## Seminar on Geometric Complex Analysis

Seminar information archive ～10/09｜Next seminar｜Future seminars 10/10～

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

**Seminar information archive**

### 2015/06/08

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Mixed Hodge structures and Sullivan's minimal models of Sasakian manifolds (Japanese)

**Hisashi Kasuya**(Tokyo Institute of Technology)Mixed Hodge structures and Sullivan's minimal models of Sasakian manifolds (Japanese)

[ Abstract ]

By the result of Deligne, Griffiths, Morgan and Sullivan, the Malcev completion of the fundamental group of a compact Kahler manifold is quadratically presented. This fact gives good advances in "Kahler group problem" (Which groups can be the fundamental groups of compact Kahler manifolds?) In this talk, we consider the fundamental groups of compact Sasakian manifolds. We show that the Malcev Lie algebra of the fundamental group of a compact 2n+1-dimensional Sasakian manifold with n >= 2 admits a quadratic presentation by using Morgan's bigradings of Sullivan's minimal models of mixed-Hodge diagrams.

By the result of Deligne, Griffiths, Morgan and Sullivan, the Malcev completion of the fundamental group of a compact Kahler manifold is quadratically presented. This fact gives good advances in "Kahler group problem" (Which groups can be the fundamental groups of compact Kahler manifolds?) In this talk, we consider the fundamental groups of compact Sasakian manifolds. We show that the Malcev Lie algebra of the fundamental group of a compact 2n+1-dimensional Sasakian manifold with n >= 2 admits a quadratic presentation by using Morgan's bigradings of Sullivan's minimal models of mixed-Hodge diagrams.

### 2015/05/25

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On uniform K-stability (Japanese)

**Tomoyuki Hisamoto**(Nagoya Univ.)On uniform K-stability (Japanese)

[ Abstract ]

It is a joint work with Sébastien Boucksom and Mattias Jonsson. We first introduce functionals on the space of test configurations, as non-Archimedean analogues of classical functionals on the space of Kähler metrics. Then, uniform K-stability is defined as a counterpart of K-energy's coercivity condition. Finally, reproving and strengthening Y. Odaka's results, we study uniform K-stability of Kähler-Einstein manifolds.

It is a joint work with Sébastien Boucksom and Mattias Jonsson. We first introduce functionals on the space of test configurations, as non-Archimedean analogues of classical functionals on the space of Kähler metrics. Then, uniform K-stability is defined as a counterpart of K-energy's coercivity condition. Finally, reproving and strengthening Y. Odaka's results, we study uniform K-stability of Kähler-Einstein manifolds.

### 2015/05/18

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On a global estimate of the Diederich–Fornaess index of Levi-flat real hypersurfaces (Japanese)

**Masanori Adachi**(Tokyo Univ. of Science)On a global estimate of the Diederich–Fornaess index of Levi-flat real hypersurfaces (Japanese)

[ Abstract ]

We give yet another proof for a global estimate of the Diederich-Fornaess index of relatively compact domains with Levi-flat boundary, namely, the index must be smaller than or equal to the reciprocal of the dimension of the ambient space. Although the Diederich-Fornaess index is originally defined for relatively compact domains in complex manifolds, our formulation reveals that it makes sense for abstract Levi-flat CR manifolds.

We give yet another proof for a global estimate of the Diederich-Fornaess index of relatively compact domains with Levi-flat boundary, namely, the index must be smaller than or equal to the reciprocal of the dimension of the ambient space. Although the Diederich-Fornaess index is originally defined for relatively compact domains in complex manifolds, our formulation reveals that it makes sense for abstract Levi-flat CR manifolds.

### 2015/05/11

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Integral Kahler Invariants and the Bergman kernel asymptotics for line bundles

**Kengo Hirachi**(The Univ. of Tokyo)Integral Kahler Invariants and the Bergman kernel asymptotics for line bundles

[ Abstract ]

On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one forms, which do not contribute to the integral. We apply this decomposition formula to describe the asymptotic expansion of the Bergman kernel for positive line bundles and to show that the CR Q-curvature on a Sasakian manifold is a divergence. This is a joint work with Spyros Alexakis (U Toronto).

