In this thesis we present a slight generalisation of the Pearl complex or relative quantum homology to some non monotone Lagrangian submanifolds. First we develop the theory for the so called almost monotone Lagrangian submanifolds, We apply it to uniruling problems as well as estimates for the relative Gromov width. In the second part we develop the theory for toric fiber in toric Fano manifolds, recovering previous computaional results of Floer homology .

Dans la première partie de cette thèse, on donne, sous certaines hypothèses, une minoration du nombre de points d’intersections d’une sous-variété Lagrangienne monotone L avec son image par une isotopie Hamiltonienne. Dans le cas où L est un espace K(pi, 1), et en particulier à courbure sectionnelle strictement négative, le minorant est 1 + beta1(L), où beta1 est le premier nombre de Betti à coefficients dans Z2. Une autre conséquence est la non-déplaçabilité d’un plongement Lagrangien monotone de RPn × K (où K est une sous-variété à courbure sectionnelle strictement négative telle que H1(K, Z) ≠ 0) dans certaines variétés symplectiques. Dans la seconde partie, on considère une sous-variété Lagrangienne monotone L non déplaçable. En utilisant l’homologie de Floer définie pour les Lagrangiennes qui sont C-1-proches de L, on obtient des informations sur son nombre de Maslov. De plus, si L peut être approchée par une suite de Lagrangiennes déplaçables, alors, sous certaines hypothèses topologiques sur L, l’énergie de déplacement des éléments de cette suite tend vers l’infini. / N the first part of the thesis, we give, under some hypotheses, a lower bound on the intersection number of a closed monotone Lagrangian submanifold L with its image by a generic Hamiltonianisotopy. For monotone Lagrangian submanifolds L which are K(pi, 1) and, in particular with negative sectional curvature, this bound is 1 + beta_1(L), where beta_1 is the first Betti number with coefficients in Z_2. Another consequence, is the non-displaceability of a monotone Lagrangian embedding of RPn x K (where K is a submanifold with negative sectional curvature such that H^1(K, Z) ≠ 0) in some symplectic manifolds. In the second part, given a closed monotone Lagrangian submanifold L, which is not displaceable, we use Floer homology defined on Lagrangians which are C^1 - close to L, to get information about it Maslov number. Besides, if L can be approached by a sequence of displaceable Lagrangians, then, under some topological assumptions on L, the displacement energy of the elements of this sequence converge to infinity.

Neste trabalho estudamos as subvariedades lagrangeanas mínimas e autossimilares do espaço paracomplexo Dn. Começamos definindo o conceito de variedade para-Kähler e, como exemplo, descrevemos o espaço projetivo paracomplexo. Em seguida, estudamos as subvariedades paracomplexas e lagrangeanas. Após mostrarmos que toda subvariedade paracomplexa não-degenerada é mínima, dedicamos a atenção ao estudo das subvariedades lagrangeanas, restringindo-nos ao ambiente Dn. Em particular, estudamos as lagrangeanas que são invariantes sob a ação canônica do grupo SO(n), e as superfícies de Castro-Chen. Em ambos os casos, analisamos a minimalidade e a autossimilaridade das mesmas. / In this work, we study minimal and self-similar Lagrangian submanifolds in the para-complex space Dn. Firstly, we define the concept of para-Kähler manifold and, to exemplify, we describe the para-complex projective space.Then, we study para-complex submanifolds and Lagrangian submanifolds. After proving that every non-degenerate para-complex submanifold is minimal, we pay attention to Lagrangian submanifolds, restricting us to the case of Dn. In particular, we study Lagrangian submanifolds which are invariant by the canonical SO(n)-action of Dn, and Castro-Chen\'s surfaces. In both cases, we analyse the minimality and self-similarity.

Neste trabalho estudamos as subvariedades lagrangeanas mínimas e autossimilares do espaço paracomplexo Dn. Começamos definindo o conceito de variedade para-Kähler e, como exemplo, descrevemos o espaço projetivo paracomplexo. Em seguida, estudamos as subvariedades paracomplexas e lagrangeanas. Após mostrarmos que toda subvariedade paracomplexa não-degenerada é mínima, dedicamos a atenção ao estudo das subvariedades lagrangeanas, restringindo-nos ao ambiente Dn. Em particular, estudamos as lagrangeanas que são invariantes sob a ação canônica do grupo SO(n), e as superfícies de Castro-Chen. Em ambos os casos, analisamos a minimalidade e a autossimilaridade das mesmas. / In this work, we study minimal and self-similar Lagrangian submanifolds in the para-complex space Dn. Firstly, we define the concept of para-Kähler manifold and, to exemplify, we describe the para-complex projective space.Then, we study para-complex submanifolds and Lagrangian submanifolds. After proving that every non-degenerate para-complex submanifold is minimal, we pay attention to Lagrangian submanifolds, restricting us to the case of Dn. In particular, we study Lagrangian submanifolds which are invariant by the canonical SO(n)-action of Dn, and Castro-Chen\'s surfaces. In both cases, we analyse the minimality and self-similarity.

