Infinitesimal variations of Hodge structure. III: Determinantal varieties and the infinitesimal invariant of normal functions.

*(English)*Zbl 0576.14009[Part I - written by the reviewer, M. Green, the author and J. Harris - appeared ibid. 50, 109-205 (1983; Zbl 0531.14006); for part II see the preceding review.]

Let H be a polarized Hodge structure of and let \(G^{\bullet}\) be the graded object associated to the Hodge filtration. Because of the Riemann- Hodge bilinear relations, the polarization descends to a bilinear form \(<.,.>\) on \(G^{\bullet}\). Let E(H) be the graded algebra of endomorphisms of \(G^{\bullet}\) which are antisymmetric with respect to \(<.,.>\). An ”infinitesimal variation” of H is a homomorphism \(\delta\) of a vector space T into E(H) whose image is abelian in the natural graded Lie algebra structure of E(H). The goal of the theory of infinitesimal variations is to extract geometric information from the pair (H,\(\delta)\) which is not available (at all or readily) from H itself. The present paper attempts to do this by defining two new invariants of (H,\(\delta)\). To describe the first, consider the map \(\delta\) \({}^{n-2k}: Sym^{n- 2k}T\to Hom^{(s)}(H^{n-k,k},H^{k,n-k})\) defined by \(\delta^{n- 2k}(\xi_ 1,...,\xi_ n)(\omega)=\delta (\xi_{n-2k})...\delta (\xi_ 1)(\omega)\). Let \(\Xi_{k,r}\) be the subvariety of \({\mathbb{P}}(T)\) defined by the condition \(rank(\delta^{n-2k}(\xi))\leq r\). This is the (k,r)-th determinantal variety” associated to an infinitesimal variation. The author shows that for certain geometrically defined variations, that the determinantal varieties carry useful geometric information. For curves \(\Xi_{0,1}\) can be identified with the bicanonical image using the notion of a Shiffer variation, thereby giving another proof of the Torelli theorem. For higher dimensional varieties \(\Xi_{0,1}\) is merely identified with a subvariety of the bicanonical image; however, it is unknown whether this subvariety is nonempty. A sharper result is given for \(\Xi_{0,g-1}\), where \(g=\dim H^{n,0}:\) this determinantal variety is identified with a certain multisecant variety of the bicanonical image, provided that the canonical series has no base points.

The second invariant is defined in terms of a normal function \(\nu\) : it is a section \(\delta\) \(\nu\) of a canonically defined bundle over a variety \(\Sigma_ r\) which fibers over the determinantal variety \(\Xi_{m,r}\) considered above, where \(2m+1\) is the weight of the Hodge structure in question. When \(\nu\) is constant, e.g., is the normal function of a primitive algebraic cycle, \(\delta\) \(\nu\) vanishes. Several applications of this invariant to curves are given, and one two- dimensional example is worked out: let \(F\subset {\mathbb{P}}^ 3\) be a quadric surface, let \(\lambda\) be the cohomology class of difference of rulings \(L_ 1-L_ 2\), and let \(\nu\) be the normal function defined over the parameter space of cubic hyperplane sections of F. Then the cycle \(L_ 1-L_ 2\) is constructed from the infinitesimal invariant \(\delta\) \(\nu\). The formalism for constructing cycles from the infinitesimal invariant exists in general; the question is whether or not the geometric objects which it defines are the right ones.

Let H be a polarized Hodge structure of and let \(G^{\bullet}\) be the graded object associated to the Hodge filtration. Because of the Riemann- Hodge bilinear relations, the polarization descends to a bilinear form \(<.,.>\) on \(G^{\bullet}\). Let E(H) be the graded algebra of endomorphisms of \(G^{\bullet}\) which are antisymmetric with respect to \(<.,.>\). An ”infinitesimal variation” of H is a homomorphism \(\delta\) of a vector space T into E(H) whose image is abelian in the natural graded Lie algebra structure of E(H). The goal of the theory of infinitesimal variations is to extract geometric information from the pair (H,\(\delta)\) which is not available (at all or readily) from H itself. The present paper attempts to do this by defining two new invariants of (H,\(\delta)\). To describe the first, consider the map \(\delta\) \({}^{n-2k}: Sym^{n- 2k}T\to Hom^{(s)}(H^{n-k,k},H^{k,n-k})\) defined by \(\delta^{n- 2k}(\xi_ 1,...,\xi_ n)(\omega)=\delta (\xi_{n-2k})...\delta (\xi_ 1)(\omega)\). Let \(\Xi_{k,r}\) be the subvariety of \({\mathbb{P}}(T)\) defined by the condition \(rank(\delta^{n-2k}(\xi))\leq r\). This is the (k,r)-th determinantal variety” associated to an infinitesimal variation. The author shows that for certain geometrically defined variations, that the determinantal varieties carry useful geometric information. For curves \(\Xi_{0,1}\) can be identified with the bicanonical image using the notion of a Shiffer variation, thereby giving another proof of the Torelli theorem. For higher dimensional varieties \(\Xi_{0,1}\) is merely identified with a subvariety of the bicanonical image; however, it is unknown whether this subvariety is nonempty. A sharper result is given for \(\Xi_{0,g-1}\), where \(g=\dim H^{n,0}:\) this determinantal variety is identified with a certain multisecant variety of the bicanonical image, provided that the canonical series has no base points.

The second invariant is defined in terms of a normal function \(\nu\) : it is a section \(\delta\) \(\nu\) of a canonically defined bundle over a variety \(\Sigma_ r\) which fibers over the determinantal variety \(\Xi_{m,r}\) considered above, where \(2m+1\) is the weight of the Hodge structure in question. When \(\nu\) is constant, e.g., is the normal function of a primitive algebraic cycle, \(\delta\) \(\nu\) vanishes. Several applications of this invariant to curves are given, and one two- dimensional example is worked out: let \(F\subset {\mathbb{P}}^ 3\) be a quadric surface, let \(\lambda\) be the cohomology class of difference of rulings \(L_ 1-L_ 2\), and let \(\nu\) be the normal function defined over the parameter space of cubic hyperplane sections of F. Then the cycle \(L_ 1-L_ 2\) is constructed from the infinitesimal invariant \(\delta\) \(\nu\). The formalism for constructing cycles from the infinitesimal invariant exists in general; the question is whether or not the geometric objects which it defines are the right ones.

Reviewer: J.A.Carlson

##### MSC:

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

14M12 | Determinantal varieties |

##### Keywords:

Hodge conjecture; infinitesimal variation of; polarized Hodge structure; determinantal variety; Torelli theorem; normal function of a primitive algebraic cycle##### References:

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