Mini-workshop on Geometry
Date: 10:00–12:00/13:30–17:20, November 26th (Thursday), 2015
Place: The building of mathematics: Room 201 (before noon), Room 107(after midday)
Organizers: Shin-ichi Matsumura, Reiko Miyaoka (contact r-miyaok@m.tohoku.ac.jp)

Program
10:00 - 10:40 Norihiko Suzuki
Stabilities of rough curvature-dimension condition
10:50 - 11:50 Masato Mimura
Strong algebraization of fixed point property
11:50 - 13:30

LUNCH

13:30 - 14:10 Takashi Komatsu
Limiting distributions of discrete-time quantum walks on the square lattice
14:20 - 15:20 Shin-ichi Matsumura
A vanishing theorem of Kollár-Ohsawa type on compact Kähler manifolds
15:30 - 16:10 Kenta Tottori
Calabi’s conjecture of the Kähler-Ricci soliton type16:20 - 17:20 Masaharu Ishikawa
Stable maps and hyperbolic volumes of 3-manifolds

Abstract..................... ..................... ..................... .....................
Stabilities of rough curvature-dimension condition
Norihiko Suzuki
Lott-Villani and Sturm defined the curvature-dimension condition on metric measurespaces. Bonciocat-Sturm introduced the rough curvature dimension condition as a generalization of the condition by taking into account the roughness parameter. This condition
characterizes the rough lower Ricci curvature bound and upper dimensional bound ofmetric measure spaces. They proved that this condition is stable under convergence withrespect to the transportation distance in the case where the roughness parameter converges to 0. We prove the stability in the case where the roughness parameter converges
to a nonzero real number. This talk is based on the joint work with Ryunosuke Ozawa in
Kyoto University.
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Strong algebraization of fixed point property
Masato Mimura
We obtain a purely algebraic criterion for fixed point properties on Banach spaces underrelative fixed point properties. This is a strengthening of Shalom’s algebraization inICM 2006, in the sense of that we completely remove any form of ”bounded generation”
assumption. Several applications may, in addition, be discussed.
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Limiting distributions of discrete-time quantum walks on the square lattice
Takashi Komatsu
We study discrete-time quantum walks on the square lattice. The notion of quantum
walks was introduced by Y. Aharonov et al. as a quantum analogue of the classicalrandom walks. Recently, quantum walks have been intensively studied in connection
with quantum computing and quantum physics. In this talk, we will propose a model of
discrete-time quantum walks on the square lattice without localization and give its limit
distribution. Furthermore, we see that the Konno function appears as the density function
with respect to radial direction in our quantum walk and give an another expression of
our result from the view of the quantum information.A vanishing theorem of Kollár-Ohsawa type on compact Kähler manifolds
Shin-ichi Matsumura
In the last decades, it has been one of the important subjects in algebraic geometry
to study the variation of the cohomology groups of vector bundles for deformations ofalgebraic varieties. In this talk, I would like to consider the higher direct images (whichcan be seen as such variations) for deformations of Kähler varieties parametrized byanalytic spaces, and give a Hodge type decomposition for them by using the theory of
harmonic integrals developed by Takegoshi. As an application, I will prove a vanishingtheorem of Kollár-Ohsawa type in combined with the L2 -method for the ∂-equation. Itgives an aﬃrmative answer for the conjecture posed by Fujino.
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Calabi’s conjecture of the Kähler-Ricci soliton type
Kenta Tottori
In this talk, we discuss Calabi’s equation of the Kähler-Ricci soliton type on a compactKähler manifold. This equation was introduced by Zhu as a generalization of Calabi’sconjecture. We give necessary and suﬃcient conditions for the unique existence of asolution for this equation on a compact Kähler manifold with a holomorphic vector field
which has a zero point. We also consider the case of a nowhere vanishing holomorphic
vector field, and give suﬃcient conditions for the unique existence of a solution for thisequation.
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Stable maps and hyperbolic volumes of 3-manifolds
Masaharu Ishikawa
A stable map plays an important role when we study topology of manifolds. For example,
a Morse function is a stable map from a manifold to the real line, which is used to obtain
the information of homology groups of manifolds. We study 3-manifolds by using stablemaps from them to the real plane. It is known by O.Saeki that there exists a stable map
without certain singular fibers if and only if the Gromov norm of the 3-manifold is zero,i.e., it is a graph manifold. In the study of complexity of 3-manifolds, F.Costantino andD.Thurston used the idea of identifying the Stein factorization of a stable map with ashadow of the 3-manifold. Note that the above singular fibers correspond to the verticesof the shadow. We prove that there exists a stable map corresponding to a given shadow
without changing the number of such singular fibers. As a consequence, we obtain an
estimation of the Gromov norm of the 3-manifold by the minimal number of such singularfibers. This is a joint work with Yuya Koda in Hiroshima University.
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