Mathematics Department, Brown University, Providence, Rhode Island 02912.
Proc Natl Acad Sci U S A. 1983 Feb;80(4):1157-8. doi: 10.1073/pnas.80.4.1157.
Let Z be an algebraic p cycle homologous to zero in an algebraic complex manifold V. Associated to Z is a linear function nu on holomorphic (2p + 1)-forms on V, modulo periods, that vanishes if Z is algebraically equivalent to zero in V. I give a formula for nu for the case of V the jacobian of an algebraic curve C and Z=C - C' (C' = "inverse" of C') in terms of iterated integrals of holomorphic 1-forms on C. If C is the degree 4 Fermat curve, I use this formula to show that C - C' is not algebraically equivalent to zero.
令 Z 为代数 p 环面，在代数复流形 V 中与零同调。与 Z 相关联的是 V 上全纯（2p + 1）-形式的线性函数 nu，模周期，如果 Z 在 V 中与零代数等价，则该函数为零。我给出了 V 为代数曲线 C 的雅可比（Jacobian）且 Z=C - C'（C' = "C 的逆"）的情况下，nu 的公式，该公式基于 C 上全纯 1-形式的迭代积分。如果 C 是四次费马曲线，我将使用此公式证明 C - C' 与零不代数等价。
