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Infinitesimal variations of Hodge structure. I. (English) Zbl 0531.14006
Because of horizontality, a generic polarized Hodge structure of weight \(w>1\) does not arise from geometry. Moreover, because the classifying space \(D\) for \(H\) is homogeneous under the action of \(G_ R\), there are no linear algebra invariants which can distinguish between geometric and non-geometric structures. A possible solution to this difficulty is given by the notion of an infinitesimal variation of Hodge structure (IVHS). Such an object, essentially the differential of a variation of Hodge structure at one point, is given formally by a complex vector space T and a homomorphism \(\delta:T\to End(Gr^ FH)\) which is (a) horizontal (of degree -1), (b) skew-symmetric \((<\delta(x)\alpha,\beta>+<\alpha,\delta(x)\beta>=0),\) and commutative \((\delta(x_ 1)\delta(x_ 2)=\delta(x_ 2)\delta(x_ 1)).\) Unlike the set of bare Hodge structures, the set of infinitesimal variations is not homogeneous under the action of \(G_ R:\) nontrivial invariants exist.
Section I introduces five invariants, the first of which is \(\delta^ n:Sym^ nT\to Hom^{(s)}(H^{n,0},H^{0,n}).\) The remaining sections explore the geometric content of \(\delta^ n\). An immediate object of interest is the space of infinitesimal Schottky relations, by definition the kernel \({\mathcal Q}\) of \(\delta^ n\). When \(\delta\) comes from the moduli of curves, \({\mathcal Q}\) is the space of quadrics through the canonical curve. In general we are able to interpret geometrically only a subspace of \({\mathcal Q}\), the Gauss linear system, which generalizes the Jacobian system for hypersurfaces. The invariant \(\delta^ n\) is also used to prove a Torelli theorem for cubic hypersurfaces of dimension 3n. Ron Donagi [Compos. Math. (to appear)] has vastly generalized this theorem in recent work using the new technique of symmetrizers.

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F20 Étale and other Grothendieck topologies and (co)homologies
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14D20 Algebraic moduli problems, moduli of vector bundles
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