On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one forms, which do not contribute to the integral. We apply this decomposition formula to describe the asymptotic expansion of the Bergman kernel for positive line bundles and to show that the CR Q-curvature on a Sasakian manifold is a divergence. This is a joint work with Spyros Alexakis (U Toronto).

### 2015/04/27

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Variational formulas for canonical differentials and application (Japanese)

**Sachiko Hamano**(Fukushima Univ.)Variational formulas for canonical differentials and application (Japanese)

[ Abstract ]

We prove the variational formulas of the second order for $L_1$- and $L_0$-canonical differentials, which with the remarkable contrast are our first example in the case of the deforming non-planar open Riemann surface. As a direct application, we show the rigidity of the Euclidean radius of the moduli disk on open torus under pseudoconvexity. The main part of this talk is a joint work with Masakazu Shiba and Hiroshi Yamaguchi.

We prove the variational formulas of the second order for $L_1$- and $L_0$-canonical differentials, which with the remarkable contrast are our first example in the case of the deforming non-planar open Riemann surface. As a direct application, we show the rigidity of the Euclidean radius of the moduli disk on open torus under pseudoconvexity. The main part of this talk is a joint work with Masakazu Shiba and Hiroshi Yamaguchi.

### 2015/04/20

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Weighted Laplacians on real and complex complete metric measure spaces (Japanese)

**Akito Futaki**(The Univ. of Tokyo)Weighted Laplacians on real and complex complete metric measure spaces (Japanese)

[ Abstract ]

We compare the weighted Laplacians on real and complex (K¥"ahler) metric measure spaces. In the compact case K¥"ahler metric measure spaces are considered on Fano manifolds for the study of K¥"ahler Ricci solitons while real metric measure spaces are considered with Bakry-¥'Emery Ricci tensor. There are twisted Laplacians which are useful in both cases but look alike each other. We see that if we consider noncompact complete manifolds significant differences appear.

We compare the weighted Laplacians on real and complex (K¥"ahler) metric measure spaces. In the compact case K¥"ahler metric measure spaces are considered on Fano manifolds for the study of K¥"ahler Ricci solitons while real metric measure spaces are considered with Bakry-¥'Emery Ricci tensor. There are twisted Laplacians which are useful in both cases but look alike each other. We see that if we consider noncompact complete manifolds significant differences appear.

### 2015/04/13

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Campana's Multiplicity and Integral Points on P^2 (English)

**Yu Yasufuku**(Nihon Univ.)Campana's Multiplicity and Integral Points on P^2 (English)

[ Abstract ]

We analyze when the complements of (possibly reducible) curves in P^2 have Zariski-dense integral points. The analysis utilizes the structure theories for affine surfaces based on logarithmic Kodaira dimension. When the log Kodaira dimension is one, an important role is played by Campana's multiplicity divisors for fibrations, but there are some subtleties. This is a joint work with Aaron Levin (Michigan State).

We analyze when the complements of (possibly reducible) curves in P^2 have Zariski-dense integral points. The analysis utilizes the structure theories for affine surfaces based on logarithmic Kodaira dimension. When the log Kodaira dimension is one, an important role is played by Campana's multiplicity divisors for fibrations, but there are some subtleties. This is a joint work with Aaron Levin (Michigan State).

### 2015/04/06

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Analytic torsion for K3 surfaces with involution (Japanese)

**Ken-ichi Yoshikawa**(Kyoto Univ.)Analytic torsion for K3 surfaces with involution (Japanese)

[ Abstract ]

In 2004, I introduced a holomorphic torsion invariant for 2-elementary K3 surfaces, i.e., K3 surfaces with involution. In the talk, I will report a recent progress in this invariant. Namely, for all possible deformation types, the holomorphic torsion invariant viewed as a function on the moduli space, is expressed as the product of an explicit Borcherds lift and an explicit Siegel modular form. If time permits, I will interpret the result in terms of the BCOV invariant, i.e., the genus-one string amplitude in B-model, for Calabi-Yau threefolds of Borcea-Voisin. This is a joint work with Shouhei Ma.

In 2004, I introduced a holomorphic torsion invariant for 2-elementary K3 surfaces, i.e., K3 surfaces with involution. In the talk, I will report a recent progress in this invariant. Namely, for all possible deformation types, the holomorphic torsion invariant viewed as a function on the moduli space, is expressed as the product of an explicit Borcherds lift and an explicit Siegel modular form. If time permits, I will interpret the result in terms of the BCOV invariant, i.e., the genus-one string amplitude in B-model, for Calabi-Yau threefolds of Borcea-Voisin. This is a joint work with Shouhei Ma.

