The 21th Symposium on Complex Geometry
Kanazawa 2015
Oct. 28
9:00-11:00 Toshiki Mabuchi (Osaka University)
“The Donaldson-Tian-Yau Conjecture and the moduli space of test configu-rations”
10:30-11:30 Charles Boyer (University of New Mexico) “Extremal Sasakian Geometry”
13:00-14:00 Hajime Tsuji (Sophia University)
“Rothe’s method in parabolic complex Monge-Ampere equation” 14:30-15:30 Takayuki Koike (University of Tokyo)
“On a neighborhood of a torus leaf of a certain class of holomorphic foliations on complex surfaces”
16:00-17:00 Kota Hattori (Keio University)
“New examples of compact special Lagrangian submanifolds embedded in hyper-K¨ahler manifolds”
Oct. 29
9:00-10:00 Yoshinori Namikawa (Kyoto University)
“A finiteness theorem for symplectic singularities” 10:30-11:30 Takeo Ohsawa (Nagoya University)
“L2拡張定理に関連する二三の問題” 13:00-14:00 Osamu Fujino (Kyoto University)
“消滅定理と半正値性定理”
14:30-15:30 Shigeharu Takayama (University of Tokyo)
“Degeneration of algebraic varieties and the metric completeness of parameter spaces”
16:00-17:00 Keizo Hasegawa (Niigata University)
“Compact locally homogeneous Kaehler and lcK manifolds” 17:30-18:30 Katsutoshi Yamanoi (Osaka University)
“Bloch-Ochiaiの定理と小林擬双曲性”
Oct. 30
9:00-10:00 Takuro Mochizuki (RIMS)
“ある調和バンドルの族の極限について”
(Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces)
10:30-11:30 Junjiro Noguchi (University of Tokyo)
“On the Levi problem for ramified Riemann domains” 13:00-14:00 Ryushi Goto (Osaka University)
“Stratified flat structures on moduli spaces of generalized complex surfaces” 14:30-15:30 Ryoichi Kobayashi (Nagoya University)
Abstract
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講演者：満渕俊樹（Toshiki Mabuchi)大阪大学
タイトル：“The Donasldson-Tian-Yau Conjecture and the moduli space of test configurations”
アブストラクト：The Donaldson-Tian- Yau Conjecture for K¨ahler-Einstein cases was recently solved affirmatively by Tian and Chen-Donaldson-Sun. In this talk, the Donaldson-Tian-Yau Conjecture for general polarizations will be discussed. This general conjecture will be shown to be deeply related to the moduli space of test configurations.
講演者：Charles Boyer, Emeritus Professor of Mathematics, University of New Mexico
タイトル：“‘Extremal Sasakian Geometry”
アブストラクト：I begin the talk by giving a brief general discussion about Sasakian geometry. I then focus attention on the problem of finding constant scalar curvature Sasaki metrics. In joint work with Hongnian Huang, Eveline Legendre, and Christina Tønessen-Friedman, we describe the relationship between the K-semistability of the affine cone of a Sasaki manifold and the existence of constant scalar curvature Sasaki metrics. We also give many examples where there are several constant scalar curvature rays in the Sasaki cone.
講演者：辻 元(Hajime Tsuji)上智大
タイトル：“Rothe’s method in parabolic complex Monge-Ampere equation”
アブストラクト：In this talk I would like to present the approximation of the solution of parabolic Monge-Ampere equations in terms of the (time) fractional equations.
I also give an application to complex geometry such as invariance of plurigenera. This is a joint work with Sebastien Boucksom.
講演者：小池 貴之(Shigeharu Takayama) 東大数理
タイトル：“On a neighborhood of a torus leaf of a certain class of holomorphic foliations on complex surfaces”
アブストラクト：Let C be a smooth elliptic curve embedded in a smooth complex surface X such that C is a leaf of a suitable holomorphic foliation of X. We investigate complex analytic properties of a neighborhood of C under some assumptions on complex dynamical properties of the holonomy function.
As an application, we give an example of (C, X) in which the line bundle [C] is formally flat along C however it does not admit a C∞ Hermitian metric with semi-positive curvature.
講演者：服部広大(Kota Hattori) 慶応大学
タイトル：“New examples of compact special Lagrangian submanifolds embedded in hyper-K¨ahler manifolds”
アブストラクト：In hyper-K¨ahler manifolds, the complex submanifolds is called holomorphic Lagrangian submanifolds if the pullback of the holomorphic symplectic structures are vanish-ing. Then they can be regarded as special Lagrangian submanifolds by rotating three complex structures appropriately.
In this talk, I will talk about the construction of compact special Lagrangian submanifolds which never come from the holomorphic Lagrangian, by applying the desingularization developed by D. Joyce.