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge - de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.

Let M be smooth symplectic closed manifold and L a closed Lagrangian submanifold of M. It was shown by Ozan that Ham(M,L): the relative Hamiltonian diffeomorphisms on M fixing the Lagrangian submanifold L setwise is a subgroup which is equal to the kernel of the restriction of the flux homomorphism to the universal cover of the identity component of the relative symplectomorphisms. In this thesis we show that Ham(M,L) is a non-simple perfect group, by adopting a technique due to Thurston, Herman, and Banyaga. This technique requires the diffeomorphism group be transitive where this property fails to exist in our case.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18444号 / 理博第4004号 / 新制||理||1577(附属図書館) / 31322 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 毅, 教授 堤 誉志雄, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In this thesis we present a slight generalisation of the Pearl complex or relative quantum homology to some non monotone Lagrangian submanifolds. First we develop the theory for the so called almost monotone Lagrangian submanifolds, We apply it to uniruling problems as well as estimates for the relative Gromov width. In the second part we develop the theory for toric fiber in toric Fano manifolds, recovering previous computaional results of Floer homology . / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

This thesis is split up into two parts each revolving around Floer homology and quantum cohomology of closed monotone symplectic manifolds. In the first part we consider symplectic manifolds obtained by symplectic reduction. Our main result is that a quantum version of an abelianization formula of Martin holds, which relates the quantum cohomologies of symplectic quotients by a group and by its maximal torus. Also we show a quantum version of the Leray-Hirsch theorem for Floer homology of Lagrangian intersections in the quotient. The second part is devoted to Floer homology of a pair of monotone Lagrangian submanifolds in clean intersection. Under these assumptions the symplectic action functional is degenerated. Nevertheless Frauenfelder defines a version of Floer homology, which is in a certain sense an infinite dimensional analogon of Morse-Bott homology. Via natural filtrations on the chain level we were able to define two spectral sequences which serve as a tool to compute Floer homology. We show how these are used to obtain new intersection results for simply connected Lagrangians in the product of two complex projective spaces. The link between both parts is that in the background the same technical methods are applied; namely the theory of holomorphic strips with boundary on Lagrangians in clean intersection. Since all our constructions rely heavily on these methods we also give a detailed account of this theory although in principle many results are not new or require only straight forward generalizations.

Cette thèse présente quelques propriétés des sous-variétés lagrangiennes monotones. On résoud d'abord une conjecture de Barraud et Cornea dans le cadre monotone en montrant que le rayon de Gromov relatif à deux lagrangiennes dans la même classe d'isotopie hamiltonienne donne une borne inférieure à la distance de Hofer entre ces deux mêmes lagrangiennes. Le cas non-monotone de cette conjecture reste ouvert encore. On définit toutes les structures nécessaires à l'énoncé et à la preuve de cette conjecture. Deuxièmement, on définit une nouvelle version d'un morphisme de Seidel relatif à l'aide des cobordismes lagrangiens de Biran et Cornea. On montre que cette version est chaîne-homotope aux différentes autres versions apparaissant dans la littérature. Que toutes ces définitions sont équivalentes fait partie du folklore mais n'apparaît pas dans la littérature. On conclut par une conjecture qui identifie un triangle exact obtenu par chirurgie lagrangienne et un autre dû à Seidel et faisant intervenir le twist de Dehn symplectique. / We present in this thesis a few properties of monotone Lagrangian submanifolds. We first solve a conjecture of Barraud and Cornea in the monotone setting by showing that the relative Gromov radius of two Hamiltonian-isotopic Lagrangians gives a lower bound on the Hofer distance between them. The general non-monotone case remains open to this day. We define all the structures relevant to state and prove the conjecture. We then define a new version of a Lagrangian Seidel morphism through the recently introduced Lagrangian cobordisms of Biran and Cornea. We show that this new version is chain-homotopic to various other versions appearing in the litterature. That all these previous versions are the same is folklore but did not appear in the litterature. We conclude with a conjecture claiming that an exact triangle obtained by Lagrangian surgery is isomorphic to an exact triangle of Seidel involving the symplectic Dehn twist.