Abel-Jacobi invariant and curves on generic abelian varieties. (English) Zbl 0837.14036

Barth, Wolf (ed.) et al., Abelian varieties. Proceedings of the international conference held in Egloffstein, Germany, October 3-8, 1993. Berlin: Walter de Gruyter. 237-249 (1995).
Let \(A\) be a generic complex abelian variety of dimension \(a\); the aim of this article is to prove that the geometric genus of a curve on \(A\) is greater than \(a (a - 1)/2\), and that the geometric genus of a curve on the Kummer variety \(K_A\) is at least \((a - 1) (a - 2)/2\).
This is done as follows. Let \(C\) be a smooth curve with a non-constant morphism \(f : C \to A\). We denote by \(C^+\) an Abel-Jacobi image of \(C\) in its Jacobian \(J(C)\), and by \(C^-\) the image of \(C^+\) by multiplication by \(-1\). The 1-cycle \(C^+ - C^-\) is homologically equivalent to 0, but it is often not algebraically equivalent to 0 (as shown by Ceresa). More precisely, if one considers the primitive intermediate Jacobian \(P(J(C))\) of \(J(C)\) (that is, the quotient of its intermediate Jacobian by a maximal abelian subvariety) and the Abel- Jacobi map \(AJ\), the element \(AJ(C^+ - C^-)\) of \(P(J(C))\) is usually non torsion. On the other hand, it was shown by Nori that for \(a\geq 4\), the Abel-Jacobi image of any homologically trivial 1-cycle in \(A\) is torsion in \(P(A)\). The idea is to use this contrast. If one considers a family \(F : {\mathcal C} \to {\mathcal A}\) of deformations of the map \(f\), over an open subset \(U\) of the Siegel space, the Abel-Jacobi maps globalize to define normal functions \(\nu_F\) and \(\nu\), attached to the cycles \(C^+_t - C^-_t\), which are sections of the families \({\mathcal P} (A) \to U\) and \({\mathcal P} (J(C)) \to U\) of primitive intermediate Jacobians. Let \(P^{p,q} (A)\) be the primitive Hodge spaces of \(A\); the tangent space to \(U\) is isomorphic to \(T = \text{Sym}^2 (H^{0,1} (A))\), and the Kodaira-Spencer map defines a map \(\beta : T \otimes P^{2,1} (A) \to P^{1,2}(A)\), whose kernel is denoted by \(K(A)\). The Griffiths infinitesimal invariant is a linear functional \(\delta \nu_F\) on \(K(A)\); we know from Nori’s theorem that it vanishes. It factorizes as \(K(A)\hookrightarrow K(J(C)) \to (\mathbb{C})\), where the second map is \(\delta \nu\). In other words, \(\delta \nu\) vanishes on \(K(A)\). Using a formula from the author’s joint paper with A. Collino in Duke Math. J. 78, No. 1, 59-88 (1995)], the author uses this fact to construct \((a - 1) (a - 2)/2\) independent elements in \(H^{1,0} (C)\) orthogonal to \(f^* H^{1,0} (A)\). This implies the bound \(g(C) \geq (a - 1) (a - 2)/2 + a\). A more refined analysis shows that \(g(C) = a(a - 1)/2 + 1\) can only happen when \(a = 4\) or 5 and \(C\) is a Prym curve, together with the statement about curves on generic Kummer varieties.
For the entire collection see [Zbl 0817.00023].


14K30 Picard schemes, higher Jacobians
14H40 Jacobians, Prym varieties
14C25 Algebraic cycles