Filtrations on cohomology arising from geometry -- Mark Andrea de Cataldo

I will report on work in progress with Luca Migliorini at Bologna. Given a map of complex algebraic varieties, the Leray spectral sequence and the perverse Leray spectral sequence induce filtrations on the (intersection) cohomology (with compact supports) of the domain. We describe these filtrations geometrically in terms of hyperplane sections and deduce various new Hodge-theoretic consequences for the homology of complex varieties. A very interesting precursor of this new point of view, concerning the Leray spectral sequence for cohomology, is due to Donu Arapura.