We begin developing a theory of morphisms of moduli spaces of pseudoholomorphic curves and discs with Lagrangian boundary conditions as Kuranishi spaces, using a modification of the procedure of Fukaya-Oh-Ohta-Ono. As an example, we consider the total space of the line bundles O(−n) and O on P1 as toric Kähler manifolds, and we construct isomorphic Kuranishi structures on the moduli space of holomorphic discs in O(−n) on P1 with boundary on a moment map fiber Lagrangian L and on a moduli space of holomorphic discs subject to appropriate tangency conditions in O. We then deform this latter Kuranishi space and use this deformation to define a Lagrangian potential for L in O(−n), and hence a superpotential for O(−n). With some conjectural assumptions regarding scattering diagrams in P1 × P, this superpotential can then be calculated tropically analogously to a bulk-deformed potential of a Lagrangian in P1 × P1.

Kuranishi atlases were introduced by McDuff and Wehrheim as a means to build a virtual fundamental cycle on moduli spaces of J-holomorphic curves and resolve some of the challenges in this field. This thesis considers genus zero Gromov-Witten invariants on a general closed symplectic manifold. We complete the construction of these invariants using Kuranishi atlases. To do so, we show that Gromov-Witten moduli spaces admit a smooth enough Kuranishi atlas to define a virtual fundamental class in any virtual dimension. In the process, we prove a stronger gluing theorem. Once we have defined genus zero Gromov-Witten invariants, we show that they satisfy the Gromov-Witten axioms of Kontsevich and Manin, a series of main properties that these invariants are expected to satisfy. A key component of this is the introduction of the notion of a transverse subatlas, a useful tool for working with Kuranishi atlases.

This thesis comprises the study of two moduli spaces of piecewise J-holomorphic curves. The main scheme is to consider a subdivision of the 2-sphere into a collection of small domains and to study collections of J-holomorphic maps into a symplectic manifold. These maps are coupled by Lagrangian boundary conditions. The work can be seen as finding a 2-dimensional analogue of the finite-dimensional path space approximation by piecewise geodesics on a Riemannian manifold (Q,g). For a nice class of target manifolds we consider tangent bundles of Riemannian manifolds and symplectizations of unit tangent bundles. Via polarization they provide a rich set of Lagrangians which can be used to define appropriate boundary value problems for the J-holomorphic pieces. The work focuses on existence theory as a pre-stage to global questions such as combinatorial refinement and the quality of the approximation. The first moduli space of lifted type is defined on a triangulation of the 2-sphere and consists of disks in the tangent bundle whose boundary projects onto geodesic triangles. The second moduli space of punctured type is defined on a circle packing domain and consists of boundary punctured disks in the symplectization of the unit tangent bundle. Their boundary components map into single fibers and at punctures the disks converge to geodesics. The coupling boundary conditions are chosen such that the piecewise problem always is Fredholm of index zero and both moduli spaces only depend on discrete data. For both spaces existence results are established for the J-holomorphic pieces which hold true on a small scale. Each proof employs a version of the implicit function theorem in a different setting. Here the argument for the moduli space of punctured type is more subtle. It rests on a connection to tropical geometry discovered by T. Ekholm for 1-jet spaces. The boundary punctured disks are constructed in the vicinity of explicit Morse flow trees which correspond to the limiting objects under degeneration of the boundary condition.

