## Seminar on Geometric Complex Analysis

Date, time & place Monday 10:30 - 12:00 128Room #128 (Graduate School of Math. Sci. Bldg.) Kengo Hirachi, Shigeharu Takayama, Ryosuke Nomura

Seminar information archive

### 2019/11/18

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Ken-ichi Yoshikawa (Kyoto Univ.)
j-invariant and Borcherds Phi-function (Japanese)
[ Abstract ]
The j-invariant is a modular function on the complex upper half plane inducing an isomorphism between the moduli space of elliptic curves and the complex plane. Besides the j-invariant itself, the difference of j-invariants has also attracted some mathematicians. In this talk, I will explain a factorization of the difference of j-invariants in terms of Borcherds Phi-function, the automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor. This is a joint work with Shu Kawaguchi and Shigeru Mukai.

### 2019/10/28

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Junjiro Noguchi (Univ. of Tokyo)
On Kiyoshi Oka's unpublished papers 1943 (Japanese)

### 2019/10/21

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Yoshihiko Matsumoto (Osaka Univ.)
Canonical almost complex structures on ACH Einstein manifolds
[ Abstract ]
Einstein ACH (asymptotically complex hyperbolic) manifolds are seen as a device that establishes a correspondence between CR geometry on the boundary and Riemannian geometry in “the bulk.” This talk concerns an idea of enriching the geometric structure of the bulk by adding some almost complex structure compatible with the metric. I will introduce an energy functional of almost complex structures and discuss an existence result of critical points when the given ACH Einstein metric is a small perturbation of the Cheng-Yau complete K?hler-Einstein metric on a bounded strictly pseudoconvex domain. The renormalized Chern-Gauss-Bonnet formula is also planned to be discussed.

### 2019/10/07

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Yusaku Chiba (Ochanomizu Univ.)
Cohomology of vector bundles and non-pluriharmonic loci (Japanese)
[ Abstract ]
We study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem. We especially study the examples of non-pluriharmonic loci in smooth toric varieties. I would like to explain the relation of non-pluriharmonic loci and polytopes.

### 2019/09/30

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Sachiko Hamano (Osaka City Univ.)
Rigidity of the directional moduli on pseudoconvex domains fibered by open Riemann surfaces
[ Abstract ]
G. Schmieder-M. Shiba observed conformal embeddings of a fixed open Riemann surface of positive finite genus into closed Riemann surfaces of the same genus, and they showed the range of each diagonal element of the period matrices. Now we shall consider a smooth deformation of open Riemann surfaces with a complex parameter. In this talk, we show the rigidity of directional moduli induced by elements of the period matrices on pseudoconvex domains fibered by open Riemann surfaces of the same topological type.

### 2019/07/08

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Hiroshi Kaneko (Tokyo University of Science)
A Riemann-Roch theorem on a weighted infinite graph (Japanese)
[ Abstract ]
A Riemann-Roch theorem on a connected finite graph was initiated by M. Baker and S. Norine, where connected graph with finite vertices was investigated and unit weight was given on each edge and vertex of the graph. Since a counterpart of the lowest exponents of the complex variable in the Laurent series was proposed as divisor for the Riemann-Roch theorem on graph, its relationships with tropical geometry were highlighted earlier than other complex analytical observations on graphs. On the other hand, M. Baker and F. Shokrieh revealed tight relationships between chip-firing games and potential theory on graphs, by characterizing reduced divisors on graphs as the solution to an energy minimization problem. The objective of this talk is to establish a Riemann-Roch theorem on an edge-weighted infinite graph. We introduce vertex weight assigned by the given weights of adjacent edges other than the units for expression of divisors and assume finiteness of total mass of graph. This is a joint work with A. Atsuji.

### 2019/07/01

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Yeping Zhang (Kyoto Univ.)
BCOV invariant and birational equivalence (English)
[ Abstract ]
Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called BCOV invariant. Now we consider a pair (X,Y), where X is a Kaehler manifold and $Y ¥subseteq X$ is a canonical divisor. In this talk, we extend the BCOV invariant to such pairs. The extended BCOV invariant is well-behaved under birational equivalence. We expect that these considerations may eventually lead to a positive answer to Yoshikawa's conjecture that the BCOV invariant for Calabi-Yau threefold is a birational invariant.

### 2019/06/24

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Atsushi Yamamori (Kogakuin University)
A certain holomorphic invariant and its applications (Japanese)
[ Abstract ]
In this talk, we first explain a Bergman geometric proof of inequivalence of the unit ball and the bidisk. In this proof, the homogeneity of the domains plays a substantial role. We next explain a recent attempt to extend our method for non-homogeneous cases.

