Claire Voisin: Mixed Hodge structures and the topology of algebraic varieties

Abstract: Deligne proved that the cohomology of a complex algebraic variety carries a functorial “mixed Hodge structure”, which generalizes the previously known results for smooth projective complex varieties. Formal properties of morphisms of Mixed Hodge structures then allowed him to deduce a number of important consequences, including the “theorem of the fixed part”. I will describe these fundamental results, and some applications to the study of Hodge classes. This lecture was given at The University of Oslo, May 22, 2013 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations. Program for the Abel Lectures 2013 1. "Hidden symmetries of algebraic varieties" by Abel Laureate 2013, professor Pierre Deligne, Institute for Advanced Study, Princeton 2. "Life Over Finite Fields" by professor Nicholas Katz, Princeton University 3. "Mixed Hodge structures and the topology of algebraic varieties" by professor Claire Voisin, École Polytechnique and CNRS 4. "Algebraic geometry and the ongoing unification of mathematics", a science lecture by professor Ravi Vakil, Stanford University