Fonctions thêta et critère de projectivité des tores complexes

This paper is devoted to some properties of the theory of theta functions on Riemann surfaces and explore some of its numerous consequences, a subject of renewed interest in recent years. We show how the meromorphic functions on the Riemann surfaces of an arbitrary genus can be constructed explicitly in terms of the multi-dimensional theta functions. We discuss the important role of the zeros of theta function and the Jacobi inversion problem which askes whether we can find a divisor that is the preimage for an arbitrary point in the Jacobian. The Lefschetz theorem on projective embeddings over the complex numbers is of utmost importance in the complex geometric theory of compact manifolds. We present an analytic proof of this theorem. We explain how positive line bundles on abelian varieties can be explicitly described in terms of multipliers and how their sections can be described by theta functions.This paper is devoted to some properties of the theory of theta functions on Riemann surfaces and explore some of its numerous consequences, a subject of renewed interest in recent years. We show how the meromorphic functions on the Riemann surfaces of an arbitrary genus can be constructed explicitly in terms of the multi-dimensional theta functions. We discuss the important role of the zeros of theta function and the Jacobi inversion problem which askes whether we can find a divisor that is the preimage for an arbitrary point in the Jacobian. The Lefschetz theorem on projective embeddings over the complex numbers is of utmost importance in the complex geometric theory of compact manifolds. We present an analytic proof of this theorem. We explain how positive line bundles on abelian varieties can be explicitly described in terms of multipliers and how their sections can be described by theta functions