An Analytic Approach to Complex Tori

Compact complex manifolds, unlike algebraic varieties, do not always admit a holomorphic embedding into a projective space. When this is possibile they are called projective, implying the existence of a positive definite integral Kähler form, which corresponds to the pullback of the Fubini-Study form on the projective space. This thesis focuses on three different approaches to find a holomorphic embedding in the context of complex tori: theta functions, holomorphic line bundles, and multiplier ideal sheaves. A complex torus that embeds into a projective space is known as an abelian variety. The first approach focuses on theta functions and their Riemann form, which corresponds to the integral Kähler form. These functions are fundamental to construct meromorphic functions on complex tori. In the second approach the framework of holomorphic line bundles is introduced. The connection with theta functions is extensively used in the Appell-Humbert Theorem and the Lefschetz Theorem. The third approach adopts a more modern perspective based on J.P. Demailly's work, using multiplier ideal sheaves. They provide a powerful analytic tool to obtain vanishing theorems and study embeddings. This approach extends beyond complex tori to general compact complex manifolds, recovering the Kodaira Embedding Theorem.