View source for Variation of Hodge structure

A variation of Hodge structure of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v0961702.png" /> on a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v0961703.png" /> is a couple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v0961704.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v0961705.png" /> is a locally constant sheaf of finitely-generated Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v0961706.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v0961707.png" /> is a finite decreasing filtration of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v0961708.png" /> by holomorphic subbundles, subject to the following conditions: i) the flat connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v0961709.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617010.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617011.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617013.png" /> local sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617015.png" />, respectively, satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617016.png" /> (Griffiths' transversality); ii) for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617017.png" />, the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617018.png" /> is a Hodge structure of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617020.png" />.

A polarization of a variation of Hodge structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617021.png" /> is a flat bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617022.png" /> which induces a polarization of the Hodge structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617023.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617024.png" />. Similar notions exist for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617025.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617026.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617027.png" /> , [[#References|[a2]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617028.png" /> is a proper smooth morphism of algebraic varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617030.png" /> is the underlying local system of a polarizable variation of Hodge structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617031.png" />. By a result of A. Borel, for a polarized variation of Hodge structure on a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617032.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617034.png" /> is compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617035.png" /> is a divisor with normal crossings, the monodromy around each local component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617036.png" /> is quasi-unipotent [[#References|[a3]]] (monodromy theorem). A polarized variation of Hodge structure over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617037.png" /> gives rise to a holomorphic period mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617038.png" /> to a classifying space of Hodge structures (see [[Period mapping|Period mapping]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617039.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617040.png" /> a compact Kähler manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617041.png" /> a divisor with normal crossings on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617042.png" />, then for a polarized variation of Hodge structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617043.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617044.png" />, the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617045.png" /> has a minimal extension to a perverse sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617046.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617048.png" /> carries a pure Hodge structure [[#References|[a4]]]–[[#References|[a6]]]. In fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096170/v09617049.png" /> is part of a polarized Hodge module [[#References|[a7]]]. Generalizations are the notions of variation of mixed Hodge structure [[#References|[a8]]], [[#References|[a9]]] and mixed Hodge module [[#References|[a10]]].

====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 {{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>