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

cohomology.

Contents

1. Introduction

2. Formalism for decompositions

3. Applications

References

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)

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Proceedings of Symposia in Pure Mathematics

Volume 80.2, 2009

c 2009 American Mathematical Society

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Proceedings of Symposia in Pure Mathematics

Volume 80.2 , 2009

c 2009 American Mathematical Society

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http://dx.doi.org/10.1090/pspum/080.2/2483945