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Homological versus algebraic equivalence in a jacobian
1Mathematics Department, Brown University, Providence, Rhode Island 02912.
Summary
This study introduces a method to detect if an algebraic cycle is algebraically equivalent to zero. For the Fermat curve, it proves a specific cycle is not algebraically equivalent to zero, advancing algebraic geometry research.
Area of Science:
- Algebraic Geometry
- Complex Manifolds
- Number Theory
Background:
- An algebraic cycle Z homologous to zero in a complex manifold V is associated with a linear function nu.
- This function nu vanishes if Z is algebraically equivalent to zero in V.
Purpose of the Study:
- To provide a formula for the linear function nu.
- To apply this formula to a specific case involving the Fermat curve to determine algebraic equivalence.
Main Methods:
- The study derives a formula for nu on the Jacobian of an algebraic curve C.
- The formula expresses nu in terms of iterated integrals of holomorphic 1-forms on C.
- The method is applied to the degree 4 Fermat curve.
Main Results:
- A formula for nu is established for the Jacobian of an algebraic curve C, specifically for the cycle C - C'.
- It is demonstrated that for the degree 4 Fermat curve, the cycle C - C' is not algebraically equivalent to zero.
Conclusions:
- The derived formula for nu is effective in distinguishing algebraic cycles.
- The result for the Fermat curve provides a concrete example of a cycle not algebraically equivalent to zero, contributing to the understanding of algebraic equivalence in complex manifolds.
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