Title:

Algebraic tori attached to algebraic varieties

Abstract:

To a compact complex manifold X defined by polynomial equations, Griffiths associated complex tori called the intermediate Jacobians of X. Their construction is of a transcendental nature. However in some cases of interest these intermediate Jacobians can indeed be defined by polynomial equations. One is then led to ask: suppose X is defined by polynomial equations with rational coefficients, then are its intermediate Jacobians also defined by polynomial equations with rational coefficients? Are there analogues to the intermediate Jacobians over finite fields? I will give an overview of recent and not-so-recent results, together with applications to classical questions in algebraic geometry.