Grant amount： \4940000 （ Direct Cost: \3800000 、 Indirect Cost： \1140000 ）

First, we extended the index theorem to fractal sets such as the Cantor set and the Sierpinski gasket. Second, by exploiting the framework of Noncommutative Geometry we generalized the Atiyah-Patodi-Singer index theorem to a Galois covering of compact manifold with boundary, which gives a formula for the pairing between K-group and cyclic cohomology. Third, we clarified the relation of the Dixmier-Douady class and the Godbillon-Vey class, which respectively appears as a characteristic class for Gerbe and foliated circle bundles. It turned out that they are connected via the Cheeger-Chern-Simons invariant. As a byproduct we succeeded to describe the universal central extension of circle diffeomorphism group in terms of the Calabi invariant.

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Grant amount： \8710000 （ Direct Cost: \6700000 、 Indirect Cost： \2010000 ）

Foliations and contact structures are studied, with a focus on those structures on 3, 4, and 5 dimensional spaces. Especially the construction of important examples and their mutual relationships are investigated. Interactions of many mathematical theories such as Milnor fibrations associated with singularities, fluid mechanics, symplectic geometry which is a refinement of classical mechanics, and several complex variables, re flected on those structures and objects are studied.
This study is also understood as looking at the tightness of those structures which is interpreted as the result of such mathematical theories are reflected on the geodesic flows of surfaces, which gives rise to a special class of flows called `Anosov flows'.

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Grant amount： \41340000 （ Direct Cost: \31800000 、 Indirect Cost： \9540000 ）

Symplectic structure is a geometric structure, which appeared in the understanding of Hamilton's equation of motion. In recent years, there has been profound development in the geometric study of symplectic structures. In particular, combined with the mathematical study on mirror symmetry, symplectic geometry
attracts attentions from many researchers. The investigator has been working on Floer theory, which plays a significant role in symplectic geometry, and its applications. In this research project, we studied Floer theory for Lagrangian torus fibers in toric manifold in a concrete way and obtained various interesting results.

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Grant amount： \2990000 （ Direct Cost: \2300000 、 Indirect Cost： \690000 ）

As a foliation is an integrable tangent plane field, in the case of 1-dimensional line fields, we treated a completely integrable vector field, which satisfies an extra integrability condition. We consider a placement problem of a closed leaf (i.e., a periodic trajectory) of the 1-dimensional foliation tangent to a completely integrable vector field on an open 3-manifold. Such a foliation is given by the inverse images of a submersion to the plane. We gave a necessary and sufficient condition for the realization for any given link, and in the case of a knot, we described the condition by the words of classical invariants.
A 2-dimensional foliation transverse to a completely integrable vector field satisfies Thurston's inequality and it gives a natural class of such foliations. Thus, we can consider such a class of foliations is a natural class on an open 3-manifold.

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Grant type：Competitive

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Grant amount： \2340000 （ Direct Cost: \2100000 、 Indirect Cost： \240000 ）

W. Thurston showed that a foliation on a 3-manifold which has no Reeb component enjoys the property that the Euler class of the tangent bundle satisfies an inequality, Thurston's inequality. On the other hand, the Reeb foliation on the three sphere satisfies Thurston's inequality and a previous research followed by this research showed that there is a class of foliations each of which has Reeb components and satisfi es Thurston's inequality.
In the research in 2006, for a class of foliations which are called spinnable foliations, we obtained a sufficient condition for the foliation satisfying Thurston's inequality. Moreover, we revealed an aspect where Thurston's inequality does not hold. They are described by properties of the monodromy diffeomorphisms which determine the spinnable foliations
In view of the research with respect to the convergence of contact structures to foliations, we studied a finer inequality, the relative version of Thurston's inequality, which deepens the research until 2006. In fact, for spinnable foliations we showed that the relative version implies the absolute version. The same statement for contact structures was known however, it does not hold in general for foliations. Also in 2007, we found the method to construct a foliation which satisfies Thurston's inequality with Euler class of infinite order. Until then, all foliations which satisfies Thurston's inequality have trivial Euler class. Indeed, we can find'a spinnable foliation whose Euler class is of infinite order by the research in 2006. Then we can perform Dehn surgery along the Reeb component and with certain condition on the original spinnable foliation we can conclude that with finitely many exceptions the resultant satisfies Thurston's inequality with Euler class of infinite order by virtue of D. Gabai's sutured manifold theory.

