## Seminar on Geometric Complex Analysis

Seminar information archive ～12/08｜Next seminar｜Future seminars 12/09～

Date, time & place | Monday 10:30 - 12:00 128Room #128 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | Kengo Hirachi, Shigeharu Takayama |

**Seminar information archive**

### 2019/04/22

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Optimal destabilizer for a Fano manifold (Japanese)

**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.

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.)

(Japanese)

**Takeo Ohsawa**(Nagoya Univ.)(Japanese)

### 2019/01/28

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Minimizing CM degree and slope stability of projective varieties (JAPANESE)

**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.

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.)

POLAR TRANSFORM AND LOCAL POSITIVITY FOR CURVES

(ENGLISH)

**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.

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.)

Newton polyhedra and order of contact on real hypersurfaces (JAPANESE)

**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.

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.)

Variational theories of complex analysis of several variables (JAPANESE)

**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.

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.)

DGA-Models of variations of mixed Hodge structures (JAPANESE)

**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.)

BCOV invariants of Calabi-Yau varieties (ENGLISH)

**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.

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.)

On the quasiconformal equivalence of Dynamical Cantor sets (JAPANESE)

**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.

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.)

On morphisms of compact Kaehler manifolds with semi-positive holomorphic sectional curvature (JAPANESE)

**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.

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)

**Masanori Adachi**(Shizuoka University)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.

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.)

Recent problems on Loewner theory (JAPANESE)

**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.

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.)

Strange Fatou components of automorphisms of $\mathbb{C}^2$ and Runge embedding of $\mathbb{C} \times \mathbb{C}^*$ into $\mathbb{C}^2$. (ENGLISH)

**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.

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.

### 2018/07/09

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Rigidity results for symplectic curvature flow (ENGLISH)

**Casey Kelleher**(Princeton University)Rigidity results for symplectic curvature flow (ENGLISH)

[ Abstract ]

We continue studying a parabolic flow of almost Kähler structure introduced by Streets and Tian which naturally extends Kähler-Ricci flow onto symplectic manifolds. In a system consisting primarily of quantities related to the Chern connection we establish clean formulas for the evolutions of canonical objects. Using this we give an extended characterization of fixed points of the flow.

We continue studying a parabolic flow of almost Kähler structure introduced by Streets and Tian which naturally extends Kähler-Ricci flow onto symplectic manifolds. In a system consisting primarily of quantities related to the Chern connection we establish clean formulas for the evolutions of canonical objects. Using this we give an extended characterization of fixed points of the flow.

### 2018/07/02

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Rigidity of certain groups of circle homeomorphisms and Teichmueller spaces (JAPANESE)

**Katsuhiko Matsuzaki**(Waseda University)Rigidity of certain groups of circle homeomorphisms and Teichmueller spaces (JAPANESE)

[ Abstract ]

In this talk, I explain a complex analytic method and its applications

for the study of quasisymmetric homeomorphisms of the circle by extending them to the unit disk quasiconformally.

In RIMS conference "Open Problems in Complex Geometry'' held in 2010,

I gave a talk entitled "Problems on infinite dimensional Teichmueller spaces", and

mentioned several problems on the fixed points of group actions on

the universal Teichmueller space and its subspaces, and the rigidity of conjugation of

certain groups of circle homeomorphisms.

I will report on the development of these problems since then.

In this talk, I explain a complex analytic method and its applications

for the study of quasisymmetric homeomorphisms of the circle by extending them to the unit disk quasiconformally.

In RIMS conference "Open Problems in Complex Geometry'' held in 2010,

I gave a talk entitled "Problems on infinite dimensional Teichmueller spaces", and

mentioned several problems on the fixed points of group actions on

the universal Teichmueller space and its subspaces, and the rigidity of conjugation of

certain groups of circle homeomorphisms.

I will report on the development of these problems since then.

### 2018/06/25

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Cornered Asymptotically Hyperbolic Spaces

**Stephen McKeown**(Princeton University)Cornered Asymptotically Hyperbolic Spaces

[ Abstract ]

This talk will concern cornered asymptotically hyperbolic spaces, which have a finite boundary in addition to the usual infinite boundary. I will first describe the construction a normal form near the corner for these spaces. Then I will discuss formal existence and uniqueness, near the corner, of asymptotically hyperbolic Einstein metrics, with a CMC-umbilic condition imposed on the finite boundary. This is analogous to the Fefferman-Graham construction for the ordinary, non-cornered setting. Finally, I will present work in progress regarding scattering on such spaces.