講演者：並河良典（Yoshinori Namikawa)京都大学
タイトル：“A finiteness theorem for symplectic singularities”
アブストラクト：The notion of a symplectic singularity was first introduced by Beauville. An affine symplectic singularity with a good C∗-action is called a conical symplectic variety. Many interesting varieties are conical symplectic varieties. For example, among them are nilpotent orbit closures of a complex semisimple Lie algebbra, Slodowy slices to such orbits, Nakajima quiver varieties, hypertoric varieties and so on.
In this talk we discuss how many such varieties exist. If we fix the dimension of conical symplectic varieties X and the maximal weight N of the minimal homogeneous generators of the coordinate ring R of X, then there are only finitely many such X up to isomorphism.
We first relate a conical symplectic variety with a log Fano klt pair, which has a contact structure. Then the boundedness result for log Fano klt pairs with fixed Cartier index assures that the family of conical symplectic varieties of a fixed dimension and with a fixed maximal weight, forms a bounded family. Finally we prove the rigidity of conical symplectic varieties by using Poisson deformations.
講演者：大沢 健夫(Takeo Ohsawa) 名古屋大学多元数理 タイトル：“L2拡張定理に関連する二三の問題”
アブストラクト：Blocki, Guan, Zhou, Cao, Berndtsson, Lempertにより、L2 拡張定理に関して この３年間にめざましい進展がなされた。そのあとを振り返り、今後どんな問題を考えると面白 そうかを考えてみたい。
講演者：藤野 修(Osamu Fujino) 京大理 タイトル：“消滅定理と半正値性定理”
アブストラクト：高次元複素射影代数多様体の研究には、川又—Viehweg消滅定理（Kawamata— Viehweg vanishing theorem）と藤田—Zucker—川又半正値性定理（Fujita—Zucker—Kawamata semi-positivity theorem）が有効に使われてきました。極小モデル理論の基本的な定理の証明には、川 又—Viehweg消滅定理と廣中の特異点解消定理を巧妙に用いたX論法と呼ばれる手法が使われてい ました。また、藤田—Zucker—川又半正値性定理は標準束公式などの形で有効に利用されてきまし た。これら二つの定理は純Hodge構造の理論の結果と見なせます。ここ数年の私の仕事で、これ らの混合Hodge構造版が得られました。この一般化を用いると、極小モデル理論の適用範囲が究 極的に広がります。この講演では、可約な多様体に対するKoll´ar—大沢型の消滅定理と、その応用 である新しい半正値性定理を説明したいです。この半正値性定理の応用としては、安定多様体の モジュライ空間の射影性の証明があります。
講演者：Keizo Hasegawa (Niigata University)
タイトル：“Compact locally homogeneous Kaehler and lcK manifolds”
アブストラクト：“Locally homogeneous structure”, or “Geometric structure” in the sense of Thurston, is now of central importance in the field of Topology and Geometry. While homoge-neous complex manifolds are in a very restricted class of complex manifolds, there are abundant examples of locally homogeneous complex manifolds, including Riemannian surfaces of genus greater than 1.
In this work we study compact locally homogeneous complex manifolds with compatible Kaehler structure or locally conformally Kaehler (l.c.K.) structure. It should be noted that almost all of non-Kaehler complex surfaces with b2 = 0, up to small deformations, admit locally
homogeneous l.c.K. structures.
講演者：山ノ井 克俊 (Katsutoshi Yamanoi) 阪大理 タイトル：“Bloch-Ochiaiの定理と小林擬双曲性”
アブストラクト：アーベル多様体Aの一般型の部分多様体 X を考える。このときX の整正則曲 線、すなわち複素平面から X への非自明な正則写像の像は、すべて、X の特殊集合とよばれる 真部分代数的集合に含まれることが知られている（Bloch, Ochiai, Kawamata,…）。
この講演では、その発展であるX の小林擬双曲性の問題、すなわちX の小林擬距離が、X の 特殊集合の外で非退化になるか、という問題について考えてみたい。
講演者：望月拓郎(Takuro Mochizuki) 京都大学数理研 タイトル：“ある調和バンドルの族の極限について”
(Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces)
アブストラクト：X をコンパクトリーマン面とし, (E, ∂E, θ, h) をX上の調和バンドルとします. 各t > 0 について,ヒッグス束(E, ∂E, tθ) に小林-ヒッチン対応を適用すると,調和バンドルの族 (E, ∂E, tθ, ht) が得られます. t→ 0 に関する極限はヒッチンやシンプソンによってよく研究され ており, ホッジ構造の変動に収束することが古典的な結果としてよく知られています. 一方, 近年 になってt→ ∞に関する挙動が研究されるようになりました. この講演では, ヒッグス場θが適当な条件を満たす場合に, (E, ∂E, tθ, ht) の t→ ∞ における 挙動に関して二つの結果を紹介する予定です. 一つ目は, スペクトル曲線が分岐する場所から離 れていれば, スペクトル曲線の分解に応じたヒッグス束の分解が, ほぼ直交している(asymptotic decoupling)という結果です. これは, (E, ∂E, tθ, ht) に随伴する平坦束のモノドロミーの挙動に関 するヒッチンWKB問題に応用されます. 二つ目は, E の階数が2 の場合に限定されますが,計量 の族ht の収束とその極限(limiting configuration)の記述に関する結果です.
