symplectic displacement energy for exact Lagrangian immersions
(JAPANESE)
[ Abstract ]
We give an inequality of the displacement energy for exact Lagrangian immersions and the symplectic area of punctured holomorphic discs. Our approach is based on Floer homology for Lagrangian immersions and Chekanov's homotopy technique of continuations. Moreover, we discuss our inequality and the Hofer--Zehnder capacity.