This talk will concern cornered asymptotically hyperbolic spaces, which have a finite boundary in addition to the usual infinite boundary. I will first describe the construction a normal form near the corner for these spaces. Then I will discuss formal existence and uniqueness, near the corner, of asymptotically hyperbolic Einstein metrics, with a CMC-umbilic condition imposed on the finite boundary. This is analogous to the Fefferman-Graham construction for the ordinary, non-cornered setting. Finally, I will present work in progress regarding scattering on such spaces.

### 2018/06/11

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Cohomology of non-pluriharmonic loci (JAPANESE)

**Yusaku Tiba**(Ochanomizu University)Cohomology of non-pluriharmonic loci (JAPANESE)

[ Abstract ]

In this talk, we study a pseudoconvex counterpart of the Lefschetz hyperplane theorem.

We show a relation between the cohomology of a pseudoconvex domain and the cohomology of the non-pluriharmonic locus of an exhaustive plurisubharmonic function.

In this talk, we study a pseudoconvex counterpart of the Lefschetz hyperplane theorem.

We show a relation between the cohomology of a pseudoconvex domain and the cohomology of the non-pluriharmonic locus of an exhaustive plurisubharmonic function.

### 2018/06/04

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Junjiro Noguchi**(The University of Tokyo)(JAPANESE)

### 2018/05/28

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A generalization of Kähler Einstein metrics for Fano manifolds with non-vanishing Futaki invariant (JAPANESE)

**Satoshi Nakamura**(Tohoku University)A generalization of Kähler Einstein metrics for Fano manifolds with non-vanishing Futaki invariant (JAPANESE)

[ Abstract ]

The existence problem of Kähler Einstein metrics for Fano manifolds was one of the central problems in Kähler Geometry. The vanishing of the Futaki invariant is known as an obstruction to the existence of Kähler Einstein metrics. Generalized Kähler Einstein metrics (GKE for short), introduced by Mabuchi in 2000, is a generalization of Kähler Einstein metrics for Fano manifolds with non-vanishing Futaki invariant. In this talk, we give the followings:

(i) The positivity for the Hessian of the Ricci Calabi functional which characterizes GKE as its critical points, and its application.

(ii) A criterion for the existence of GKE on toric Fano manifolds from view points of an algebraic stability and an analytic stability.

The existence problem of Kähler Einstein metrics for Fano manifolds was one of the central problems in Kähler Geometry. The vanishing of the Futaki invariant is known as an obstruction to the existence of Kähler Einstein metrics. Generalized Kähler Einstein metrics (GKE for short), introduced by Mabuchi in 2000, is a generalization of Kähler Einstein metrics for Fano manifolds with non-vanishing Futaki invariant. In this talk, we give the followings:

(i) The positivity for the Hessian of the Ricci Calabi functional which characterizes GKE as its critical points, and its application.

(ii) A criterion for the existence of GKE on toric Fano manifolds from view points of an algebraic stability and an analytic stability.

### 2018/05/21

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Kähler-Ricci soliton, K-stability and moduli space of Fano

manifolds (JAPANESE)

**Eiji Inoue**(The University of Tokyo)Kähler-Ricci soliton, K-stability and moduli space of Fano

manifolds (JAPANESE)

[ Abstract ]

Kähler-Ricci soliton is a kind of canonical metrics on Fano manifolds and is a natural generalization of Kähler-Einstein metric in

view of Kähler-Ricci flow.

In this talk, I will explain the following good geometric features of Fano manifolds admitting Kähler-Ricci solitons:

1. Volume minimization, reductivity and uniqueness results established by Tian&Zhu.

2. Relation to algebraic (modified) K-stability estabilished by Berman&Witt-Niström and Datar&Székelyhidi.

3. Moment map picture for Kähler-Ricci soliton (‘real side’)

4. Moduli stack (‘virtual side’) and moduli space of them

A result in 1 is indispensable for the formulation in 3 and 4, and explains why we should consider solitons, beyond Einstein metrics. I also show an essential idea in the construction of the moduli space of Fano manifolds admitting Kähler-Ricci solitons and give some remarks on technical key point.