### 2019/06/17

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Andrei Pajitnov (Universite de Nantes)
Inoue surfaces and their generalizations (English)
[ Abstract ]
In 1972 M. Inoue constructed complex non-algebraic surfaces that proved very important for classification of surfaces via the Enriques-Kodaira scheme. Inoue surface is the quotient of H ¥times C by action of a discreet group associated to a given matrix in SL(3, Z). In 2005 K. Oeljeklaus and M. Toma generalized Inoue’s construction to higher dimensions. Oeljeklaus-Toma manifold is the quotient of H^s ¥times C^n by action of a discreet group, associated to the maximal order of a given algebraic number field.
In this talk, I will give a brief overview of these works and related results. Then I will discuss a new generalization of Inoue surfaces to higher dimensions. The manifold in question is the quotient of H ¥times C^n by action of a discreet group associated to a given matrix in SL(2n+1, Z). This is joint work with Hisaaki Endo.

### 2019/05/27

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Takayuki Koike (Osaka City Univ.)
Gluing construction of K3 surfaces (Japanese)
[ Abstract ]
Arnol'd showed the uniqueness of the complex analytic structure of a small neighborhood of an elliptic curve embedded in a surface whose normal bundle satisfies "Diophantine condition" in the Picard variety. By applying this theorem, we construct a K3 surface by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of anti-canonical curves of blow-ups of the projective planes at general nine points. Our construction has 19 complex dimensional degrees of freedom. For general parameters, the resulting K3 surface is neither Kummer nor projective. By the argument based on the concrete computation of the period map, we also investigate which points in the period domain correspond to K3 surfaces obtained by such construction. (Based on joint work with Takato Uehara)

### 2019/05/20

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Tomohiro Okuma (Yamagata Univ.)
Cohomology and normal reduction numbers of normal surface singularities (Japanese)
[ Abstract ]
The normal reduction number of a normal surface singularity relates the maximal degree of the generators of associated graded algebra for certain line bundles on resolution spaces. We show fundamental properties of this invariant and formulas for some special cases. This talk is based on the joint work with Kei-ichi Watanabe and Ken-ichi Yoshida.

### 2019/05/13

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Homare Tadano (Tokyo Univ. of Science)
Some Bonnet--Myers Type Theorems for Transverse Ricci Solitons on Complete Sasaki Manifolds (Japanese)
[ Abstract ]
The aim of this talk is to discuss the compactness of complete Ricci solitons and its generalizations. Ricci solitons were introduced by R. Hamilton in 1982 and are natural generalizations of Einstein manifolds. They correspond to self-similar solutions to the Ricci flow and often arise as singularity models of the flow. The importance of Ricci solitons was demonstrated by G. Perelman, where they played crucial roles in his affirmative resolution of the Poincare conjecture.
In this talk, after we review basic facts on Ricci solitons, I would like to introduce some Bonnet--Myers type theorems for complete Ricci solitons. Our results generalize the previous Bonnet--Myers type theorems due to W. Ambrose (1957), J. Cheeger, M. Gromov, and M. Taylor (1982), M. Fernandez-Lopez and E. Garcia-Rio (2008), M. Limoncu (2010, 2012), Z. Qian (1997), Y. Soylu (2017), and G. Wei and W. Wylie (2009). Moreover, I would also like to extend such Bonnet--Myers type theorems to the case of transverse Ricci solitons on complete Sasaki manifolds. Our results generalize the previous Bonnet--Myers type theorems for complete Sasaki manifolds due to I. Hasegawa and M. Seino (1981) and Y. Nitta (2009).

### 2019/04/22

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Tomoyuki Hisamoto (Nayoya Univ.)
Optimal destabilizer for a Fano manifold (Japanese)
[ Abstract ]
Around 2005, S. Donaldson asked whether the lower bound of the Calabi functional is achieved by a sequence of the normalized Donaldson-Futaki invariants.
For a Fano manifold we construct a sequence of multiplier ideal sheaves from a new geometric flow and answer to Donaldson's question.

### 2019/04/15

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Takeo Ohsawa (Nagoya Univ.)
(Japanese)

### 2019/01/28

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Kentaro Ohno (University of Tokyo)
Minimizing CM degree and slope stability of projective varieties (JAPANESE)
[ Abstract ]
Chow-Mumford (CM) line bundle is considered to play an important role in moduli problem for K-stable Fano varieties. In this talk, we consider a minimization problem of the degree of the CM line bundle among all possible fillings of a polarized family over a punctured curve. We show that such minimization implies the slope semistability of the fiber if the central fiber is smooth.

