Complex Geometry Seminar

Let $f:X\to Y$ be a projective map, and assume for simplicity $X$ to be smooth. The Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber states that the (derived direct image of the constant sheaf ${\bf Q}_X$ is isomorphic to a direct sum of shifted intersection cohomology complexes of local systems. In the talk I will describe a recent proof of the theorem for complex varieties, developed by M.A. de Cataldo and myself (preprint math.AG0306030 version 2), based on classical Hodge Theory. In this approach a major role is played by a series of ``intersection forms along the strata'' which are naturally associated to $f$, and the Decomposition Theorem is shown to be equivalent with their non-degeneracy. We show that these forms are in fact definite, a generalization of Grauert criterion for contractibility of curves on a surface. This result is achieved by studying the nilpotent endomorphism on the rational cohomology of $X$ given by the cup product with the pullback of an ample line bundle on $Y$, and relating its Jordan canonical form to the topology of the map. In the talk I will illustrate how this approach works on some lower dimensional geometric examples, especially contractions, where this sort of ``Decomposition Theorem with signs'' seems to carry interesting geometric information.