The Intermediate Jacobians of the Theta Divisors of Four-dimensional Principally Polarized Abelian Varieties

Let A be a principally polarized abelian variety of dimension four and let Θ ⊂ A be a symmetric theta-divisor, which we assume to be smooth. Using the Hodge structure on H(Θ) we associate to A two abelian subvarieties J(K) ⊂ J(H) of the intermediate jacobian J(Θ) of Θ of dimensions five and nine respectively. We show that J(H) is generated by the image under the Abel-Jacobi map of the family F of Prym-embedded curves in Θ and that there is a commutative diagram F −→ J(Θ) ↓ ↓ P−1(A) −→ J(Q) where J(Q) is the dual abelian variety of J(K), P : R5 −→ A4 is the Prym map, the two vertical arrows are onto and the image of P−1(A) generates J(Q).