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====References====
<table><TR><TD valign="top">[a1a]</TD> <TD valign="top"> P. Griffiths,   "Periods of integrals on algebraic manifolds I" ''Amer. J. Math.'' , '''90''' (1968) pp. 568–626</TD></TR><TR><TD valign="top">[a1b]</TD> <TD valign="top"> P. Griffiths,   "Periods of integrals on algebraic manifolds II" ''Amer. J. Math.'' , '''90''' (1968) pp. 808–865</TD></TR><TR><TD valign="top">[a1c]</TD> <TD valign="top"> P. Griffiths,   "Periods of integrals on algebraic manifolds III" ''Publ. Math. IHES'' , '''38''' (1970) pp. 228–296</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Deligne,   "Travaux de Griffiths" , ''Sem. Bourbaki Exp. 376'' , ''Lect. notes in math.'' , '''180''' , Springer (1970) pp. 213–235</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Schmid,   "Variation of Hodge structure: the singularities of the period mapping" ''Invent. Math.'' , '''22''' (1973) pp. 211–319</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Cattani,   A. Kaplan,   W. Schmid,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617050.png" /> and intersection cohomologies for a polarizable variation of Hodge structure" ''Invent. Math.'' , '''87''' (1987) pp. 217–252</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Kashiwara,   T. Kawai,   "The Poincaré lemma for variations of polarized Hodge structures" ''Publ. R.I.M.S. Kyoto Univ.'' , '''23''' (1987) pp. 345–407</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> S. Zucker,   "Hodge theory with degenerating coefficients: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617051.png" />-cohomology in the Poincaré metric" ''Ann. of Math.'' , '''109''' (1979) pp. 415–476</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> M. Saito,   "Modules de Hodge polarisables" ''Publ. R.I.M.S. Kyoto Univ.'' , '''24''' (1988) pp. 849–995</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J. Steenbrink,   S. Zucker,   "Variation of mixed Hodge structure, I" ''Invent. Math.'' , '''80''' (1985) pp. 489–542</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> M. Kashiwara,   "A study of a variation of mixed Hodge structure" ''Publ. R.I.M.S. Kyoto Univ.'' , '''22''' (1986) pp. 991–1024</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> M. Saito,   "Mixed Hodge modules" ''Publ. R.I.M.S. Kyoto Univ.'' , '''26''' (1990) pp. 221–333</TD></TR></table>
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<table><TR><TD valign="top">[a1a]</TD> <TD valign="top"> P. Griffiths, "Periods of integrals on algebraic manifolds I" ''Amer. J. Math.'' , '''90''' (1968) pp. 568–626 {{MR|0229641}} {{ZBL|0169.52303}} </TD></TR><TR><TD valign="top">[a1b]</TD> <TD valign="top"> P. Griffiths, "Periods of integrals on algebraic manifolds II" ''Amer. J. Math.'' , '''90''' (1968) pp. 808–865 {{MR|0233825}} {{ZBL|0183.25501}} </TD></TR><TR><TD valign="top">[a1c]</TD> <TD valign="top"> P. Griffiths, "Periods of integrals on algebraic manifolds III" ''Publ. Math. IHES'' , '''38''' (1970) pp. 228–296 {{MR|0282990}} {{ZBL|0212.53503}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Deligne, "Travaux de Griffiths" , ''Sem. Bourbaki Exp. 376'' , ''Lect. notes in math.'' , '''180''' , Springer (1970) pp. 213–235 {{MR|}} {{ZBL|0208.48601}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Schmid, "Variation of Hodge structure: the singularities of the period mapping" ''Invent. Math.'' , '''22''' (1973) pp. 211–319 {{MR|0382272}} {{ZBL|0278.14003}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Cattani, A. Kaplan, W. Schmid, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617050.png" /> and intersection cohomologies for a polarizable variation of Hodge structure" ''Invent. Math.'' , '''87''' (1987) pp. 217–252 {{MR|870728}} {{ZBL|0611.14006}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Kashiwara, T. Kawai, "The Poincaré lemma for variations of polarized Hodge structures" ''Publ. R.I.M.S. Kyoto Univ.'' , '''23''' (1987) pp. 345–407 {{MR|0890924}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> S. Zucker, "Hodge theory with degenerating coefficients: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617051.png" />-cohomology in the Poincaré metric" ''Ann. of Math.'' , '''109''' (1979) pp. 415–476 {{MR|534758}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> M. Saito, "Modules de Hodge polarisables" ''Publ. R.I.M.S. Kyoto Univ.'' , '''24''' (1988) pp. 849–995 {{MR|1000123}} {{ZBL|0691.14007}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J. Steenbrink, S. Zucker, "Variation of mixed Hodge structure, I" ''Invent. Math.'' , '''80''' (1985) pp. 489–542 {{MR|0791673}} {{MR|0791674}} {{ZBL|0626.14007}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> M. Kashiwara, "A study of a variation of mixed Hodge structure" ''Publ. R.I.M.S. Kyoto Univ.'' , '''22''' (1986) pp. 991–1024 {{MR|866665}} {{ZBL|0621.14007}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> M. Saito, "Mixed Hodge modules" ''Publ. R.I.M.S. Kyoto Univ.'' , '''26''' (1990) pp. 221–333 {{MR|1047741}} {{MR|1047415}} {{ZBL|0727.14004}} {{ZBL|0726.14007}} </TD></TR></table>