(Let (E, ∂E, θ) be a stable Higgs bundle of degree 0 on a compact connected Riemann surface.
Once we fix a flat metric hdet(E) on the determinant of E, we have the harmonic metrics ht
(t > 0) for the stable Higgs bundles (E, ∂E, tθ) such that det(ht) = hdet(E).
In this talk, we will discuss two results on the behaviour of ht when t goes to ∞. First,
we show that the Hitchin equation is asymptotically decoupled under some asumption for the Higgs field. We apply it to the study of the so called Hitchin WKB-problem. Second, we discuss the convergence of the sequence (E, ∂E, θ, ht) in the case where the rank of E is 2. We
introduce a rule to determine the parabolic weights of a “limiting configuration”, and we show the convergence of the sequence to the limiting configuration in an appropriate sense.)
講演者：野口 潤次郎(Junjiro Noguchi)
タイトル：“On the Levi problem for ramified Riemann domains”
アブストラクト：The Levi Problem asks if the pseudoconvexity (the local Steinness at boundaries) implies the holomorphic convexity or the global Steinness.
• K. Oka solved it first for general subdomains of C2 in Oka VI 1942, and for unramified Riemann domains over Cn in 1943 (reports to T. Takagi, unpublished), and in Oka IX 1953 (Oka ’s Theorem).
• In 1954 H.J. Bremermann and F. Norguet proved it independently for subdomains of Cn.
• Oka’s Theorem was generalized for unramified Riemann domains over Pn(C) by R. Fujita 1963 and by A. Takeuchi 1964.
For the ramified case we know:
• H. Grauert obtained a counter-example of a finitely sheeted ramified Riemann domain over Pn(C) in 1962.
• J.E. Fornæss constructed a counter-example of a 2-sheeted ramified Riemann domain over C2 (and hence over Cn) in 1978.
• R. Narasimhan observed in 1978 that a combination of works by Andreotti-Narasimhan (1964) and Elencwag (1975) implies an affirmative solution of the Levi Problem for rela-tively compact domains of a non-singular Riemann domain ramified over Cn. (Therefore, the problem is at the infinity!)
We here deal with the Levi Problem for ramified Riemann domains over Cn. We consider two geometric conditions, “Cond. A” and “Cond. B” for a non-singular Riemann domain X ramified over Cn: Cond. A is the triviality of the holomorphic cotangent bundle by closed abelian differentials, and Cond. B is the localization condition at the ideal boundaries of X over Cn. We prove: If a finitely sheeted ramified non-singular domain X satisfying Cond. A and Cond. B is pseudoconvex, then X is Stein. We discuss the relation with the counter-examples of Grauert and Fornæss.
The key is the use of the inverse map of Abelian integrals, and the introduction of a positive scalar function ρ(a; Ω) for subdomains Ω of X, which plays the role of Hartogs’ radius.
The proof also leads to a new proof of the classical Behnke-Stein Theorem (1949): Every open Riemann surface is Stein.
Reference
J. Noguchi, Analytic Function Theory of Several Variables, Asakura, 2014; English translation, Springer, in preparation, 2015.
−−−−, A scalar associated with the inverse of some Abelian integrals and a ramified Riemann domain, preprint, ArXiv 2015.
講演者：後藤 竜司(Ryushi Goto)阪大理
タイトル：“Stratified flat structures on moduli spaces of generalized complex surfaces”
アブストラクト：The 2 dimensional complex projective space P2 is rigid as a complex manifold, however P2 admits 2 dimensional moduli spaces of generalized complex structures which has a torsion free flat connection on its open strata. We show that a logarithmic generalized complex structure with smooth elliptic curve as type changing loci has unobstructed deformations which are parametrized by an open set of the second de Rham cohomology group of the complement of elliptic curve. Then we will construct moduli spaces of generalized del pezzo surfaces.
We further investigate deformations of logarithmic generalized complex structures in the cases of type changing loci with singularities. By using types of singularities, we obtain a stratification of moduli spaces of generalized complex structures on complex surfaces and it turns out that each strata corresponding to nodes and cusps admits a flat torsion free connection.
講演者：小林 亮一(Ryoichi Kobayashi)名大多元数理 タイトル：“Legendre duality for general polarization”
アブストラクト This talk is an attempt of generalizing the Legendre duality interpretation of the Calabi-Yau theorem proposed by R. Berman.