### 2015/02/02

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Inverse of an Abelian Integral on open Riemann Surfaces and a Proof of Behnke-Stein's Theorem

**Junjiro Noguchi**(University of Tokyo)Inverse of an Abelian Integral on open Riemann Surfaces and a Proof of Behnke-Stein's Theorem

[ Abstract ]

Let $X$ be an open Riemann surface and let $\Omega \Subset X$ be a relatively compact domain of $X$. We firstly introduce a scalar function $\rho(a, \Omega)>0$ for $a \in \Omega$ by means of an Abelian integral, which is a sort of convergence radius of the inverse of the Abelian integral, and heuristically measures the distance from $a$ to the boundary $\partial \Omega$. We prove a theorem of Cartan-Thullen type with $\rho(a, \Omega)$ for a holomorphically convex hull $\hat{K}_\Omega$ of a compact subset $K \Subset \Omega$; in particular, $-\log \rho(a, \Omega)$ is a continuous subharmonic function in $\Omega$. Secondly, we give another proof of Behnke-Stein's Theorem (the Steiness of $X$), one of the most basic facts in the theory of Riemann surfaces, by making use of the obtained theorem of Cartan--Thullen type with $\rho(a, \Omega)$, and Oka's Jôku-Ikô together with Grauert's Finiteness Theorem which is now a rather easy consequence of Oka-Cartan's Fundamental Theorem, particularly in one dimensional case.

Let $X$ be an open Riemann surface and let $\Omega \Subset X$ be a relatively compact domain of $X$. We firstly introduce a scalar function $\rho(a, \Omega)>0$ for $a \in \Omega$ by means of an Abelian integral, which is a sort of convergence radius of the inverse of the Abelian integral, and heuristically measures the distance from $a$ to the boundary $\partial \Omega$. We prove a theorem of Cartan-Thullen type with $\rho(a, \Omega)$ for a holomorphically convex hull $\hat{K}_\Omega$ of a compact subset $K \Subset \Omega$; in particular, $-\log \rho(a, \Omega)$ is a continuous subharmonic function in $\Omega$. Secondly, we give another proof of Behnke-Stein's Theorem (the Steiness of $X$), one of the most basic facts in the theory of Riemann surfaces, by making use of the obtained theorem of Cartan--Thullen type with $\rho(a, \Omega)$, and Oka's Jôku-Ikô together with Grauert's Finiteness Theorem which is now a rather easy consequence of Oka-Cartan's Fundamental Theorem, particularly in one dimensional case.

### 2015/01/26

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On the uniform birationality of the pluriadjoint maps (Japanese)

**Tomoki Arakawa**(Sophia Univeristy)On the uniform birationality of the pluriadjoint maps (Japanese)

[ Abstract ]

In this talk, we investigate higher dimensional polarized manifolds by using singular hermitian metrics and multiplier ideal sheaves. In particular, we show the uniform birationality of the pluriadjoint maps.

In this talk, we investigate higher dimensional polarized manifolds by using singular hermitian metrics and multiplier ideal sheaves. In particular, we show the uniform birationality of the pluriadjoint maps.

### 2015/01/19

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Hyperbolic span and pseudoconvexity (Japanese)

**Hiroshi Yamaguchi**(Shia University, Prof. emeritus)Hyperbolic span and pseudoconvexity (Japanese)

[ Abstract ]

We show that the hyperbolic span for open torus (which is introduced by M. Shiba in 1993) has the intimate relation with the pseudoconvexity.

We show that the hyperbolic span for open torus (which is introduced by M. Shiba in 1993) has the intimate relation with the pseudoconvexity.