This thesis comprises the study of two moduli spaces of piecewise J-holomorphic curves. The main scheme is to consider a subdivision of the 2-sphere into a collection of small domains and to study collections of J-holomorphic maps into a symplectic manifold. These maps are coupled by Lagrangian boundary conditions. The work can be seen as finding a 2-dimensional analogue of the finite-dimensional path space approximation by piecewise geodesics on a Riemannian manifold (Q,g). For a nice class of target manifolds we consider tangent bundles of Riemannian manifolds and symplectizations of unit tangent bundles. Via polarization they provide a rich set of Lagrangians which can be used to define appropriate boundary value problems for the J-holomorphic pieces. The work focuses on existence theory as a pre-stage to global questions such as combinatorial refinement and the quality of the approximation. The first moduli space of lifted type is defined on a triangulation of the 2-sphere and consists of disks in the tangent bundle whose boundary projects onto geodesic triangles. The second moduli space of punctured type is defined on a circle packing domain and consists of boundary punctured disks in the symplectization of the unit tangent bundle. Their boundary components map into single fibers and at punctures the disks converge to geodesics. The coupling boundary conditions are chosen such that the piecewise problem always is Fredholm of index zero and both moduli spaces only depend on discrete data. For both spaces existence results are established for the J-holomorphic pieces which hold true on a small scale. Each proof employs a version of the implicit function theorem in a different setting. Here the argument for the moduli space of punctured type is more subtle. It rests on a connection to tropical geometry discovered by T. Ekholm for 1-jet spaces. The boundary punctured disks are constructed in the vicinity of explicit Morse flow trees which correspond to the limiting objects under degeneration of the boundary condition.

Dans ce travail, nous définissons des objets composés de disques complexes marqués reliés entre eux par des segments de droite munis d’une longueur. Nous construisons deux séries d’espaces de module de ces objets appelés clus- ters, une qui sera dite non symétrique, la version ⊗, et l’autre qui est dite symétrique, la version •. Cette construction permet des choix de perturba- tions pour deux versions correspondantes des trajectoires de Floer introduites par Cornea et Lalonde ([CL]). Ces choix devraient fournir une nouvelle option pour la description géométrique des structures A∞ et L∞ obstruées étudiées par Fukaya, Oh, Ohta et Ono ([FOOO2],[FOOO]) et Cho ([Cho]). Dans le cas où L ⊂ (M, ω) est une sous-variété lagrangienne Pin± mono- tone avec nombre de Maslov ≥ 2, nous définissons une structure d’algèbre A∞ sur les points critiques d’une fonction de Morse générique sur L. Cette struc- ture est présentée comme une extension du complexe des perles de Oh ([Oh]) muni de son produit quantique, plus récemment étudié par Biran et Cornea ([BC]). Plus généralement, nous décrivons une version géométrique d’une catégorie de Fukaya avec seul objet L qui se veut alternative à la description (relative) hamiltonienne de Seidel ([Sei]). Nous vérifions la fonctorialité de notre construction en définissant des espaces de module de clusters occultés qui servent d’espaces sources pour des morphismes de comparaison. / We define objects made of marked complex disks connected by metric line seg- ments and construct two sequences of moduli spaces of these objects, referred as the ⊗ version (nonsymmetric) and the • version (symmetric). This allows choices of coherent perturbations over the corresponding versions of the Floer trajectories proposed by Cornea and Lalonde ([CL]). These perturbations are intended to lead to an alternative geometric description of the (obstructed) A∞ and L∞ structures studied by Fukaya, Oh, Ohta and Ono ([FOOO2],[FOOO]) and Cho ([Cho]). Given a Pin± monotone lagrangian submanifold L ⊂ (M, ω) with mini- mal Maslov number ≥ 2, we define an A∞ -algebra structure from the critical points of a generic Morse function on L. We express this structure as a cochain complex extending the pearl complex introduced by Oh ([Oh]) and further ex- plicited by Biran and Cornea ([BC]), equipped with its quantum product. This could also be seen as an alternative geometric description of a Fukaya cate- gory of (M, ω) with L as its only object, a hamiltonian relative version appear- ing in [Sei]. Using spaces of quilted clusters, we verify, using more general quilted cluster spaces, that this defines a functor from a homotopy category of Pin± monotone lagrangian submanifolds hL mono,± (M, ω) to the homotopy category of cochain complexes hK(Λ-mod) where Λ is an appropriate Novikov ring.

