Proceedings of Symposia in Pure Mathematics
Hodge-theoretic aspects of the Decomposition Theorem
Mark Andrea A. de Cataldo and Luca Migliorini
Abstract. Given a projective morphism of compact, complex, algebraic vari-
eties and a relatively ample line bundle on the domain we prove that a suitable
choice, dictated by the line bundle, of the decomposition isomorphism of the
Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber, yields
isomorphisms of pure Hodge structures. The proof is based on a new coho-
mological characterization of the decomposition isomorphism associated with
the line bundle. We prove some corollaries concerning the intersection form in
intersection cohomology, the natural map from cohomology to intersection co-
homology, projectors and Hodge cycles, and induced morphisms in intersection
1. Introduction
2. Formalism for decompositions
3. Applications
1. Introduction
Let f : X Y be a projective map of proper, complex, algebraic varieties. The
Decomposition Theorem predicts that the derived direct image complex Rf∗ICX of
the rational intersection cohomology complex ICX of X splits into the direct sum
of shifted intersection cohomology complexes on Y. This splitting is not canonical.
When viewed in hypercohomology, it yields decompositions of the rational intersec-
tion cohomology groups IH(X, Q) into the direct sum of intersection cohomology
groups with twisted coefficients of closed subvarieties of Y.
The Decomposition Theorem is the deepest known fact concerning the ho-
mology of complex algebraic varieties and it has far-reaching consequences. The
following consideration may give a measure of the importance as well as of the spe-
cial character of this result. The splitting behavior of Rf∗ICX over Y is dictated
in part by the one over any open subset U Y. This remarkable fact is special to
complex algebraic geometry, e.g. it fails for complex analytic geometry. More pre-
cisely: let U Y be a Zariski-dense open subset, S U be a closed submanifold,
c 0000 (copyright holder)
Proceedings of Symposia in Pure Mathematics
Volume 80.2, 2009
c 2009 American Mathematical Society
Proceedings of Symposia in Pure Mathematics
Volume 80.2 , 2009
c 2009 American Mathematical Society
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