The 13th Kagoshima Algebra-Analysis-Geometry Seminar

February 13-16, 2018
Room 101, Bldg. 1, Faculty of Science, Kagoshima University

ORGANIZERS:
• Shoji Yokura (Kagoshima University)
• Kiyoshi Takeuchi (University of Tsukuba)
• Shunichi Kimura (Hiroshima University)
• Masaaki Murakami (Kagoshima University)
• Hiroyuki Nakaoka (Kagoshima University)

SUPPORTED BY:
• Grant-in-Aid for Scientific Research (B) 17H02848 (Kiyoshi Takeuchi)
• Grant-in-Aid for Scientific Research (B) 16H03936 (Shoji Yokura)
• Grant-in-Aid for Scientific Research (C) 15K04825 (Masaaki Murakami)
• Grant-in-Aid for Young Scientists (B) 15K17538 (Takefumi Kondo)

PROGRAM

February 13 (Tuesday)
• 12:30-13:00 : registration

• 13:00-13:50 : Eiichi Sato (Kyushu University)
"On surjective holomorphic maps between Fano manifolds"   (abstract)

• 14:00-14:50 : Masaharu Ishikawa (Tohoku University)
"Singularity at infinity of real polynomial maps of two variables"   (abstract)

• 15:10-16:00 : Kazuhiro Kawamura (University of Tsukuba)
"Hochschild cohomologies of Lipschitz algebras over compact Riemannian manifolds"   (abstract)

• 16:10-17:00 : Tadashi Ashikaga (Tohoku Gakuin University)
"Castelnuovo-Horikawa index of genus 3 via signature divisor"   (abstract)

February 14 (Wednesday)
• 10:00-10:50 : Dominik Wrazidlo (Kyushu University)
"Fold maps, positive topological field theories, and exotic spheres"   (abstract)

• 11:00-11:50 : Shinnosuke Okawa (Osaka University)
"Derived equivalence and Grothendieck ring of varieties"   (abstract)

• 13:30-14:20 : Yoshinori Gongyo (University of Tokyo)
"ネフ反標準束と有理連結ファイブレーション"   (abstract)

• 14:30-15:20 : Kohei Yahiro (University of Tokyo)
"Integral transforms of D-modules on partial flag varieties"   (abstract)

• 15:40-16:30 : Thomas Reichelt (Universitaet Heidelberg)
"Hodge theory of GKZ systems"   (abstract)

• 16:40-17:30 : Osamu Saeki (Kyushu University)
"Simplifying indefinite fibrations and trisections of 4-manifolds"   (abstract)

February 15 (Thursday)
• 10:00-10:50 : Yujiro Kawamata (University of Tokyo)
"DK予想について"   (abstract)

• 11:00-11:50 : Wim Veys (KU Leuven)
"Zeta functions and oscillatory integrals for meromorphic functions"   (abstract)

• 13:30-14:20 : Toshitake Kohno (University of Tokyo)
"Higher category extensions of holonomy maps and representations of braid cobordisms"   (abstract)

• 14:30-15:20 : Takato Uehara (Saga University)
"超越的な複素K3曲面の構成"   (abstract)

• 15:40-16:30 : Manabu Akaho (Tokyo Metropolitan University)
"J-holomorphic curves in symplectic topology"   (abstract)

• 16:40-17:30 : Mutsuo Oka (Tokyo University of Science)
"Smooth mixed projective curves and a conjecture"   (abstract)

• 19:00- conference dinner

February 16 (Friday)
• 10:30-11:20 : Ken-ichi Yoshikawa (Kyoto University)
"Enriques manifolds and analytic torsion"   (abstract)

• 11:30-12:20 : Kanehisa Takasaki (Kindai University)
"$$3\mathrm{D}$$ Young diagrams and Gromov-Witten theory of $$\mathbb{C}\mathbb{P}^1$$"   (abstract)

ABSTRACTS

Manabu Akaho   "J-holomorphic curves in symplectic topology"

In 1985 Gromov introduced a technique of J-holomorphic curves and obtained many deep results in symplectic geometry and topology. In this talk I will explain very basic properties of J-holomorphic curves and show the existence of periodic orbits of Hamiltonian vector fields in a slightly different way to Gromov.

Tadashi Ashikaga   "Castelnuovo-Horikawa index of genus 3 via signature divisor"

We determine explicitly the Castelnuovo-Horikawa index of any fiber germ of non-hyperelliptic fibrations of genus 3. This is an extension of the result of Horikawa in 1976 for genus 2 fibrations, and is closely related to the geography of surfaces near the Castelnuovo line. Fot this purpose, we analyze en effect of an automorphism of a stable curve to its Kuranishi space, and calculate the related orbifold signature.

