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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Neste trabalho apresento a teoria de Gromov-Witten, cohomologia quÃntica e mapas estÃveis e uso estas ferramentas para obter alguns resultados enumerativos. Em particular, provo a fÃrmula de Kontsevich para curvas racionais projetivas planas de grau d. FaÃo um estudo introdutÃrio dos espaÃos de Mumford-Knudsen e construo os espaÃos de Kontsevich a fim de definir os invariantes de Gromov-Witten. Estes sÃo usados para definir o anel de cohomologia quÃntica. Em seguida, aplico a teoria geral para o caso do plano projetivo e, usando a associatividade do produto quÃntico, obtenho a fÃrmula de Kontsevich. TambÃm estudo a fronteira do espaÃo modulli de mapas estÃveis e descrevo o grupo de Picard destes. Com isso, seguindo as ideias de Pandharipand, especialmente o algoritmo por este desenvolvido, calculo alguns nÃmeros caracterÃsticos de curvas no espaÃo projetivo. / In this work, I present the Gromov-Witten theory, quantum cohomology and stable maps and use these tools to obtain some enumerative results. In particular, I proof the Kontsevich formula to projective rational plane curves of degree d. I do an introductory study of Mumford-Knudsen spaces and construct the Kontsevich spaces in order to define gromov-witten invariants. These are used to define the quantum cohomology ring. Next, I apply the general theory to the case of the projective plane and, using the the associativity of the quantum product, I obtain the Kontsevich formula. I also study the boundary of the modulli space of stable maps and describe its Picard group. Following the ideas of Pandharipand, especially the algorithm he developed, I calculate some characteristic numbers of curves in the projective space.

Thesis advisor: Dawei Chen / We investigate the count of meromorphic differentials on the Riemann sphere pos-sessing a single zero, multiple poles with prescribed orders, and fixed residues at each pole. Gendron and Tahar previously examined this problem with respect to general residues using flat geometry, while Sugiyama approached it from the perspective of fixed-point multipliers of polynomial maps in the case of simple poles. In our study, we employ intersection theory on compactified moduli spaces of differentials, enabling us to handle arbitrary residues and pole orders, which provides a complete solution to this problem. We also determine interesting combinatorial properties of the solution formula. This thesis is organized as follows: In Chapter 1 we give an introduction to the problem and summarize the main results obtained. In Chapter 2 we review the compactification of moduli spaces of differentials and introduce various divisor classes. In Section 2.3 we explain how to identify the universal line bundle class with the divisor class of the locus of differentials satisfying a general given residue tuple and prove Theorem 1.0.1 (i). In Section 2.4 we impose exactly one independent partial sum vanishing condition to the residues and prove Theorem 1.0.1 (ii). In Section 2.5 we give a polynomial expression in terms of the zero order for the degree of mixed products between powers of the dual tautological class and the psi-class of the zero. Finally in Chapter 3 we prove Theorem 1.0.2 for arbitrary residues and investigate combinatorial properties of the solution formula. We have also verified our formula numerically for a number of cases by using the software package [CMZ2]. / Thesis (PhD) — Boston College, 2024. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Cette thèse est consacrée à l'étude de la géométrie des variétés de Fano complexes en utilisant les propriétés des sous-faisceaux du fibré tangent et la géométrie du diviseur fondamental. Les résultats principaux compris dans ce texte sont : (i) Une généralisation de la conjecture de Hartshorne: une variété lisse projective est isomorphe à un espace projectif si et seulement si son fibré tangent contient un sous-faisceau ample.(ii) Stabilité du fibré tangent des variétés de Fano lisses de nombre de Picard un : à l'aide de théorèmes d'annulation sur les espaces hermitiens symétriques irréductibles de type compact M, nous montrons que pour presque toute intersection complète générale dans M, le fibré tangent est stable. La même méthode nous permet de donner une réponse sur la stabilité de la restriction du fibré tangent de l'intersection complète à une hypersurface générale.(iii) Non-annulation effective pour des variétés de Fano et ses applications : nous étudions la positivité de la seconde classe de Chern des variétés de Fano lisses de nombre de Picard un. Ceci nous permet de montrer un théorème de non-annulation pour les variétés de Fano lisses de dimension n et d'indice n-3. Comme application, nous étudions la géométrie anticanonique des variétés de Fano et nous calculons les constantes de Seshadri des diviseurs anticanoniques des variétés de Fano d'indice grand.(iv) Diviseurs fondamentaux des variétés de Moishezon lisses de dimension trois et de nombre de Picard un : nous montrons l'existence d'un diviseur lisse dans le système fondamental dans certain cas particulier. / This thesis is devoted to the study of complex Fano varieties via the properties of subsheaves of the tangent bundle and the geometry of the fundamental divisor. The main results contained in this text are:(i) A generalization of Hartshorne's conjecture: a projective manifold is isomorphic to a projective space if and only if its tangent bundle contains an ample subsheaf.(ii) Stability of tangent bundles of Fano manifolds with Picard number one: by proving vanishing theorems on the irreducible Hermitian symmetric spaces of compact type M, we establish that the tangent bundles of almost all general complete intersections in M are stable. Moreover, the same method also gives an answer to the problem of stability of the restriction of the tangent bundle of a complete intersection on a general hypersurface.(iii) Effective non-vanishing for Fano varieties and its applications: we study the positivity of the second Chern class of Fano manifolds with Picard number one, this permits us to prove a non-vanishing result for n-dimensional Fano manifolds with index n-3. As an application, we study the anticanonical geometry of Fano varieties and calculate the Seshadri constants of anticanonical divisors of Fano manifolds with large index.(iv) Fundamental divisors of smooth Moishezon threefolds with Picard number one: we prove the existence of a smooth divisor in the fundamental linear system in some special cases.