# 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].