Kähler-Ricci soliton is a kind of canonical metrics on Fano manifolds and is a natural generalization of Kähler-Einstein metric in

view of Kähler-Ricci flow.

In this talk, I will explain the following good geometric features of Fano manifolds admitting Kähler-Ricci solitons:

1. Volume minimization, reductivity and uniqueness results established by Tian&Zhu.

2. Relation to algebraic (modified) K-stability estabilished by Berman&Witt-Niström and Datar&Székelyhidi.

3. Moment map picture for Kähler-Ricci soliton (‘real side’)

4. Moduli stack (‘virtual side’) and moduli space of them

A result in 1 is indispensable for the formulation in 3 and 4, and explains why we should consider solitons, beyond Einstein metrics. I also show an essential idea in the construction of the moduli space of Fano manifolds admitting Kähler-Ricci solitons and give some remarks on technical key point.

### 2018/05/14

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Harmonic map and the Einstein equation in five dimension (JAPANESE)

**Sumio Yamada**(Gakushuin University)Harmonic map and the Einstein equation in five dimension (JAPANESE)

[ Abstract ]

We present a new method in constructing 5-dimensional stationary solutions to the vacuum Einstein equation. In 1917, H. Weyl expressed the Schwarzschild black hole solution using a cylindical coordinate system, and consequently realized that the metric is completely determined by a harmonic function. Since then, the relation between harmonic maps and the Einstein equation has been explored mostly by physicists, which they call the sigma model of the Einstein equation. In this talk, after explaining the historical background, we demonstrate that in 5D, the Einstein spacetimes can have a wide range of black hole horizons in their topological types. In particular we establish an existence theorem of harmonic maps, which subsequently leads to constructions of 5D spacetimes with black hole horizons of positive Yamabe types, namely $S^3$, $S^2 \times S^1$, and the lens space $L(p,q)$. This is a joint work with Marcus Khuri and Gilbert Weinstein.

We present a new method in constructing 5-dimensional stationary solutions to the vacuum Einstein equation. In 1917, H. Weyl expressed the Schwarzschild black hole solution using a cylindical coordinate system, and consequently realized that the metric is completely determined by a harmonic function. Since then, the relation between harmonic maps and the Einstein equation has been explored mostly by physicists, which they call the sigma model of the Einstein equation. In this talk, after explaining the historical background, we demonstrate that in 5D, the Einstein spacetimes can have a wide range of black hole horizons in their topological types. In particular we establish an existence theorem of harmonic maps, which subsequently leads to constructions of 5D spacetimes with black hole horizons of positive Yamabe types, namely $S^3$, $S^2 \times S^1$, and the lens space $L(p,q)$. This is a joint work with Marcus Khuri and Gilbert Weinstein.

### 2018/05/07

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Proper holomorphic mappings and generalized pseudoellipsoids (JAPANESE)

**Atsushi Hayashimoto**(National Institute of Technology, Nagano College)Proper holomorphic mappings and generalized pseudoellipsoids (JAPANESE)

[ Abstract ]

We study the classification of proper holomorphic mappings between generalized pseudoellipsoids of different dimensions.

Huang proved some classification theorems of proper holomorphic mappings between balls of different dimensions, which are called gap theorems. Our present theorems are their weakly pseudoconvex versions.

In the theorem, classified mapping is so-called a variables splitting mapping and each component is derived from a homogeneous proper polynomial mapping between balls.

The essential methods are the ''good'' decompositions of CR vector bundle and reduction the mapping under consideration to the mapping of balls. By this reduction, we can apply Huang's gap theorem.

We study the classification of proper holomorphic mappings between generalized pseudoellipsoids of different dimensions.

Huang proved some classification theorems of proper holomorphic mappings between balls of different dimensions, which are called gap theorems. Our present theorems are their weakly pseudoconvex versions.

In the theorem, classified mapping is so-called a variables splitting mapping and each component is derived from a homogeneous proper polynomial mapping between balls.

The essential methods are the ''good'' decompositions of CR vector bundle and reduction the mapping under consideration to the mapping of balls. By this reduction, we can apply Huang's gap theorem.

