Hodge Theory and Algebraic Curves

A fundamental tool in studying the geometry of complex manifolds is represented by Hodge theory. The goal of this thesis is to understand how do Hodge structures arise naturally from manifolds and to analyze their role in studying their geometry, with particular attention to the case of algebraic curves. Firstly, some fundamental properties of Riemann surfaces are summarized. Holomorphic and meromorphic functions, maps and 1-forms over such manifolds are intoroduced together with the main results concerning them, such as the Hurwitz's formula, the Stokes theorem and the residue theorem. Subsequently, the tool of divisors is introduced, to be used particularly for stating the Riemann-Roch theorem, the Clifford's theorem and the Serre's duality theorem for Riemann surfaces. Finally, an overview of smooth algebraic curves is given. The Castelnuovo's bound theorem and the Max Noether's theorem are proved, and the subsequent correspondence of categories between compact Riemann surfaces and smooth algebraic projective curves is stated. In the third chapter, our point of view is turned to complex manifolds of any dimension n, which are briefly introduced together with holomorphic functions over them and their properties inherited from the functions defined on the complex plane. Main features of vector bundles are stated, with special interest in the real tangent and cotangent bundles with their complexified. Differential forms are defined together with the differential operators acting on them. Moreover analogous objects are presented with values on any vector bundle, in order to define the Dolbealut complexes. The last section of the third chapter finally concerns the class of complex varieties we will be mostly interested in: the Kähler manifolds. Note that, via the Fubini-Study metric, every projective complex manifold is showed to be Kähler. The fourth chapter finally presents the main topic of this thesis work: Hodge structures. Firstly, two abstract definitions of a pure Hodge structure on a finitely generated abelian group, one via the Hodge decomposition and the other via the Hodge filtration, are given, and indeed proven to be equivalent. In order to obtain new Hodge structures, one can resort to basic algebraic constructions, such as direct sums and tensor products. Moreover, Hodge substructures are defined through morphisms between known structures. Subsequently, the Hodge operator is defined, in order to obtain formal adjoints to the differential operators with respect to the L^2 metric on a complex manifold X. The Laplacians are defined together with the respective harmonic k-forms. The space of harmonic k-forms project isomorphically to the de Rham cohomology groups, thanks to the Hodge's theorem. As a consequence the Poincaré's and Serre's duality theorems are stated. In the particular case of Kähler manifolds, by introducing the Lefschetz operator and its adjoint, it is shown that the laplacians are all equal up to a constant. Studying the commutation relations concerning all the above operators, the Hodge decomposition theorem is proved, allowing us to put a pure Hodge structure over the cohomology groups of any compact Kähler manifold. Then the primitive differential forms are defined, to be used for obtaining the Lesfchetz decomposition. This unique decomposition result follows from the Hard lefschetz theorem. Finally, the Hodge index theorem is proved. Thanks to these results, polarized Hodge structures are introduced, and it is shown that a compact Kähler manifold equipped with a Kähler form naturally induces such a structure on its cohomology groups. The chapter is concluded by the definition of the first Chern class map, satisfying the Lefschetz's theorem on (1,1)-classes, which is a crucial step for the Kodaira embeddig theorem, subsequently stated. The fifth chapter is devoted to extending the construction of Hodge structures on the cohomology groups of complex manifolds which are not necessarily Kähler, via the introduction of mixed Hodge structures. After giving the abstract definition of such a structure on a finitely generated abelian group, together with its morphisms, it is shown how mixed Hodge structures form an abelian cateogory. It follows a brief algebraic excursus on hypercohomology and spectral sequences associated to a decreasing filtration on a complex of abelian groups. Subsequently, a decreasing Hodge filtration and an increasing weight filtration are defined on the logarithmic de Rham complex of an open complex manifold. In this setting, by making use of the tool of spectral sequences, The Deligne's theorem is proved, allowing the existence of a mixed Hodge structure on any open manifold. Finally, the Frölicher spectral sequence is defined, and its degeneration at the first term guarantees a result almost equivalent to the Hodge decomposition on compact Kähler manifolds. The latter section of the chapter concerns the study of the variations of Hodge structure derived from a family of deformations of a complex manifold. It is shown how the first order deformation is classified by the Kodaira-Spencer map. Subsequently, local systems and corresponding vector bundles equipped with flat connections are introduced, with particular interest in the Gauss-Manin connection, computed explicitly via the Cartan-Lie formula. The case of a compact Kähler central fibre carries some interesting results. In fact the upper semicontinuity of the Hodge numbers implies that they are locally constant in this setting. Moreover, the fibres in a neighbourhood of a compact Kähler manifold must be Kähler as well. In the last chapter our focus is moved toward algebraic curves. Firstly, the local period maps are defined for a family of deformations of a compact Kähler manifold, and are proved to be holomorphic and satisfying the Griffiths transversality condition. Then the period domain is briefly discussed, both in the non-polarized and in the polarized case. In order to extend our considerations globally, it is useful to consider the Gauss-Manin connection introduced in the previous chapter. The infinitesimal variation of Hodge structure at a point is defined and an explicit representation of the differential of the period map is given. By applying these results to the case of a general smooth algebraic curve and a universal deformation, via the theorems due to Max Noether and Petri, the infinitesimal Torelli theorem for curves and the generic Torelli theorem for curves of genus at least 5 are proven. The last section shows how to explicitly calculate the mixed Hodge structure in the cases of a singular curve arising from the union of two smooth curves intersecting transversally, and of a nodal curve, via the use of Mayer-Vietoris sequences. Finally, it is described carefully how to achieve an analytic description of a genus 2 algebraic curve with a nodal point, starting from its mixed Hodge structure