Variation of Hodge structure

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