Herbert Clemens

An Analogue of Abel’s Theorem

This work makes a parallel construction for curves on threefolds to a “current-theoretic” proof of Abel’s theorem giving the rational equivalence of divisors P and Q on a Riemann surface when Q - P is (equivalent to) zero in the Jacobian variety of the Riemann surface. The parallel construction is made for homologous “sub-canonical” curves P and Q on a general class of threefolds. If P and Q are algebraically equivalent and Q - P is zero in the (intermediate) Jacobian of a threefold, the construction “almost” gives rational equivalence.