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Mixed Hodge modules. (English) Zbl 0727.14004
In his previous paper in Publ. Res. Inst. Math. Sci. 24, No.6, 849-995 (1988; Zbl 0691.14007) the author showed that a polarizable Hodge module with strict support (i.e. such that the underlying perverse sheaf is an intersection complex) is generically a polarizable variation of Hodge structure. In the paper under review the author shows the converse, viz. any polarizable variation of Hodge structure defined on a Zariski open subset can uniquely (and functorially) be extended to a polarizable Hodge module with strict support. Thus the intersection cohomology \(IH^*(X,L)\) with coefficients in L is endowed with a natural Hodge structure provided that X is compact and bimeromorphic to a compact Kähler manifold \(\tilde X\) and L is a local system on a Zariski open smooth subset of X and underlies a polarizable variation of Hodge structure. The author also introduces and studies functorial properties of mixed Hodge modules corresponding to perverse mixed complexes. He verifies that (polarizable) mixed Hodge modules on a point can be identified with (polarizable) \({\mathbb{Q}}\)-mixed Hodge structures. He proceeds with showing that polarizable mixed Hodge modules are stable with respect to external product. A special section is devoted to mixed Hodge Modules on algebraic varieties.
Reviewer: F.L.Zak (Moskva)

14D07 Variation of Hodge structures (algebro-geometric aspects)
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
32J27 Compact Kähler manifolds: generalizations, classification
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
55N33 Intersection homology and cohomology in algebraic topology
Full Text: DOI
[1] Beilinson, A.A., On the derived category of perverse sheaves (preprint). · Zbl 0652.14008
[2] , Notes on absolute Hodge cohomology, Contemporary Mathematics, 55 (1986), 35-68.
[3] Beilinson, A.A., Bernstein, J. and Deligne, P., Faisceaux pervers, Asterisque, 100 (1982), 5-171.
[4] Bernstein, J., Algebraic theory of ^-Modules (preprint).
[5] Borel, A., Algebraic on ^-Modules, Academic Press, Boston, 1987. · Zbl 0642.32001
[6] Cattani, E. and Kaplan, A., Polarized Mixed Hodge structures and the local monodromy of a variation of Hodge structure, Invent. Math., 67(1982), 101-115. · Zbl 0516.14005 · doi:10.1007/BF01393374 · eudml:142901
[7] Cattani, E., Kaplan, A. and Schmid, W., Degeneration of Hodge structure, Ann. of Math., 123 (1986), 457-535. · Zbl 0617.14005 · doi:10.2307/1971333
[8] [9] Deligne, P., Theorie de Hodge I, Actes Congres Intern. Math., (1970), 425-430; II, Pub I. Math. IHES, 40 (1971), 5-58; III, ibid., 44 (1974), 5-77.
[9] , La conjecture de Weil II, Publ. Math. IHES, 52 (1980), 137-252. |-11-| ^ Equations differentielles a points singuliers reguliers, Led. Notes in Math., 163, Springer, Berlin, 1970.
[10] Kashiwara, M., Vanishing cycle sheaves and holonomic systems of differential equations, in Lect. Notes in Math., 1016, 134-142, Springer, Berlin, 1983. · Zbl 0566.32022
[11] , A study of variation of mixed Hodge structure, Publ. RIMS, 22 (1986), 991-1024. · Zbl 0621.14007 · doi:10.2977/prims/1195177264
[12] Kashiwara, M. and Kawai, T., On the holonomic system of microdifferential equations III, Publ. RIMS, 17 (1981), 813-979. · Zbl 0505.58033 · doi:10.2977/prims/1195184396
[13] , The Poincare lemma for a variation of Hodge structure, Publ. RIMS, 23 (1987), 345-407.
[14] , Hodge structure and holonomic systems, Proc. Japan Acad., 62 Ser. A (1985), 1-4. · Zbl 0629.14006
[15] MacPherson, R. and Vilonen, K., Elementary construction of perverse sheaves, Invent. Math., 84 (1986), 403-435. · Zbl 0597.18005 · doi:10.1007/BF01388812 · eudml:143344
[16] Mebkhout, Z, Une autre equivalence de categories, Comp. Math., 51 (1984), 63-88.
[17] Saito, M, Modules de Hodge polarisables, Publ. RIMS, 24 (1988), 849-995. · Zbl 0691.14007 · doi:10.2977/prims/1195173930
[18] , Mixed Hodge Modules, Proc. Japan Acad., 62 Ser. A (1986), 360-363. · Zbl 0635.14008 · doi:10.3792/pjaa.62.360
[19] , On the derived categories of Mixed Hodge Modules, Proc. Japan Acad., 62 Ser. A (1986), 364-366. · Zbl 0635.14009 · doi:10.3792/pjaa.62.364
[20] Steenbrink, J and Zucker, S., Variation of mixed Hodge structure I, Invent. Math., 80 (1985), 485-542. · Zbl 0626.14007 · doi:10.1007/BF01388729 · eudml:143242
[21] Verdier, J.-L., Extension of a perverse sheaf over a closed subspace, Asterisque, 130 (1985), 210-217.
[22] Deligne, P., Theoreme de Lefschetz et criteres de degenerescence de suites spectrales., Publ Math. IHES, 35 (1968), 107-126. · Zbl 0159.22501 · doi:10.1007/BF02698925 · numdam:PMIHES_1968__35__107_0 · eudml:103884
[23] Saito, M., Duality for vanishing cycle functors, Publ. RIMS., 25 (1989), 889-921 £26] , Decomposition theorem for proper Kahler morphisms (IHES preprint July 1988).
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