## 複素解析幾何セミナー

開催情報 月曜日　10:30～12:00　数理科学研究科棟(駒場) 128号室 平地 健吾, 高山 茂晴

### 2020年02月17日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Precompactness of the moduli space of pseudo-normed graded algebras
[ 講演概要 ]
Graded algebras (such as canonical rings) coming from the spaces of sections of polarized algebraic varieties are studied by many mathematicians. On the other hand, the pseudo-norm project proposed by S.-T. Yau and C.-Y. Chi gives us a new differential geometric aspect of the Torelli type theorem.
In this talk, we give the details of how the geometry of pseudo-normed graded algebras allows us to obtain a natural compactification of the moduli space of pseudo-normed graded algebras.
(1) For a sequence of pseudo-normed graded algebras (of the same type), the above precompactness gives us some limit different from the Gromov-Hausdorff limit in Riemannian geometry.
(2) As an example of our construction, we have the Deligne-Mumford compactification, in which the notion of the orthogonal direct sum of pseudo-normed spaces comes up naturally. We also have a higher dimensional analogue by using weight filtration.

### 2020年01月27日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Canonical measure and it’s applications
[ 講演概要 ]
The canonical measure is a natural generalization of K\”ahler-Einstein metrics to the case of projective manifolds with nonnegative Kodaira dimension. In this talk we consider the variation of canonical measures under projective deformations and give some applications.

### 2020年01月20日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Diederich-Fornaess and Steinness indices for abstract CR manifolds
[ 講演概要 ]
The Diederich-Fornaes and Steinness indices are estimated for weakly pseudoconvex domains in complex manifolds in terms of the D'Angelo 1-form of the boundary CR manifolds. In particular, CR invariance of these indices is shown when the domain is Takeuchi 1-convex. This is a joint work with Jihun Yum (Pusan National University).

### 2019年12月16日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

A simplified proof of the optimal L^2 extension theorem and its application (Japanese)
[ 講演概要 ]
I will explain a simplified proof of an optimal version of the　Ohsawa-Takegoshi L^2-extension theorem. In the proof, I use a method of Berndtsson-Lempert and skip some argument by the method of McNeal-Varolin. As an application, I will explain a result on extensions from possibly non-reduced varieties.

### 2019年12月11日(水)

16:00-17:00   数理科学研究科棟(駒場) 156号室
Joel Merker 氏 (Paris Sud)
Einstein-Weyl structures (English)
[ 講演概要 ]
On a conformal 3D manifold with electromagnetic field, Einstein-Weyl equations are the counterpart of Einstein's classical field equations. In 1943, Elie Cartan showed, using abstract arguments, that the general solution depends on 4 functions of 2 variables. I will present families of explicit solutions depending on 9 functions of 1 variable, much beyond what was known before. Such solutions are generic in the sense that the Cotton tensor is nonzero. This is joint work with Pawel Nurowski.

### 2019年12月09日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Analytic torsions associated with the Rumin complex on contact spheres (Japanese)
[ 講演概要 ]
Rumin 複体は接触多様体上に定まる，実数体の定数層の分解であり，実数体のde Rham 複体の部分複体である．本講演では，球面上のRumin ラプラシアンの固有値を書き下し，Rumin 複体の解析的捩率関数がRiemann のゼータ関数を用いて書き表されることを示す．特に，その関数の原点で消滅していることと，その解析的捩率が具体的に計算できることを紹介する．

### 2019年12月02日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Moishezonツイスター空間の分類に向けて
[ 講演概要 ]
ツイスター空間は4次元共形幾何から生じる3次元複素多様体であり、コンパクトな場合、ほとんどが非ケーラーであることが知られている。一方、コンパクトなツイスター空間でMoishezonであるものの例は数多く知られている。そのような空間の位相構造はかなり限定されたものになるため、Moishezonツイスター空間を分類しそれらの構造を記述することは必ずしも不可能とは言えないと思われる。本講演では、ある単純な仮定を満たすMoishezonツイスター空間の分類結果についてお話しする。なお、この仮定を満たさないMoishezonツイスター空間の例は知られていない。

### 2019年11月18日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

j-invariant and Borcherds Phi-function (Japanese)
[ 講演概要 ]
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年10月28日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

On Kiyoshi Oka's unpublished papers 1943 (Japanese)
[ 講演概要 ]
いわゆる岡の解決した不分岐リーマン領域に対する3大問題(Oka IX, 1953)は、実はこの未発表論文(VII～XI)で終わっている。　Oka　VII、VIIIで示された連接性、不定域イデアルの理論はこれ等を、分岐リーマン領域へ確立しようとする試みより生まれたことが、この未発表論文から明らかになる。この講演では、この未発表論文で擬凸問題がどのように解決されたかを紹介する。 分岐リーマン領域の場合の擬凸問題は、Fornaessによる反例が与えられたとはいえ、情況は不明で未解決問題として今も残っている（岡の夢）ことにも言及したい。

### 2019年10月21日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Canonical almost complex structures on ACH Einstein manifolds
[ 講演概要 ]
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年10月07日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Cohomology of vector bundles and non-pluriharmonic loci (Japanese)
[ 講演概要 ]
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年09月30日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Rigidity of the directional moduli on pseudoconvex domains fibered by open Riemann surfaces
[ 講演概要 ]
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年07月08日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

[ 講演概要 ]
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年07月01日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
Yeping Zhang 氏 (京都大学)
BCOV invariant and birational equivalence (English)
[ 講演概要 ]
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年06月24日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

A certain holomorphic invariant and its applications (Japanese)
[ 講演概要 ]
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年06月17日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
Andrei Pajitnov 氏 (Universite de Nantes)
Inoue surfaces and their generalizations (English)
[ 講演概要 ]
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年05月27日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Gluing construction of K3 surfaces (Japanese)
[ 講演概要 ]
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年05月20日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Cohomology and normal reduction numbers of normal surface singularities (Japanese)
[ 講演概要 ]
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年05月13日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Some Bonnet--Myers Type Theorems for Transverse Ricci Solitons on Complete Sasaki Manifolds (Japanese)
[ 講演概要 ]
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年04月22日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Optimal destabilizer for a Fano manifold (Japanese)
[ 講演概要 ]
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年04月15日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

ある種の完備ケーラー多様体上のL2評価とその応用 (Japanese)
[ 講演概要 ]
しばらく前にL2拡張定理の応用として西野の剛性定理の別証を得たが, 最近これがL2消滅定理だけからも導けることが判明した. その結果, 剛性定理がシュタイン族に対してだけでなく完備ケーラー族に対しても成立することが わかった. この議論を用いると, ある種の完備ケーラー多様体が二つの多様体の直積に分解するための一つの条件を書くことができる. また, 同様の方法によりRadoの定理を高次元に一般化することが可能になる.

### 2019年01月28日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Minimizing CM degree and slope stability of projective varieties (JAPANESE)
[ 講演概要 ]
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年01月21日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室
Nicholas James McCleerey 氏 (Northwestern University)
POLAR TRANSFORM AND LOCAL POSITIVITY FOR CURVES (ENGLISH)
[ 講演概要 ]
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年12月17日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

Newton polyhedra and order of contact on real hypersurfaces (JAPANESE)
[ 講演概要 ]
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年12月03日(月)

10:30-12:00   数理科学研究科棟(駒場) 128号室

[ 講演概要 ]