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) 1 Volume 80.2, 2009 c 2009 American Mathematical 489 493 501 504
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