Modules de Hodge Polarisables

The decomposition theorem, perverse sheaves and the topology of algebraic maps

“…The hard Lefschetz theorem for the intersection cohomology of projective varieties is proved in [9]. In the 1980's, M. Saito ( [156,157]) proved that in the projective case these groups admit a pure Hodge structure (i.e. a (p, q)-Hodge decomposition), re-proved that they satisfy the hard Lefschetz theorem and proved the Hodge-Riemann bilinear relations.…”

“…(Not all local systems as above are of geometric origin.) In his remarkable work on the subject, M. Saito answered the first question in the affirmative in [156] and the second question in the affirmative in the case of IC X in [158]; we refer the reader to M. Saito's paper for the precise formulations in the Kähler context. In fact, he developed in [157] a general theory of compatibility of mixed Hodge structures with the various functors, and in the process he completed the extension of the Hodge-Lefschetz theorems from the cohomology of projective manifolds, to the intersection cohomology of projective varieties.…”

“…The work of M. Saito [156,157] settled these issues completely with the use of mixed Hodge modules. The reader interested in the precise statements and generalizations is referred to Saito's papers (for brief summaries, see [70] and §3.2).…”

“…All these questions have been answered in the work of M. Saito [156,157] in the 1980's. The case of IC X (i.e.…”

“…If X is a possibly singular projective variety, then we obtain the hard Lefschetz theorem in intersection cohomology ( §1.4). Remark 1.6.4 Theorems 1.6.1 and 1.6.3 also apply to Rf * IC X (L) for certain classes of local systems L (see [9,156]). Example 1.6.5 Let X = P 1 C × C and Y be the space obtained collapsing the set P 1 C × {o} to a point.…”