Ce mémoire traite de la question suivante: est-ce que les cobordismes lagrangiens préservent l'uniréglage? Dans les deux premiers chapitres, on présente en survol la théorie des courbes pseudo-holomorphes nécessaire. On examine d'abord en détail la preuve que les espaces de courbes $ J $-holomorphes simples est une variété de dimension finie. On présente ensuite les résultats nécessaires à la compactification de ces espaces pour arriver à la définition des invariants de Gromov-Witten. Le troisième chapitre traite ensuite de quelques résultats sur la propriété d'uniréglage, ce qu'elle entraine et comment elle peut être démontrée. Le quatrième chapitre est consacré à la définition et la description de l'homologie quantique, en particulier celle des cobordismes lagrangiens, ainsi que sa structure d'anneau et de module qui sont finalement utilisées dans le dernier chapitre pour présenter quelques cas ou la conjecture tient. / In this dissertation we study the following question: do Lagrangian cobordisms preserve uniruling? In the two first chapters, the necessary pseudoholomorphic curves theory is quickly presented. We first study in detail the proof that the spaces of simple $ J $-holomorphic curves is a manifold of finite dimension. We then present the necessary results to produce the appropriate compactification of these spaces to get to the definition of Gromov-Witten invariants. In the third chapter then some results on the property of uniruling are presented: what are its consequences, how can it be obtained. In the fourth chapter quantum homology is defined, in particular for Lagrangian cobordism, and its ring and module structures are studied which are finally used in the last chapter to present examples of cobordisms which preserves uniruling.

Dans ce travail, nous définissons des objets composés de disques complexes marqués reliés entre eux par des segments de droite munis d’une longueur. Nous construisons deux séries d’espaces de module de ces objets appelés clus- ters, une qui sera dite non symétrique, la version ⊗, et l’autre qui est dite symétrique, la version •. Cette construction permet des choix de perturba- tions pour deux versions correspondantes des trajectoires de Floer introduites par Cornea et Lalonde ([CL]). Ces choix devraient fournir une nouvelle option pour la description géométrique des structures A∞ et L∞ obstruées étudiées par Fukaya, Oh, Ohta et Ono ([FOOO2],[FOOO]) et Cho ([Cho]). Dans le cas où L ⊂ (M, ω) est une sous-variété lagrangienne Pin± mono- tone avec nombre de Maslov ≥ 2, nous définissons une structure d’algèbre A∞ sur les points critiques d’une fonction de Morse générique sur L. Cette struc- ture est présentée comme une extension du complexe des perles de Oh ([Oh]) muni de son produit quantique, plus récemment étudié par Biran et Cornea ([BC]). Plus généralement, nous décrivons une version géométrique d’une catégorie de Fukaya avec seul objet L qui se veut alternative à la description (relative) hamiltonienne de Seidel ([Sei]). Nous vérifions la fonctorialité de notre construction en définissant des espaces de module de clusters occultés qui servent d’espaces sources pour des morphismes de comparaison. / We define objects made of marked complex disks connected by metric line seg- ments and construct two sequences of moduli spaces of these objects, referred as the ⊗ version (nonsymmetric) and the • version (symmetric). This allows choices of coherent perturbations over the corresponding versions of the Floer trajectories proposed by Cornea and Lalonde ([CL]). These perturbations are intended to lead to an alternative geometric description of the (obstructed) A∞ and L∞ structures studied by Fukaya, Oh, Ohta and Ono ([FOOO2],[FOOO]) and Cho ([Cho]). Given a Pin± monotone lagrangian submanifold L ⊂ (M, ω) with mini- mal Maslov number ≥ 2, we define an A∞ -algebra structure from the critical points of a generic Morse function on L. We express this structure as a cochain complex extending the pearl complex introduced by Oh ([Oh]) and further ex- plicited by Biran and Cornea ([BC]), equipped with its quantum product. This could also be seen as an alternative geometric description of a Fukaya cate- gory of (M, ω) with L as its only object, a hamiltonian relative version appear- ing in [Sei]. Using spaces of quilted clusters, we verify, using more general quilted cluster spaces, that this defines a functor from a homotopy category of Pin± monotone lagrangian submanifolds hL mono,± (M, ω) to the homotopy category of cochain complexes hK(Λ-mod) where Λ is an appropriate Novikov ring.