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Grant type：Competitive

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Grant amount： \3700000 （ Direct Cost: \3700000 ）

The head Mitsumatsu and an investigator Miyoshi colaborated with others to study the euler class of tangent bundles to foliations and (so called Thurston-Winkelnkemper's) contact structures which are associate with spinnable structures, as a typical class of convergences of contact structures to foliations. Especially they studied the (non-)vanishing of the euler class and the violation of Thurston's inequality and Bennequin's inequlity, from the topological view point of monodromies. As a consequence, a certain class of mapping classes of a surface with boundary can be presented neither as a product of only right-handed Dehn twists nor as that of only right-handed ones. This result was presented in several symposiums including the annual meeting of MSJ in March 2006 as a special invited talk by Miyoshi. The paper is under submission.
The investigator Ono studied the symplectec homology from Floer theory as well as from Seiberg-Witten theory. Including the solution to the Flux conjecture as well as the detemination of the symplectic filling of the link of simple singularities, his contributions to this area are profound.
The investigator Tsuboi studied the relationship between foliation theory and that of contact structures from the view point of the group of contact diffeomorphisms. The investigator Matsumoto stepped further to the foliation theory and studied the ends of Lie foliations.
The head investigator also studied the incompressible fluid dynamics in the framework of the geometry of volume preserving diffeomorphisms and infinite dimensional Hamiltonian systems. He proved that looking from the point of view of such global differential geometry is still valid even to viscous fluides with dissipations. This study is presented in many symposiums, especially as a special project talk in the annual meeting of MSJ in September 2005.

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Grant type：Competitive

Grant amount： \1000000

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Grant amount： \8600000 （ Direct Cost: \8600000 ）

Based on the notion of asymptotic linking, the head investigator proposed the framework where the study on contact structures and that on foliations would be unified, and began the research. Op the side of foliations, it turned out that exotic classes and the 1st foliated cohomology are strongly related to this framework. On the other side, the torsion invariant of contact structures has a deep relation with it. For algebraic Anosov foliations, we also established the computation of its 1st foliated cohomology and found its relation to local orbit rigidity.
A research group including Miyoshi and Mitsumatsu investigated Thurston's inequality for foliations on the boundary of compact Stein surfaces and established the absolute version of the inequality for certain cases. The relative ersion and more general case are left for the future as important subject. Tsuboi and Mitsumatsu worked on the perfectness of groups of diffeomorphisms preservein certain geometric structures. Especially Tsuboi provednthe perfectness for contact diffeomorphisms and analytic diffeomorphisms of certain manifolds. Tsuboi also classified regular bi-contact structures on Seifert fibered spaces.
Another group including Ono and Ohta, mainly working on contact/symlectic topology, characterized the symplectic diffeo-types of the filling of the link of simple and hyper-elliptic singularities. Also they got started the construction of obstruction theory for Lagrangian Floer homology theory.

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Grant type：Competitive

Grant amount： \1100000

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Grant amount： \3300000 （ Direct Cost: \3300000 ）

We developed the theory of toric varieties from the topological viewpoint. In these several years I worked with Professor Akio Hattori and found that geometrical properies of a torus manifold can be described in terms of a combinatorial object called a multi-fan. In particular, we found a neat formula describing the elliptic genus of a torus manifold in terms of the multi-fan associated with the torus manifold, and obtained a vanishing theorem saying that the level N elliptic genus of a torus manifold vanishes if the 1st Chern class of the manifold is divisible by N. As a corollary of this vanishing theorem, we obtained a result that if the 1st Chern class of a complete toric variety M of complex dimension n is divisible by N, then N must be less than or equal to n+1, and in case N=n+l, M is isomorphic to the complex protective space. This is a toric version of the famous Kobayashi-Ochiai or Mori's theorem.
I invited Taras Panov from Moscow State University for a month and studied the equivariant cohomology of a torus manifold M and the cohomology of its orbit space. As a result, it turned out that when the cohomology ring of M is generated in degree two, the equivariant cohomology of M is a Stanley-Reisner ring and the orbit space of M has the same form as a convex polytope from a cohomological point of view. We also studied the case where M has vanishing odd degree cohomology. It turns out that this case is obtained by blowing down the previous case. Interestingly, the equivariant cohomology of M in this case provides a generalization of the Stanley-Reisner ring. The ring like this was already introduced by Stanley about ten years ago but we may think of our results as giving a geometrical meaning of the ring. Along this line, I proved a conjecture by Stanley about the h-vector of a Gorenstein* simplicial poset. The proof is purely algebraic but the idea stems from topology and this shows a close connection between combinatorics, commutative algebra and topology.