Yoshinori Gongyo  "ネフ反標準束と有理連結ファイブレーション"

ネフ(相対的)反標準束の飯高小平次元について調べる。特に江尻氏とHacon--Mckernanの2つの質問に肯定的な解答を与える。これは江尻祥氏との共同研究である。

Masaharu Ishikawa   "Singularity at infinity of real polynomial maps of two variables"

It is known by Coste and de la Puente that singularities at infinity of　real polynomial maps of two variables are characterized by two kinds of　phenomena: cleaving and vanishing. They also gave an algorithm to determine the bifurcation set, which is the set of critical values at infinity. We　determine the bifurcation set of such a polynomial map in Newton　non-degenerate case by using toric compactification. The two phenomena can　be observed by checking the positions of the singular fibers (in the sense of at infinity'') near the divisors at infinity, not only in the case of　Newton non-degenerate but also in general case. We may use toroidal　resolutions successively in Newton degenerate case. In consequence, the　toric compactification and toroidal resolutions give us another algorithm　to determine the bifurcation set. This is a joint work with Nguyen Tat　Thang and Pham Tien Son.

Yujiro Kawamata   "DK予想について"

Kazuhiro Kawamura  "Hochschild cohomologies of Lipschitz algebras over compact Riemannian manifolds"

For a compact Riemannian manifold $$M$$ with the induced metric, LipM denotes the commutative Banach algebra of complex-valued Lipschitz functions on M. Motivated by a classical work due to de Leeuw, we introduce a compact space $$\hat M$$, a non-metrizable analogue of “the space of directions” and show that the Hochschild cohomology H^{n}(LipM,C(\hat M)) (in the sense of B.E. Johnson and A.Y. Helemskii) has infinite rank as a LipM-module, for each n \geq 1. In particular, the global cohomological dimension of Lip M is infinite, an analogue of an old result on C^1-function algebras due to Pugach and Kleshchev. On the other hand we have H^{1}(Lip M, C(M)) = 0.

Toshitake Kohno   "Higher category extensions of holonomy maps and representations of braid cobordisms "

We explain a method to construct higher category extensions of holonomy representations of homotopy path groupoids by means of Chen's formal homology connections. As an application, using a 2-functor from the path 2-groupoid of configuration spaces, we construct representations of the 2-category of braid cobordisms. We also discuss categorification of KZ connections.

Mutsuo Oka   "Smooth mixed projective curves and a conjecture"

Let $$f(\mathbf{z},\bar{\mathbf{z}})$$ be a strongly mixed homogeneous polynomial of $$3$$ variables $$\mathbf{z}=(z_1,z_2,z_3)$$ of polar degree $$q$$ with an isolated singularity at the origin. It defines a smooth Riemann surface $$C$$ in the complex projective space $$\mathbb{P}^{2}$$. The fundamental group of the complement $$\mathbb P^2\setminus C$$ is a cyclic group of order $$q$$ if $$f$$ is a homogeneous polynomial without $$\bar{\mathbf{z}}$$. We propose a conjecture that this may be even true for mixed homogeneous polynomials by giving several supporting examples.

Shinnosuke Okawa   "Derived equivalence and Grothendieck ring of varieties"

Two smooth projective varieties X and Y are said to be D-equivalent if they have equivalent bounded derived category of coherent sheaves. X and Y are said to be L-equivalent if the difference [X] - [Y] of their classes in the Grothendieck ring of varieties is annihilated by the class of an affine space. Recently many examples of pairs (X, Y) which are D-equivalent, L-equivalent, and [X] - [Y] is non-zero have been discovered by several people. In this talk I will briefly survey this topic and give a first example consisting of complex symplectic varieties in any possible dimension.

Thomas Reichelt   "Hodge theory of GKZ systems"

GKZ hypergeometric systems were introduced by Gelfand, Kapranov and Zelevinsky as a generalization of Gauss hypergeometric differential equation. It can be shown that for certain parameters the GKZ-systems carry the structure of a mixed Hodge module in the sense of Morihiko Saito. We will discuss the Hodge and weight filtration of these D-modules.

Osamu Saeki  "Simplifying indefinite fibrations and trisections of 4-manifolds"

A broken Lefschetz fibration (BLF, for short) is a smooth map of a closed oriented 4-manifold onto a closed surface whose singularities consist of Lefschetz critical points together with indefinite folds (or round singularities). Such a class of maps was first introduced by Auroux-Donaldson-Katzarkov (2005) in relation to near-symplectic structures. In this talk, we give a set of explicit moves for BLFs, and give an elementary and constructive proof to the fact that any map into the 2-sphere is homotopic to a BLF with embedded round image. Such an algorithm allows us to give a purely topological and constructive proof of a theorem of Auroux-Donaldson-Katzarkov on the existence of broken Lefschetz pencils with embedded round image on near-symplectic 4-manifolds. We moreover establish a correspondence between BLFs and Gay-Kirby trisections of 4-manifolds, and show the existence of simplified trisections on all 4-manifolds. This is a joint work with R. Inanc Baykur (University of Massachusetts).

