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Zbl 0743.14006
Peters, C.A.M.
Rigidity for variations of Hodge structure and Arakelov-type finiteness theorems. (English)
[J] Compos. Math. 75, No.1, 113-126 (1990). ISSN 0010-437X; ISSN 1570-5846/e

Let $S$ be any smooth complex quasi-projective variety with a polarized variation of Hodge structure over it and $p:S\to\Gamma\backslash D$ be the corresponding period map. Here $D$ is a suitable period domain and $\Gamma$ is the monodromy group of the variation. If $p$ fits into a family of maps over the unit disk $\Delta$ yielding a period map $S\times\Delta\to\Gamma\backslash D$ we say that we have a deformation of period maps. This situation arises naturally when considering deformations of families over a fixed base $S$, such as in the Arakelov theorem, which states that there are no non-trivial deformations of families of curves over punctured curves $S$, except isotrivial ones. Any such deformation of period maps gives a horizontal vectorfield $Y$ along the image of $S$ in $\Gamma\backslash D$, the corresponding infinitesimal deformation. This field $Y$ turns out to commute with any local holomorphic vector field tangent $Z$ to the image of $S$ in $\Gamma\backslash D$. Using this, a local calculation on $D$ shows that the holomorphic bisectional curvature evaluated at $(Y,Z)$ is non- positive.\par Using standard estimates on the behaviour of the Hodge metrics in the limit, it follows that the section $Y$ determines a flat section of the local system $\hbox{End}(H\sb S)$ of endomorphisms of the local system $H\sb S$ (skew with respect to the polarization $Q$) defining the variation of Hodge structure. Moreover this flat section has pure type (- 1,1), so in particular, if there are not global skew endomorphisms of the local system $H\sb S$ of type $(-1,1)$ the system must be rigid. --- Together with Deligne's finiteness result [{\it P. Deligne} in Discrete groups in geometry and analysis, Prog. Math. 67, 1-19 (1987; Zbl 0656.14010)] and a form of Torelli's theorem one can get Arakelov-type finiteness results:\par There are at most finitely many families of polarized $K3$ surfaces over a fixed quasi-projective variety $S$ for which $\hbox{End}(H\sb S\sp{- 1,1})=0$, where $H\sb S$ is the local system on $S$ of primitive cohomology of the fibres.\par Very recently, {\it M. Saito} and {\it S. Zucker} obtained a complete classification of rigid and non-rigid families of $K3$-surfaces [see Math. Ann. 289, No. 1, 1-31 (1991; Zbl 0697.14024)]. There is also a useful result saying that $Y$ commutes with any $\overline X$, where $X$ is the image in $\hbox{End}(H\sb S)$ of any vector tangent to $S$. This has applications to the maximal rank of non-rigid deformations in weights 1 and 2 [see the author, ``On the rank of non-rigid variations of period maps in the weight one and two case'', in Complex Algebraic Varieties, Proc. Conf. Bayreuth 1990, Lect. Notes Math. 1507, 157-165 (1992)]. For example, a family of $g$-dimensional polarized abelian varieties having ${1\over 2}g(g-1)+1$ or more moduli is rigid and any family of $K3$- surfaces or Enriques surfaces, whose period map has rank two or more is rigid.
[C.A.M.Peters (Leiden)]

MSC 2000:: *14C30 Transcendental methods
14D15 (Formal) deformations
32G20 Period matrices
14D07 Variation of Hodge structures
14J27 Elliptic surfaces
14J28 K3-surfaces, etc.

Keywords: rigid family of $g$-dimensional polarized abelian varieties; polarized variation of Hodge structure; non-rigid deformation of period maps; Arakelov theorem; non-trivial deformations of families of curves over punctured curves; $K3$-surfaces; Enriques surfaces

Citations: Zbl 0656.14010; Zbl 0697.14024; Zbl 0707.14034

Cited in: Zbl 0791.14002

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