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Grant type：Competitive

Grant amount： \1000000

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Grant type：Competitive

Grant amount： \1200000

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Grant amount： \3300000 （ Direct Cost: \3300000 ）

We developed the theory of toric varieties from topological viewpoint. The theory of toric varieties says that there is a one-to-one correspondence between "toric varieties" (an object in algebraic geometry) and "fans" (an object in combinatorics). In our project, we studied "torus manifolds" or "torus orbifolds" which are topological counterparts to toric varieties and a wider object than that of toric varieties, and constructed a correspondence from those extended objects to an extended combinatorial object called "multi-fans". One of the fundamental problems in our correspondence is to characterize geometrically obtained multi-fans, and we completely characterized the multi-fans obtained form torus orbifolds. Moreover, we described signatures and T_y-genera of torus manifolds in terms of multi-fans. There is another fundamental correspondence given by moment maps. We introduced a notion of multi-polytopes, which appear as images of moment maps, and generalized Ehrhart polynomials and Khovanskii-Pukhlikov formula for convex polytopes to multi-polytopes.

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Grant type：Competitive

Grant amount： \1300000

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Grant type：Competitive

Grant amount： \1100000

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Grant amount： \3100000 （ Direct Cost: \3100000 ）

Mitsumatsu and Mizutani studied and constructed many examples of bi-contact structures with a research group of foliations. Especially, they constructed a bicontact structure on the 3-sphere which consists both of over-twisted ones. Still the realization problem of homotopy class of plane fields as such structures remains to be studied.
Ono has established in a colaboration with Fukaya foundamental theory in applying the J-curves to symplectic topology, overcoming the notorious problem of negative multiples. Major consequences from this are the definition of Gromov-Witten invariants for general symplectic manifolds and the positive solution for a version of the Arnold conjecture for the same class. He and Kanda also worked on applying Seiberg-witten theory to contact topology, colaborating with Ohta, and got toplogical constraints on the symplectic filling 4-manifolds around simple singularities. This streem of works continues and is expected to make a further progress, especially in relation with the last subject of study below.
Kanda studied contact structures in more toplogical way, and classified tight contact structures on 3-torus and showed nonexactness of Bennequin's inequality.
Takakura and Mitsumatsu have been searching for the formalism to study contact topology by using Lagrangian/Legendrian torus, instead of looking at J-curves in the symplectization. This is based on the theory of geometric quantization, on which Takakura has been working. They found that most of major concepts in the theory of algebraic functions in one variable can be suitably translated and planted in this framework. However, studying contact topology through this remains as the next step of research to go.

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Grant amount： \6600000 （ Direct Cost: \6600000 ）

1. Conformably flat hypersurfaces. We studied conformally flat, hypersurfaces in the space forms of dimension 4, and found a good structure on the 4-dimensional standard sphere for each hypersurface. According to the structure, the set of conformally flat hypersurfaces is divided into three classes : the parabolic class, the elliptic class, and the hyperbolic class. We showed that the classes are invariant under conformal transformations of the sphere and the respective class consists of conformally flat hypersurfaces constructed by surfaces of constant curvature in one of the 3-dimensional space forms : the Euclidean space, the hyperbolic space, or the sphere.
2. Conformal-projective transformations of statistical manifolds. In this study, we obtained the following result : A conformal-projective transformation of a statistical manifold leaves all umbilical points and the skew-symmetric component of the Ricci curvature of any hypersurfaces ; moreover, this property characterizes the conformal-projective transformations when the dimension of the statistical manifold is greater than 2. We also found a tensor field that is invariant under any conformal-projective transformations and that reduces to the conformal curvature tensor if the underlying statistical manifold is a usual Riemannian manifold.
3. A representation formula of surfaces with constant mean curvature (CMC surfaces) in a 3-dimensional space form and their Gauss map. The existence problem of harmonic maps was studied in the case where the destination is a non-complete Riemannian space with non-positive curvature unbounded from below. In this situation, we showed tile existence and the uniqueness theorems of harmonic maps for a Dirichlet problem at infinity. As an application, we constructed CMC surfaces in the 3-dimensional hyperbolic space form.
4. An extension of the class of CMC surfaces from the viewpoint of the theory of integrable systems. We defined surfaces with harmonic inverse mean curvature (HIMC surfaces) in the 3-dimentional space forms, and showed that there exists a correspondence among the HIMC surfaces similar to the Lawson correspondence, one of the features of the class of CMC surfaces. We also studied the relation between the class of HIMC surfaces and the class of H-surfaces, which is an extension of the class of CMC surfaces from the variational viewpoint. As a result, we proved that HIMC surfaces are obtained from the gauge-theoretic equation for H-surfaces with a certain condition of reduction.