Eiichi Sato  "On surjective holomorphic maps between Fano manifolds"

We consider the structure of surjective holomorphic maps $$f: X \rightarrow Y$$ between $$n$$-dimensional smooth projective varieties. In case that $$X, Y$$ are of general type such problems is very important even now. In our talk we focus on Fano manifolds $$X, Y$$ and study the conditions of the (non-)existence of surjective map $$f: X \rightarrow Y$$. For example it is well-known that when $$X$$ is $$P^{n}$$, so is $$Y$$. We state several results in case of Fano complete intersections.

Kanehisa Takasaki  "$$3\mathrm{D}$$ Young diagrams and Gromov-Witten theory of $$\mathbb{C} \mathbb{P}^1$$"

The melting crystal model is a model of statistical mechanics for random $$3\mathrm{D}$$ Young diagrams. The partition function of this model may be thought of as a $$q$$-deformation of the generating function of stationary Gromov-Witten invariants of $$\mathbb{C} \mathbb{P}^1$$ studied by Okounkov and Pandharipande. We consider these generating functions in the perspectives of integrable systems and quantum spectral curves. A main issue is how to capture the limit to the Gromov-Witten theory of $$\mathbb{C} \mathbb{P}^1$$ as $$q \to 1$$.

Takato Uehara   "超越的な複素K3曲面の構成"

K3曲面は, 幾何学的な立場からの興味のみならず, 近年エント ロピー正の自己同型写像の存在が示されたことで, 力学系的な立場からも興味の 対象となってきている. ここでは, 複素射影平面の9点ブローアップで得られる2 つの有理曲面を用意して, 2つの有理曲面を貼り合わせることでK3曲面が構成で きることを紹介する. この構成方法でえられるK3曲面の族は, 超越的なK3曲面を 含む大きな次元となっていて, 周期写像についても計算できることを示す. さら に時間が許せば, エントロピー正の超越的な自己同型写像との関係についても言 及する.

Wim Veys   "Zeta functions and oscillatory integrals for meromorphic functions"

This is joint work with Wilson Zuniga-Galindo. In the 70's, Igusa developed a uniform theory for local zeta functions and oscillatory integrals attached to polynomials with coefficients in a local field of characteristic zero. Over the real and complex field, there is related work of Varchenko. We extend this theory to the case of rational functions, or, more generally, meromorphic functions f/g, with coeffcients in any local field of characteristic zero. This generalization is far from straightforward, due to the fact that several new geometric phenomena appear. Also, the oscillatory integrals have two different asymptotic expansions: the `usual' one when the norm of the paramete r tends to infinity, and another one when the norm of the parameter tends to zero. The first asymptotic expansion is controlled by the (negative) poles of all the twisted local zeta functions associated to the meromorphic functions f/g - c, for certain special geometric values c. The second expansion is controlled by the (positive) poles of all the twisted local zeta functions associated to f/g.

Dominik Wrazidlo   "Fold maps, positive topological field theories, and exotic spheres"

Using ideas from theoretical physics as well as the concept of Eilenberg completeness of semirings from computer science, Markus Banagl has recently proposed the rigorous framework of so-called positive topological field theories (PTFTs). While capturing topological information in a state sum that obeys Atiyah’s gluing axiom, PTFTs have the potential to inspire the construction of innovative (differential) topological invariants for manifolds of arbitrary dimension. In this talk, we explain Banagl’s construction of a concrete PTFT which extracts combinatorial data from the singular pattern of certain fold maps of bordisms into the plane. By eliminating the technical key condition Banagl imposes on fold maps, we attack the computation of the state sum. In particular, beyond a theorem of Saeki, we present our result on the detection of certain Kervaire spheres via indefinite folds.

Kohei Yahiro  "Integral transforms of D-modules on partial flag varieties"

Beilinson and Bernstein applied the theory of D-modules to the study of representations of semisimple Lie algebras. They gave another proof of Casselman’s submodule theorem using certain integral transforms of D-modules on full flag varieties (intertwining functors). We define intertwining functors for partial flag varieties and prove that they are equivalences of derived categories. Intertwining functors give rise to equivalences of some equivariant derived categories for any smooth algebraic variety with an action of a semisimple algebraic group. We explain some of applications of these equivalences to representations of semisimple Lie algebras.

Ken-ichi Yoshikawa  "Enriques manifolds and analytic torsion"

We introduce a holomorphic torsion invariant for Enriques manifolds and discuss its applications to their moduli space. If time allows, we will also discuss some examples related to automorphic forms.