The Hodge theory of algebraic maps 
-- Mark Andrea de Cataldo, September 19, 2003

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 I will discuss recent results of my joint work with L. Migliorini
at Bologna.
I will try to give a flavor of the results which are concerned
with the structure induced on the cohomology H*(X,Q)
of a projective manifold X by a projective map f: X --> Y.

In particular:

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- we decompose H*(X,Q) as a double direct sum of Hodge structures
polarized by the intersection form on X (this generalizes the Primitive
Lefschetz Decomposition and the Hodge-Riemann Bilinear Relations),


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- we determine the refined intersection form on the homology
of the fibers of the map f (this generalizes the Grauert-Mumford
criterion for the contraction of curves on surfaces).

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These results imply directly a refined version of the
Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber
for arbitrary proper algebraic maps.