##
**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.

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) |

### Keywords:

Hodge conjecture; Torelli problem; Bloch semiregularity; infinitesimal variation of Hodge structure
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\textit{P. Griffiths} and \textit{J. Harris}, Compos. Math. 50, 207--264 (1983; Zbl 0576.14008)

### References:

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