Harmonic Volume, Symmetric Products, and the Abel-Jacobi Map

Abstract

The author generalizes B. Harris’ definition of harmonic volume to the algebraic cycle ${W_k} - W_k^-$ for $k > 1$ in the Jacobian of a nonsingular algebraic curve $X$ . We define harmonic volume, determine its domain, and show that it is related to the image $\nu$ of ${W_k} - W_k^-$ in the Griffiths intermediate Jacobian. We derive a formula expressing harmonic volume as a sum of integrals over a nested sequence of submanifolds of the $k$-fold symmetric product of $X$ . We show that $\nu$ , when applied to a certain class of forms, takes values in a discrete subgroup of ${\mathbf {R}}/{\mathbf {Z}}$ and hence, when suitably extended to complexvalued forms, is identically zero modulo periods on primitive forms if $k \geq 2$. This implies that the image of ${W_k} - W_k^-$ is identically zero in the Griffiths intermediate Jacobian if $k \geq 2$. We introduce a new type of intermediate Jacobian which, like the Griffiths intermediate Jacobian, varies holomorphically with moduli, and we consider a holomorphic torus bundle on Torelli space with this fiber. We use the relationship mentioned above between $\nu$ and harmonic volume to compute the variation of $\nu$ when considered as a section of this bundle. This variational formula allows us to show that the image of ${W_k} - W_k^-$ in this intermediate Jacobian is nondegenerate.