Torelli Theorem for K3 Surfaces

This thesis investigates K3 surfaces by starting from their projective realizations and moving towards a global Hodge-theoretic perspective. These surfaces are defined as compact complex surfaces with a trivial canonical bundle and vanishing first Betti number, representing the simplest examples of irreducible holomorphic symplectic manifolds. The first part of the work establishes the geometric foundations by examining concrete constructions such as complete intersections in Fano varieties, double covers, elliptic fibrations, and Kummer surfaces. For each of these models, the study describes how geometric degrees of freedom relate to the Picard rank of the surface. Central to this analysis is the lattice structure of H2(X, Z) determined by the cup-product pairing, which remains independent of the complex structure. The second part introduces the Hodge-theoretic framework, focusing on the period map and the construction of the period domain for marked K3 surfaces. It explores the variation of Hodge structures and the role of the Neron-Severi lattice, particularly how the Picard rank jumps along Noether-Lefschetz loci. The core of the dissertation is the Global Torelli theorem, which asserts that the Hodge structure on H2(X, Z) determines the isomorphism class of a marked K3 surface, provided that the positivity conditions of the Kahler cone are satisfied. Finally, the proof of the theorem is discussed through a twistor-based approach, which highlights the existence of Calabi-Yau metrics and complements classical arguments within a differential-geometric framework.