## Infinitesimal variations of Hodge structure. II: An infinitesimal invariant of Hodge classes.(English)Zbl 0576.14008

[Part I - written by the reviewe, M. Green and the authors - appeared ibid. 50, 109-205 (1983; Zbl 0531.14006); see also the following review.]
The authors study the locus in the moduli space of a projective variety $$X_ 0$$ of complex dimension 2p for which a fixed integral class $$\gamma$$ of type (p,p) remains of type (p,p). The principal tool is an invariant defined in terms of the infinitesimal variation of Hodge structure of $$X_ 0$$, to be described below. Numerous applications of the invariant are given, among which are the following: (a) a Hodge theoretic criterion for a homology class on a projective hypersurface to be the class of a linear subspace; (b) a numerical criterion for a Hodge class on a hypersurface of dimension 2 to be effective; (c) a result of Torelli type: the Fermat surface of degree d can be reconstructed from the associated infinitesimal variation, from which it follows that the period map for surfaces of degree d is of degree one onto its image; (d) a criterion for smooth curves on surfaces in $${\mathbb{P}}^ 3$$ to have indecomposable normal bundle [see also K. Hulek, Math. Ann. 258, 201-206 (1981; Zbl 0458.14011); (e) a special case of the infinitesimal form of the Hodge conjecture for hypersurfaces in $${\mathbb{P}}^ 5$$. This last result, which may be viewed as a criterion for certain Hodge cycles to be semiregular in the sense of Bloch, has recently been generalized by J. H. M. Steenbrink [”Some remarks about the Hodge conjecture” (preprint, Leiden)].
The infinitesimal invariant is a space $$H^{p+1,p-1}(-\gamma)$$ defined as follows. Let $$T=H^ 1(X_ 0,\Theta)$$ be the infinitesimal deformation space of $$X_ 0$$, and for $$\xi\in T$$, consider the bilinear pairing $$H^{p+1,p-1}\times T\to {\mathbb{C}}$$ given by $$(\psi,\xi)\mapsto <\delta (\xi)\psi,\gamma >$$, where $$\delta$$ is the Kodaira-Spencer map, and where the pairing is cup-product. Then $$H^{p+1,p-1}(-\gamma)$$ is the left kernel of this pairing. If $$T_{\gamma}$$ is the right kernel and $$T^{\perp}_{\gamma}$$ is the annihilator in the dual space $$T_{\gamma}$$, then there is a natural isomorphism $$H^{p+1,p- 1}/H^{p+1,p-1}(-\gamma)\cong T^{\perp}_{\gamma}$$. The space $$T^{\perp}_{\gamma}$$ may therefore be viewed as the conormal space to the locus $$U_{\gamma}$$ in a polydisk neighborhood of $$X_ 0$$ in the moduli space where $$\gamma$$ remains of type (p,p). If $$\gamma$$ is the class of an effective algebraic cycle $$\Gamma$$ of codimension p, then one can define the space $$H^{p-1}(\Omega^{p+1}(-\Gamma))$$ to be the image of $$H^{p-1}(\Omega^{p+1}\otimes I_{\Gamma})$$ in $$H^{p+1,p-1}$$, where $$I_{\Gamma}$$ is the ideal of the support of $$\Gamma$$. Then $$H^{p+1,p-1}(-\gamma)$$ contains $$H^{p-1}(\Omega^{p+1}(-\Gamma))$$. Equality holds when all infinitesimal deformations of $$\gamma$$ as a Hodge class arise from deformations of $$\Gamma$$ as an effective algebraic cycle, hence the connection with the variational version of the Hodge conjecture. Cycles for which equality holds are semiregular in the sense of Bloch.
Reviewer: J.A.Carlson

### MSC:

 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)

### Citations:

Zbl 0576.14009; Zbl 0471.14013; Zbl 0531.14006; Zbl 0458.14011
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### References:

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