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].

MSC:

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