# 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
How to Cite This Entry:
Variation of Hodge structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variation_of_Hodge_structure&oldid=51181
This article was adapted from an original article by J. Steenbrink (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article