### 2014/12/15

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

The limits of Kähler-Ricci flows

**Hajime Tsuji**(Sophia University)The limits of Kähler-Ricci flows

### 2014/12/08

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On maximal ideal cycle and fundamental cycle of normal two-dimensional quasi-homogeneous singularities, and singularities with star-shaped resolution (JAPANESE)

**Masatake Tomari**(Nihon University)On maximal ideal cycle and fundamental cycle of normal two-dimensional quasi-homogeneous singularities, and singularities with star-shaped resolution (JAPANESE)

### 2014/12/01

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Effective and noneffective extension theorems (Japanese)

**Takeo Ohsawa**(Nagoya University)Effective and noneffective extension theorems (Japanese)

[ Abstract ]

As an effective extension theorem, I will review the sharp $L^2$ extension theorem explaining the ideas of its proofs due to Blocki and Guan-Zhou. A new proof using the Poincare metric with be given, too. As a noneffective extension theorem, I will talk about an extension theorem from semipositive divisors. It is obtained as an application of an isomorphism theorem which is essentially contained in my master thesis.

As an effective extension theorem, I will review the sharp $L^2$ extension theorem explaining the ideas of its proofs due to Blocki and Guan-Zhou. A new proof using the Poincare metric with be given, too. As a noneffective extension theorem, I will talk about an extension theorem from semipositive divisors. It is obtained as an application of an isomorphism theorem which is essentially contained in my master thesis.

### 2014/11/17

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On strong K-stability of polarized algebraic manifolds (JAPANESE)

**Yasufumi Nitta**(Tokyo Institute of Technology)On strong K-stability of polarized algebraic manifolds (JAPANESE)

### 2014/11/10

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On a convex level set of a plurisubharmonic function and the support of the Monge-Ampere current (JAPANESE)

**Yusaku Tiba**(Tokyo Institute of Technology)On a convex level set of a plurisubharmonic function and the support of the Monge-Ampere current (JAPANESE)

[ Abstract ]

In this talk, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Ampere equation and has a convex level set. By using our results and Lempert's results, we show a relation between the supports of the Monge-Ampere currents and complex $k$-extreme points of closed balls for the Kobayashi distance in a bounded convex domain in $C^n$.

In this talk, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Ampere equation and has a convex level set. By using our results and Lempert's results, we show a relation between the supports of the Monge-Ampere currents and complex $k$-extreme points of closed balls for the Kobayashi distance in a bounded convex domain in $C^n$.

### 2014/10/27

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles (JAPANESE)

**Takayuki Koike**(University of Tokyo)On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles (JAPANESE)

[ Abstract ]

We apply Ueda theory to a study of singular Hermitian metrics of a (strictly) nef line bundle $L$. Especially we study minimal singular metrics of $L$, metrics of $L$ with the mildest singularities among singular Hermitian metrics of $L$ whose local weights are plurisubharmonic. In some situations, we determine a minimal singular metric of $L$. As an application, we give new examples of (strictly) nef line bundles which admit no smooth Hermitian metric with semi-positive curvature.

We apply Ueda theory to a study of singular Hermitian metrics of a (strictly) nef line bundle $L$. Especially we study minimal singular metrics of $L$, metrics of $L$ with the mildest singularities among singular Hermitian metrics of $L$ whose local weights are plurisubharmonic. In some situations, we determine a minimal singular metric of $L$. As an application, we give new examples of (strictly) nef line bundles which admit no smooth Hermitian metric with semi-positive curvature.

### 2014/10/20

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Kodaira dimension of modular variety of type IV (JAPANESE)

**Shouhei Ma**(Tokyo Institute of Technology)Kodaira dimension of modular variety of type IV (JAPANESE)

### 2014/07/14

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Higher dimensional analogues of fake projective planes (ENGLISH)

**Gopal Prasad**(University of Michigan)Higher dimensional analogues of fake projective planes (ENGLISH)

[ Abstract ]

A fake projective plane is a smooth projective complex algebraic surface which is not isomorphic to the complex projective plane but whose Betti numbers are that of the complex projective plane. The fake projective planes are algebraic surfaces of general type and have smallest possible Euler-Poincare characteristic among them. The first fake projective plane was constructed by D. Mumford using p-adic uniformization, and it was known that there can only be finitely many of them. A complete classification of the fake projective planes was obtained by Sai-Kee Yeung and myself. We showed that there are 28 classes of them, and constructed at least one explicit example in each class. Later, using long computer assisted computations, D. Cartwright and Tim Steger found that the 28 families altogether contain precisely 100 fake projective planes. Using our work, they also found a very interesting smooth projective complex algebraic surface whose Euler-Poincare characteristic is 3 but whose first Betti

number is 2. We have a natural notion of higher dimensional analogues of fake projective planes and to a large extent determined them. My talk will be devoted to an exposition of this work.

