Curves in Jacobians to Non - Jacobians Ii

This is a second paper where we introduce deformation theory methods which can be applied to finding curves in families of principally polarized abelian varieties (ppav) containing jacobians. One of our motivations for finding interesting and computationally tractable curves in ppav is to solve the Hodge conjecture for the primitive cohomology of the theta divisor which we explain below. For other motivations we refer to the prequel [5] to this paper. Let (A,Θ) be a ppav over C of dimension g ≥ 4 such that Θ is smooth. Since any abelian variety (over C) is isogenous to such an abelian variety, the Hodge conjectures for arbitrary abelian varieties are equivalent to the Hodge conjectures for principally polarized abelian varieties with smooth theta divisors. The primitive part K(Θ,Q) of the cohomology of Θ can be defined as the kernel of the map H(Θ,Q) −→ H(A,Q) obtained by Poincaré Duality from push-forward on homology. The space K(Θ,Q) defines a Hodge substructure of the cohomology of Θ of level g − 3 (see page 562 of [8]; the proof there works also for g > 4). The generalized Hodge conjecture then predicts that there is a family of curves in Θ such that K(Θ,Q) is contained in the image of its Abel-Jacobi map. The Abel-Jacobi map for a family of curves can be defined as follows. Let C → S be a family of curves with S smooth, complete and irreducible of dimension d such that there is a diagram C q −→ Θ p ↓ S .