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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" />.
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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]]).
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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]]].
+
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} ^ {\bullet } ) $
 +
where  $  {\mathcal V} _ {\mathbf Z }  $
 +
is a locally constant sheaf of finitely-generated Abelian groups on  $  {\mathcal S} $,
 +
and  $  {\mathcal F} ^ {\bullet } $
 +
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} ^ {\bullet } ( s)) $
 +
is a Hodge structure of weight  $  w $.
 +
 
 +
A polarization of a variation of Hodge structure  $  ( {\mathcal V} _ {\mathbf Z }  , {\mathcal F} ^ {\bullet } ) $
 +
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} ^ {\bullet } ) $
 +
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====
 
====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>
+
<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, "$L^2$ 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: $L_2$-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>

Latest revision as of 18:39, 2 January 2021


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} ^ {\bullet } ) $ where $ {\mathcal V} _ {\mathbf Z } $ is a locally constant sheaf of finitely-generated Abelian groups on $ {\mathcal S} $, and $ {\mathcal F} ^ {\bullet } $ 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} ^ {\bullet } ( s)) $ is a Hodge structure of weight $ w $.

A polarization of a variation of Hodge structure $ ( {\mathcal V} _ {\mathbf Z } , {\mathcal F} ^ {\bullet } ) $ 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 $, [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 [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).

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} ^ {\bullet } ) $ 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 [a4][a6]. In fact, $ IC( {\mathcal V} _ {\mathbf Z } ) $ 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, "$L^2$ 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: $L_2$-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