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A variation of Hodge structure of weight  $  w $
on a complex manifold  $  {\mathcal S} $
is a couple  $  {\mathcal V} =( {\mathcal V} _ {\mathbf Z }  , {\mathcal F} ^ {bold \cdot } ) $
where  $  {\mathcal V} _ {\mathbf Z }  $
is a locally constant sheaf of finitely-generated Abelian groups on  $  {\mathcal S} $, 
and  $  {\mathcal F} ^ {bold \cdot } $
is a finite decreasing filtration of  $  V= {\mathcal V} _ {\mathbf Z }  \otimes _ {\mathbf Z }  {\mathcal O} _  {\mathcal S}   $
by holomorphic subbundles, subject to the following conditions: i) the flat connection  $  \nabla $
on  $  V $
defined by  $  \nabla ( v\otimes f  )= v\otimes df $, 
for  $  v $, 
$  f $
local sections of  $  {\mathcal V} _ {\mathbf Z }  $
and  $  {\mathcal O} _  {\mathcal S}   $, 
respectively, satisfies  $  \nabla ( {\mathcal F}  ^ {p} )\subset  {\mathcal F} ^ {p- 1 } \otimes \Omega _  {\mathcal S}    ^ {1} $(
Griffiths' transversality); ii) for each  $  s \in {\mathcal S} $, 
the pair  $  ( {\mathcal V} _ {\mathbf Z ,s }  , {\mathcal F} ^ {bold \cdot } ( s)) $
is a Hodge structure of weight  $  w $.

A polarization of a variation of Hodge structure  $  ( {\mathcal V} _ {\mathbf Z }  , {\mathcal F} ^ {bold \cdot } ) $
is a flat bilinear form  $  {\mathcal V} _ {\mathbf Z }  \otimes {\mathcal V} _ {\mathbf Z }  \rightarrow \mathbf Z _  {\mathcal S}   $
which induces a polarization of the Hodge structure  $  {\mathcal V} _ {\mathbf Z ,s }  $
for each  $  s \in {\mathcal S} $. 
Similar notions exist for  $  \mathbf Z $
replaced by  $  \mathbf Q $
or  $  \mathbf R $, 
[[#References|[a2]]]. If  $  f:  X \rightarrow S $
is a proper smooth morphism of algebraic varieties over  $  \mathbf C $, 
then  $  R  ^ {m} f _ {*} \mathbf Z _ {X} $
is the underlying local system of a polarizable variation of Hodge structure on  $  {\mathcal S} $. 
By a result of A. Borel, for a polarized variation of Hodge structure on a complex manifold  $  S $
of the form  $  \overline{S}\; \setminus  D $, 
where  $  \overline{S}\; $
is compact and  $  D\subset  \overline{S}\; $
is a divisor with normal crossings, the monodromy around each local component of  $  D $
is quasi-unipotent [[#References|[a3]]] (monodromy theorem). A polarized variation of Hodge structure over  $  S $
gives rise to a holomorphic period mapping from  $  S $
to a classifying space of Hodge structures (see [[Period mapping|Period mapping]]).

If  $  {\mathcal S} = \overline{S}\; \setminus  D $
with  $  \overline{S}\; $
a compact Kähler manifold and  $  D $
a divisor with normal crossings on  $  \overline{S}\; $, 
then for a polarized variation of Hodge structure  $  ( {\mathcal V} _ {\mathbf Z }  , {\mathcal F} ^ {bold \cdot } ) $
on  $  S $, 
the sheaf  $  {\mathcal V} _ {\mathbf Z }  $
has a minimal extension to a perverse sheaf  $  IC( {\mathcal V} _ {\mathbf Z }  ) $
on  $  \overline{S}\; $
and  $  IH  ^ {*} ( \overline{S}\; , IC( {\mathcal V} _ {\mathbf Z }  )) $
carries a pure Hodge structure [[#References|[a4]]]–[[#References|[a6]]]. In fact,  $  IC( {\mathcal V} _ {\mathbf Z }  ) $
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>