A fake projective plane is a smooth projective complex algebraic surface which is not isomorphic to the complex projective plane but whose Betti numbers are that of the complex projective plane. The fake projective planes are algebraic surfaces of general type and have smallest possible Euler-Poincare characteristic among them. The first fake projective plane was constructed by D. Mumford using p-adic uniformization, and it was known that there can only be finitely many of them. A complete classification of the fake projective planes was obtained by Sai-Kee Yeung and myself. We showed that there are 28 classes of them, and constructed at least one explicit example in each class. Later, using long computer assisted computations, D. Cartwright and Tim Steger found that the 28 families altogether contain precisely 100 fake projective planes. Using our work, they also found a very interesting smooth projective complex algebraic surface whose Euler-Poincare characteristic is 3 but whose first Betti

number is 2. We have a natural notion of higher dimensional analogues of fake projective planes and to a large extent determined them. My talk will be devoted to an exposition of this work.

### 2014/06/30

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Primitive automorphisms of positive entropy of rational and Calabi-Yau threefolds (JAPANESE)

**Keiji Oguiso**(Osaka University)Primitive automorphisms of positive entropy of rational and Calabi-Yau threefolds (JAPANESE)

### 2014/06/23

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

A remark to the division algorithm in the proof of Oka's First Coherence Theorem (JAPANESE)

**Junjiro Noguchi**(University of Tokyo)A remark to the division algorithm in the proof of Oka's First Coherence Theorem (JAPANESE)

[ Abstract ]

The problem is the local finite generation of a relation sheaf $R(f_1, \ldots, f_q)$ in $\mathcal{O}_n=\mathcal{O}_{C^n}$. After $f_j$ reduced to Weierstrass' polynomials in $z_n$, it is the key to apply the induction in $n$ to show that elements of $R(f_1, \ldots, q)$ are expressed by $z_n$-polynomial-like elements of degree at most $p=\max_j\deg f_j$ over $\mathcal{O}_n$. In that proof one is used to use a divison by $f_j$ of $\deg f_j=p$ (Oka '48, Cartan '50, Hörmander, Demailly, . . .). In this talk we shall confirm that the division abve works by making use of $f_k$ of the minimum degree $\min_j \deg f_j$. This proof is natrually compatible with the simple case when some $f_j$ is a unit, and gives some improvement in the degree estimate of generators.

The problem is the local finite generation of a relation sheaf $R(f_1, \ldots, f_q)$ in $\mathcal{O}_n=\mathcal{O}_{C^n}$. After $f_j$ reduced to Weierstrass' polynomials in $z_n$, it is the key to apply the induction in $n$ to show that elements of $R(f_1, \ldots, q)$ are expressed by $z_n$-polynomial-like elements of degree at most $p=\max_j\deg f_j$ over $\mathcal{O}_n$. In that proof one is used to use a divison by $f_j$ of $\deg f_j=p$ (Oka '48, Cartan '50, Hörmander, Demailly, . . .). In this talk we shall confirm that the division abve works by making use of $f_k$ of the minimum degree $\min_j \deg f_j$. This proof is natrually compatible with the simple case when some $f_j$ is a unit, and gives some improvement in the degree estimate of generators.

### 2014/06/16

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

New examples of weighted Bergman kernels on a certain non-homogeneous Siegel domain (JAPANESE)

**Hideyuki Ishi**(Nagoya University)New examples of weighted Bergman kernels on a certain non-homogeneous Siegel domain (JAPANESE)

### 2014/06/09

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Modified Kähler-Ricci flow on projective bundles (JAPANESE)

**Ryosuke Takahashi**(Nagoya University)Modified Kähler-Ricci flow on projective bundles (JAPANESE)

### 2014/06/02

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Generalized pseudoellipsoids and proper holomorphic mappings between them (JAPANESE)

**Atsushi Hayashimoto**(Nagano National College of Technology)Generalized pseudoellipsoids and proper holomorphic mappings between them (JAPANESE)

### 2014/05/19

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On degenerations of Ricci-flat Kähler manifolds (JAPANESE)

**Shigeharu Takayama**(University of Tokyo)On degenerations of Ricci-flat Kähler manifolds (JAPANESE)