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 for in the Jacobian of a nonsingular algebraic curve . We define harmonic volume, determine its domain, and show that it is related to the image of 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 -fold symmetric product of . We show that , when applied to a certain class of forms, takes values in a discrete subgroup of and hence, when suitably extended to complexvalued forms, is identically zero modulo periods on primitive forms if . This implies that the image of is identically zero in the Griffiths intermediate Jacobian if . 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 and harmonic volume to compute the variation of when considered as a section of this bundle. This variational formula allows us to show that the image of in this intermediate Jacobian is nondegenerate.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1075380-8

Keywords:
Harmonic volume,
algebraic equivalence

Article copyright:
© Copyright 1993
American Mathematical Society