Revision as of 17:02, 15 April 2012

A variation of Hodge structure of weight on a complex manifold is a couple where is a locally constant sheaf of finitely-generated Abelian groups on , and is a finite decreasing filtration of by holomorphic subbundles, subject to the following conditions: i) the flat connection on defined by , for , local sections of and , respectively, satisfies (Griffiths' transversality); ii) for each , the pair is a Hodge structure of weight .

A polarization of a variation of Hodge structure is a flat bilinear form which induces a polarization of the Hodge structure for each . Similar notions exist for replaced by or , [a2]. If is a proper smooth morphism of algebraic varieties over , then is the underlying local system of a polarizable variation of Hodge structure on . By a result of A. Borel, for a polarized variation of Hodge structure on a complex manifold of the form , where is compact and is a divisor with normal crossings, the monodromy around each local component of is quasi-unipotent [a3] (monodromy theorem). A polarized variation of Hodge structure over gives rise to a holomorphic period mapping from to a classifying space of Hodge structures (see Period mapping).

If with a compact Kähler manifold and a divisor with normal crossings on , then for a polarized variation of Hodge structure on , the sheaf has a minimal extension to a perverse sheaf on and carries a pure Hodge structure [a4][a6]. In fact, is part of a polarized Hodge module [a7]. Generalizations are the notions of variation of mixed Hodge structure [a8], [a9] and mixed Hodge module [a10].

References

[a1a] P. Griffiths, "Periods of integrals on algebraic manifolds I" Amer. J. Math. , 90 (1968) pp. 568–626 MR0229641 Zbl 0169.52303
[a1b] P. Griffiths, "Periods of integrals on algebraic manifolds II" Amer. J. Math. , 90 (1968) pp. 808–865 MR0233825 Zbl 0183.25501
[a1c] P. Griffiths, "Periods of integrals on algebraic manifolds III" Publ. Math. IHES , 38 (1970) pp. 228–296 MR0282990 Zbl 0212.53503
[a2] P. Deligne, "Travaux de Griffiths" , Sem. Bourbaki Exp. 376 , Lect. notes in math. , 180 , Springer (1970) pp. 213–235 Zbl 0208.48601
[a3] W. Schmid, "Variation of Hodge structure: the singularities of the period mapping" Invent. Math. , 22 (1973) pp. 211–319 MR0382272 Zbl 0278.14003
[a4] E. Cattani, A. Kaplan, W. Schmid, " and intersection cohomologies for a polarizable variation of Hodge structure" Invent. Math. , 87 (1987) pp. 217–252 MR870728 Zbl 0611.14006
[a5] M. Kashiwara, T. Kawai, "The Poincaré lemma for variations of polarized Hodge structures" Publ. R.I.M.S. Kyoto Univ. , 23 (1987) pp. 345–407 MR0890924
[a6] S. Zucker, "Hodge theory with degenerating coefficients: -cohomology in the Poincaré metric" Ann. of Math. , 109 (1979) pp. 415–476 MR534758
[a7] M. Saito, "Modules de Hodge polarisables" Publ. R.I.M.S. Kyoto Univ. , 24 (1988) pp. 849–995 MR1000123 Zbl 0691.14007
[a8] J. Steenbrink, S. Zucker, "Variation of mixed Hodge structure, I" Invent. Math. , 80 (1985) pp. 489–542 MR0791673 MR0791674 Zbl 0626.14007
[a9] M. Kashiwara, "A study of a variation of mixed Hodge structure" Publ. R.I.M.S. Kyoto Univ. , 22 (1986) pp. 991–1024 MR866665 Zbl 0621.14007
[a10] M. Saito, "Mixed Hodge modules" Publ. R.I.M.S. Kyoto Univ. , 26 (1990) pp. 221–333 MR1047741 MR1047415 Zbl 0727.14004 Zbl 0726.14007
How to Cite This Entry:
Variation of Hodge structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variation_of_Hodge_structure&oldid=15012
This article was adapted from an original article by J. Steenbrink (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article