×

zbMATH — the first resource for mathematics

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)

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[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).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.