### 2019/01/21

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Nicholas James McCleerey (Northwestern University)
POLAR TRANSFORM AND LOCAL POSITIVITY FOR CURVES
(ENGLISH)
[ Abstract ]
Using the duality of positive cones, we show that applying the polar transform from convexanalysis to local positivity invariants for divisors gives interesting and new local positivity invariantsfor curves. These new invariants have nice properties similar to those for divisors. In particular, thisenables us to give a characterization of the divisorial components of the non-K¨ahler locus of a big class. This is joint worth with Jian Xiao.

### 2018/12/17

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Joe Kamimoto (Kyushu University)
Newton polyhedra and order of contact on real hypersurfaces (JAPANESE)
[ Abstract ]
This talk will concern some issues on order of contact on real hypersurfaces, which was introduced by D'Angelo. To be more precise, a sufficient condition for the equality of regular type and singular type is given. This condition is written by using the Newton polyhedron of a defining function. Our result includes earlier known results concerning convex domains, pseudoconvex Reinhardt domains and pseudoconvex domains whose regular types are 4. Furthermore, under the above condition, the values of the types can be directly seen in a simple geometrical information from the Newton polyhedron.

The technique of using Newton polyhedra has many significant applications in singularity theory. In particular, this technique has been great success in the study of the Lojasiewicz exponent. Our study about the types is analogous to some works on the Lojasiewicz exponent.

### 2018/12/03

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Genki Hosono (University of Tokyo)
Variational theories of complex analysis of several variables (JAPANESE)
[ Abstract ]
In complex analysis, there are some values and functions which are subharmonic under pseudoconvex variations.
For example, the variation of Robin constant (Yamaguchi) and of Bergman kernels (Maitani-Yamaguchi) were studied.
As a generalization, the curvature positivity of spaces of $L^2$ holomorphic functions is proved by Berndtsson.
These theories are known to have some relations with $L^2$ extension theorems.
In this talk, I will explain known results and discuss the variation problem of the Azukawa pseudometric, which is a generalization of the Robin constant.

### 2018/11/26

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Hisashi Kasuya (Osaka University)
DGA-Models of variations of mixed Hodge structures (JAPANESE)

### 2018/11/19

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Gerard Freixas i Montplet (Centre National de la Recherche Scientifique)
BCOV invariants of Calabi-Yau varieties (ENGLISH)
[ Abstract ]
The BCOV invariant of Calabi-Yau threefolds was introduced by Fang-Lu-Yoshikawa, themselves inspired by physicists Bershadsky-Cecotti-Ooguri-Vafa. It is a real number, obtained from a combination of holomorphic analytic torsion, and suitably normalized so that it only depends on the complex structure of the threefold. It is conjecturaly expected to encode genus 1 Gromov-Witten invariants of a mirror Calabi-Yau threefold. In order to confirm this prediction for a remarkable example, Fang-Lu-Yoshikawa studied the asymptotic behavior for degenerating families of Calabi-Yau threefolds acquiring at most ordinary double point (ODP) singularities. Their methods rely on former results by Yoshikawa on the singularities of Quillen metrics, together with more classical arguments in the theory of degenerations of Hodge structures and Hodge metrics. In this talk I will present joint work with Dennis Eriksson (Chalmers) and Christophe Mourougane (Rennes), where we extend the construction of the BCOV invariant to any dimension and we give precise asymptotic formulas for degenerating families of Calabi-Yau manifolds. Under several hypothesis on the geometry of the singularities acquired, our general formulas drastically simplify and prove some conjectures or predictions in the literature (Liu-Xia for semi-stable minimal families in dimension 3, Klemm-Pandharipande for ODP singularities in dimension 4, etc.). For this, we slightly improve Yoshikawa's results on the singularities of Quillen metrics, and we also provide a complement to Schmid's asymptotics of Hodge metrics when the monodromy transformations are non-unipotent.

### 2018/11/05

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Hiroshige Shiga (Tokyo Institute of Technology)
On the quasiconformal equivalence of Dynamical Cantor sets (JAPANESE)
[ Abstract ]
Let $E$ be a Cantor set in the Riemann sphere $\widehat{\mathbb C}$, that is, a totally disconnected perfect set in $\widehat{\mathbb C}$.
The standard middle one-thirds Cantor set $\mathcal{C}$ is a typical example.
We consider the complement $X_{E}:=\widehat{\mathbb C}\setminus E$ of the Cantor set $E$.
It is an open Riemann surface with uncountable many boundary components.
We are interested in the quasiconformal equivalence of such surfaces.

In this talk, we discuss the quasiconformal equivalence for the complements of Cantor sets given by dynamical systems.