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Grant type：Competitive

Grant amount： \1200000

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Grant amount： \1700000 （ Direct Cost: \1700000 ）

本研究の補助金の執行期間中には以下の3回のENCOUNTER with MATHEMATICSの本会議が開催された。何れも中央大学理工学部に於いて金曜日の14:30または15:00から続く土曜日の
17:00頃までの一日半である。
1. 第8回目‘TORIC幾何''98.6.12(金)・13(土)
トーリック幾何への招待I、II、小田忠雄氏(東北大・理)・佐藤拓氏(東北大・理)
トポロジーから見たトーリック多様体論I、II、枡田幹也氏(阪市大・理)、
log scheme理論入門 諏訪紀幸氏(中大・理工)
2. 第9回目‘実1次元力学系"98.10.23(金)・24日(土)
実1次元力学系I、II、III坪井俊氏(東大・数理)、
有界オイラー類と円周の同相群の共役 松元重則氏(日大・理工)、
双曲結晶群の\S^1\のPL同相群への表現:初等的な構成 皆川宏之氏(北大・理)
3. 第10回‘応用特異点論'99.2.5(金)・6日(土)
応用特異点論概説、一階偏微分方程式への応用、微分幾何学への応用 泉屋秀一氏(北大・理)、
応用特異点論の基礎I、II、石川剛郎氏(北大・理)、
微分位相幾何学への応用 佐伯修氏(広島大・理)
各回とも100〜150人程度の参加者を得、また、講演も専門的になりすぎずに、この集会の本来の意義を十分に達成したと考えられる。
上記の本会議以外には、各回の準備、及び、将来に向けての準備会議を何回か行った。また、代表者はドイツObervolfach数学研究所で毎年開かれているトポロイジー研究集会に出席し、Wolfgang Luck,Elmar Vogtの新旧主催者と定期的に集会を開催することの意義や問題点について意見交換を行った。本年度の問題点として、開催テーマで順延になったものがあったため、主催側に幾何系の人間が多いこともあって、急遽幾何系のテーマが繰り上がり、結果として、開催したテーマの分野に偏りがでたことがあげられる。今後の課題である。

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Grant amount： \1000000 （ Direct Cost: \1000000 ）

本研究課題において具体的な目標としていたことのうち、1.2次元多様体上の平坦接続のモジュライ空間の滑層構造の解明とD-加群の構成、2.幾何学的量子化の一般化としてのコホモロジー理論における消滅定理、について得られた結果と、その過程で明らかになった今後の課題等を以下に記す。
1.については、平坦接続のモジュライ空間の幾何学的量子化の次元に関するフェアリンデの分解公式の、シンプレクティック幾何学的な証明を与えることと並行させて研究を進めた。その結果、「ハミルトン的群用の下での幾何学的量子化に関する重複度公式」を応用することにより、上記分解公式が自然に得られることが判った。その際、モジュライ空間の滑層構造は、運動量写像の臨界値におけるシンプレクティック商の滑層構造という形で一般的に考察が可能であり、Kirwanによる部分的特異点解消が有効であることを示したMeinrenken-Sjamaarの結果が鍵となった。なお、本研究者が有限次元のトーラス作用を用いているのに対し、Meinrenken-Woodwardは無限次元のループ群作用を解析して同公式を証明した。両者の相関の解明は、未達成のまま残ったD-加群の構成等に関連して興味深いと考えられ、今後の課題として挙げておく。
2.に関連して、シンプレクティック・トーリック多様体上に退化付き不変偏極の族を構成し、同伴する幾何学的量子化の推移に祭し、特殊なラグランジュ部分多様体上への局所化現象が起こることを示した。さらに、この不変偏極による幾何学的量子化を層係数のコホモロジーとして記述し、上記局所化に対応するコホモロジーの消滅を示すことができた。昨年度得られたシンプレクティック・トーラスに対する同様の結果を踏まえつつ、これらをより広いクラスのシンプレクティック多様体に拡張することは、引続き今後の課題としておく。

