The Hard Lefschetz Theorem and the topology of semismall maps

By M. A. Cataldo, L. Migliorini. 2000

114 citations recorded by the source corpus.

Abstract A line bundle on a complex projective manifold is said to be lef if one of its powers is globally generated and defines a semismall map in the sense of Goresky–MacPherson. As in the case of ample bundles the first Chern class of lef line bundles satisfies the Hard Lefschetz Theorem and the Hodge–Riemann Bilinear Relations. As a consequence, we prove a generalization of the Grauert contractibility criterion: the Hodge Index Theorem for semismall maps , Theorem 2.4.1. For these maps the Decomposition Theorem of Beilinson, Bernstein and Deligne is equivalent to the non-degeneracy of certain intersection forms associated with a stratification. This observation, joint with the Hodge Index Theorem for semismall maps gives a new proof of the Decomposition Theorem for the direct image of the constant sheaf. A new feature uncovered by our proof is that the intersection forms involved are definite.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/267b888b0e785d379c405d13e9de2c13f73fb1d9

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The Hodge theory of algebraic maps (2003, methodology): “In this paper, in the spirit of our paper [5], we give a geometric proof of the Relative Hard Lefschetz, Decomposition and Semisimplicity isomorphisms (3), (4) and (5).”
The decomposition theorem, perverse sheaves and the topology of algebraic maps (2007, methodology): “In the 2000’s, we re-proved these results in [48, 51].”