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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Harmonic volume, symmetric products, and the Abel-Jacobi map

Author: William M. Faucette
Journal: Trans. Amer. Math. Soc. 335 (1993), 303-327
MSC: Primary 14H40; Secondary 14C34, 14K20, 32G20
MathSciNet review: 1075380
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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.

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Keywords: Harmonic volume, algebraic equivalence
Article copyright: © Copyright 1993 American Mathematical Society