### 2018/10/29

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Shin-ichi Matsumura (Tohoku University)
On morphisms of compact Kaehler manifolds with semi-positive holomorphic sectional curvature (JAPANESE)
[ Abstract ]
In this talk, we consider a smooth projective variety $X$ with semi-positive holomorphic "sectional" curvature, motivated by generalizing Howard-Smyth-Wu's structure theorem and Mok's result for compact Kaehler manifold with semi-positive "bisectional" curvature.
We prove that, if $X$ admits a holomorphic maximally rationally connected fibration $X ¥to Y$, then the morphism is always smooth (that is, a submersion), that the image $Y$ admits a finite ¥'etale cover $T ¥to Y$ by a complex
torus $T$, and further that all the fibers $F$ are isomorphic.
This gives a structure theorem for $X$ when $X$ is a surface.
Moreover we show that $X$ is rationally connected, if the holomorphic sectional curvature is quasi-positive.
This result gives a generalization of Yau's conjecture.

### 2018/10/22

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
On certain hyperconvex manifolds without non-constant bounded holomorphic functions (JAPANESE)
[ Abstract ]
For each compact Riemann surface of genus > 1, we can construct a Riemann sphere bundle over the given Riemann surface using the projective structure induced by its uniformization.
The total space of this bundle is divided into two 1-convex domains by a closed Levi-flat real hypersurface. Although these two domains are not biholomorphic, we see that they have several function theoretic properties in common. In this talk, I would like to explain these common properties: hyperconvexity and expressions for certain Green function, and Liouville property and growth estimates of holomorphic functions.

### 2018/10/15

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Ikkei Hotta (Yamaguchi University)
Recent problems on Loewner theory (JAPANESE)
[ Abstract ]
Loewner Theory, which goes back to the parametric representation of univalent functions introduced by Loewner in 1923, has recently undergone significant development in various directions, including Schramm’s stochastic version of the Loewner differential equation and the new intrinsic approach suggested by Bracci, Contreras, Diaz-Madrigal and Gumenyuk.

In this talk, we firstly review the theory of Loewner equations in classical and modern treatments. Then we discuss some recent problems on the theory:
(i) Quasiconformal extensions of Loewner chains;
(ii) Hydrodynamics of multiple SLE.

### 2018/07/23

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Filippo Bracci (University of Rome Tor Vergata)
Strange Fatou components of automorphisms of $\mathbb{C}^2$ and Runge embedding of $\mathbb{C} \times \mathbb{C}^*$ into $\mathbb{C}^2$. (ENGLISH)
[ Abstract ]
The classification of Fatou components for automorphisms of the complex space of dimension greater than $1$ is an interesting and difficult task. Recent deep results prove that the one-dimensional setting is deeply different from the higher dimensional one. Given an automorphism F of $\mathbb{C}^k$, the first bricks in the theory that one would like to understand are invariant Fatou components, namely, those connected open sets $U$, completely invariant under $F$, where the dynamics of $F$ is not chaotic. Among those, we consider “attracting” Fatou components, that is, those components on which the iterates of $F$ converge to a single point. Attracting Fatou components can be recurrent, if the limit point is inside the component or non-recurrent. Recurrent attracting Fatou components are always biholomorphic to $\mathbb{C}^k$, since it was proved by H. Peters, L. Vivas and E. F. Wold that in such a case the point is an attracting (hyperbolic) fixed point, and the Fatou component coincides with the global basin of attraction. Also, as a consequence of works of Ueda and Peters-Lyubich, it is know that all attracting non-recurrent Fatou components of polynomial automorphisms of $\mathbb{C}^2$ are biholomorphic to $\mathbb{C}^2$. One can quite easily find non-simply connected non-recurrent attracting Fatou components in $\mathbb{C}^3$ (mixing a two- dimensional dynamics with a dynamics with non-isolated fixed points in one- variable). In this talk I will explain how to construct a non-recurrent attracting Fatou component in $\mathbb{C}^2$ which is biholomorphic to $\mathbb{C}\times\mathbb{C}^*$. This“fantastic beast” is obtained by globalizing, using a result of F. Forstneric, a local construction due to the speaker and Zaitsev, which allows to create a global basin of attraction for an automorphism, and a Fatou coordinate on it. The Fatou coordinate turns out to be a fiber bundle map on $\mathbb{C}$, whose fiber is $\mathbb{C}^*$, then the global basin is biholomorphic to $\mathbb{C}\times\mathbb{C}^*$. The most subtle point is to show that such a basin is indeed a Fatou component. This is done exploiting Poschel's results about existence of local Siegel discs and suitable estimates for the Kobayashi distance.

Since attracting Fatou components are Runge, it turns out that this construction gives also an example of a Runge embedding of $\mathbb{C}\times\mathbb{C}^*$ into $\mathbb{C}^2$. Moreover, this example shows an automorphism of $\mathbb{C}^2$ leaving invariant two analytic discs intersecting transversally at the origin.

The talk is based on a joint work with J. Raissy and B. Stensones.