### 2018/04/23

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Degeneration and bifurcation of quadratic endomorphisms of $\mathbb{P}^2$ towards a Hénon map (JAPANESE)

**Yûsuke Okuyama**(Kyoto Institute of Technology)Degeneration and bifurcation of quadratic endomorphisms of $\mathbb{P}^2$ towards a Hénon map (JAPANESE)

[ Abstract ]

The space of quadratic holomorphic endomorphisms of P^2 (over C) is

canonically identified with the complement of the zero locus of the

resultant form on P^{17}, and all H¥'enon maps, which are (the only)

interesting ones among all the quadratic polynomial automorphisms of C^2,

live in this zero locus.

We will talk about our joint work with Fabrizio Bianchi (Imperial College,

London) on the (algebraic) degeneration of quadratic endomorphisms of C^2

towards H¥'enon maps in terms of Berteloot-Bianchi-Dupont's

bifurcation/unstability theory of holomorphic families of endomorphisms of P^k,

which mostly generalizes Ma¥~n¥'e-Sad-Sullivan, Lyubich, and DeMarco's seminal

and similar theory on P^1.

Some preliminary knowledge on ergodic theory and pluripotential theory

would be desirable, but not be assumed.

The space of quadratic holomorphic endomorphisms of P^2 (over C) is

canonically identified with the complement of the zero locus of the

resultant form on P^{17}, and all H¥'enon maps, which are (the only)

interesting ones among all the quadratic polynomial automorphisms of C^2,

live in this zero locus.

We will talk about our joint work with Fabrizio Bianchi (Imperial College,

London) on the (algebraic) degeneration of quadratic endomorphisms of C^2

towards H¥'enon maps in terms of Berteloot-Bianchi-Dupont's

bifurcation/unstability theory of holomorphic families of endomorphisms of P^k,

which mostly generalizes Ma¥~n¥'e-Sad-Sullivan, Lyubich, and DeMarco's seminal

and similar theory on P^1.

Some preliminary knowledge on ergodic theory and pluripotential theory

would be desirable, but not be assumed.

### 2018/04/16

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Metrics on a closed surface which maximize the first eigenvalue of the Laplacian (JAPANESE)

**Shin Nayatani**(Nagoya University)Metrics on a closed surface which maximize the first eigenvalue of the Laplacian (JAPANESE)

[ Abstract ]

In this talk, I will focus on recent progress on metrics that maximize the first eigenvalue of the Laplacian (under area normalization) on a closed surface. First, I introduce Hersch-Yang-Yau's inequality (1970, 1980), which was the starting point of the above problem. This is an inequality indicating that the first eigenvalue (precisely, the product of it with the area) is bounded from above by a constant depending only on the genus of the surface. Then I will outline the recent progress on the existence problem for maximizing metrics together with the relation with minimal surfaces in the sphere. Finally, I will discuss Jacobson-Levitin-Nadirashvili-Nigam-Polterovich's conjecture, which explicitly predicts maximizing metrics in the case of genus two, and the affirmative resolution of it (joint work with Toshihiro Shoda).

In this talk, I will focus on recent progress on metrics that maximize the first eigenvalue of the Laplacian (under area normalization) on a closed surface. First, I introduce Hersch-Yang-Yau's inequality (1970, 1980), which was the starting point of the above problem. This is an inequality indicating that the first eigenvalue (precisely, the product of it with the area) is bounded from above by a constant depending only on the genus of the surface. Then I will outline the recent progress on the existence problem for maximizing metrics together with the relation with minimal surfaces in the sphere. Finally, I will discuss Jacobson-Levitin-Nadirashvili-Nigam-Polterovich's conjecture, which explicitly predicts maximizing metrics in the case of genus two, and the affirmative resolution of it (joint work with Toshihiro Shoda).

### 2018/01/22

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Recent topics on the study of the Gauss images of minimal surfaces

**Yu Kawakami**(Kanazawa University)Recent topics on the study of the Gauss images of minimal surfaces

[ Abstract ]

In this talk, we give a survey of recent advances on the study of the images of the Gauss maps of complete minimal surfaces in Euclidean space.

In this talk, we give a survey of recent advances on the study of the images of the Gauss maps of complete minimal surfaces in Euclidean space.