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Grant amount： \2100000 （ Direct Cost: \2100000 ）

1.どのような共形平坦な多様体が,定曲率空間の超曲面として実現されるかという問題を研究し,4次元以上の(ある種の)共形平坦な多様体に関して,それらの多様体から定曲率空間への共形的はめ込みの具体的構成法を発見した。更に,上の構成ではめ込み可能な多様体の共形類を決定するために,それらの超曲面から球面への展開写像の構成を行った。
2.射影平坦で捩れをもたないアフィン接続が与えられた単連結多様体の射影展開写像について研究し,次ぎの結果を得た。3次元以上で接続に関して極を持つ多様体のリッチ曲率が対称で負定値ならば,その展開写像は単射であり,像は射影空間の凸集合となる。
3.シンプレクティック・トーリック多様体上に,退化した不変偏極の族を構成し,同伴する幾何学的量子化の推移において,特殊なラグランジュ部分多様体上への極所化現象が起こることを示した。
4.閉曲面上の平坦接続のモジュライ空間の幾何学的量子化(一般化されたテ-タ関数の空間)の次元に関するフェアリンデの分解公式の,シンプレクティック幾何的な証明について研究した。その方法は,ハミルトン的群作用の下での幾何学的量子化に関する重複度公式を応用するというものである。

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Grant amount： \1000000 （ Direct Cost: \1000000 ）

本研究課題において具体的な目標としていたことのうち、1.テ-タ関数の理論の偏極シンプレクティックトーラスへの一般化、2.1の結果の、リーマン面およびそのヤコビ多様体への応用、3.偏極シンプレクティック多様体のコホモロジーに関する消滅定理に対して、ある程度の結果と新たな知見を得ることができた。
1については、必ずしもケーラー的でない偏極に対しても、対応するテ-タ関数を定義できること、異なる偏極を用いて得られるテ-タ関数の空間の間には(ベクトル空間としての)同型対応が存在することが示された。実際、同型写像を2通りの方法で構成することができる。これらの間の関係を調べることは未達成であるが、同型のユニタリ性の問題等も含めて興味深い。なお、多少修正が必要だが、シンプレクティック・トーリック多様体に対しても同様の結果が得られると思われる。
2に関しては、リーマン面および偏極の退化という点に対しては、2次元閉多様体上の平坦G主束の同型類の空間(ただしGは一般のコンパクト単純リー群)を含めて統一的に扱うことができることがわかり、さらにG=U(1)の場合にはリーマン面の退化との関連がかなり明確に記述できた。
3に対しては、偏極シンプレクティック多様体の層係数コホモロジーについて小平消滅定理の拡張が成り立つことが判ったが、退化のないきれいな(中間)偏極を許容するものはかなり限られたものしかなく、応用上は、偏極に退化を許すあるいは滑層構造を持つシンプレクティック空間としての偏極まで範囲を広げて考察することが重要と思われ、今後の課題として挙げておく。
なお、J.E.Andersenも上記1、3の結果の一部を独立に得ていることをコメントしておく。

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Grant amount： \1100000 （ Direct Cost: \1100000 ）

リーマン多様体の研究における1つの大きなテーマとして、“局所的性質から、その大域的構造がどの程度決定されるか?"という問題がある。ここでいう局所的性質とは、他様体の曲り方を表わす曲率を指し、大域的構造とは位相構造あるいは微分構造による分類の問題を指す。この研究では主として、次の2つのテーマに関して研究を行った。
1.微分可能球面定理の研究
位相的球面定理を精密化して、微分構造を決定する微分可能球面定理の研究を行い、重要な成果を上げた。
定理:完備、単連結なリーマン多様体(M,g)の断面曲率Kが、0.654<K<1となるとき、Mは標準的球面と微分同相である。
2.共形構造を持つ多様体の研究
リーマン多様体(M,g)が共形構造を持つとは、ワイルの共形曲率テンソルが消えるときをいう。この研究では、その様な多様体を余次元1でユークリッド空間に共形的に埋め込む問題を考えた。もしその様な埋め込みが存在するならば、(M,g)はショトキ-多様体に限られる。逆に、“全てのショトキ-多様体は共形的埋め込みを持つか"という問題である。これについては一部成果を上げているが継続して研究中である。

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