The perverse filtration and the Lefschetz hyperplane theorem

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

“…(4)) are Hodge substructures of the natural Hodge structure on H * (X). The geometric description of the perverse filtration in [55] (see §2.4) implies that this fact holds for every algebraic map, proper or not, to a quasi projective variety, and the proof in [55] is independent of the decomposition theorem. It follows that the geometric description of the perverse filtration in [55] can therefore be used to yield a considerable simplification of the line of reasoning in [51] for it endows, at the outset, the perverse cohomology groups H a b (X) with a natural Hodge structure, compatible with the primitive Lefschetz decompositions stemming from (18), and with respect to which the cup product maps L : H * (Y, P i ) → H * +2 (Y, P i ) and η : P k H * (X) → P k−2 H * +2 (X) are Hodge maps of type (1, 1).…”

“…• The extension to the quasi projective context of the results in [51,54] is contained in [45], which builds on [55]. Since these papers deal with non compact varieties, the statements involve mixed Hodge structures.…”

“…Although in general, the map H n (f −1 (0)) → H n (X) is not injective, the weight miracle is used to prove that the map H 0 (i ! P 0 ) → H n 0 (X) is an injection with image a pure Hodge substructure of the Hodge structure we have on H n 0 (X) (by virtue of the geometric description of the perverse filtration [55] mentioned above). Since this image lands automatically in the L ′ -primitive part, we conclude that the descended intersection form polarizes this image, hence ι is an isomorphism and we have the desired splitting into intersection cohomology complexes.…”

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“…More precisely, the results for projective varieties and the maps between them (in this case, all Hodge structures are pure) are found in [51,54] and the extension to quasi-projective varieties and the proper maps between them is found in [45], which builds heavily on [55].…”

“…In [55], we give a geometric description of the perverse filtration on the cohomology and on the cohomology with compact supports of a constructible complex on a quasi-projective variety. The paper [45] gives an alternative proof with the applications to mixed Hodge theory mentioned in §1.9; the paper [46] proves similar results for the standard filtration on cohomology with compact supports.…”