Complex Algebraic Geometry

On a topological space, algebraic topology and homological algebra endow cohomology groups with various filtrations. In the case of algebraic varieties, one may wonder if such filtrations, e.g. the Grothendieck filtration, the Leray filtration, the perverse filtration etc., admit a geometric description. I shall report on recent and on-going work with L. Migliorini where we give such a description and draw some consequences.