Prym Differentials Research Articles

An analogue of the rauch variational formula for prym differentials

In [5], H. E. Rauch discovered a formula for the first variation of an abelian differential on a Riemann surface and its periods with respect to the change of complex structure induced by a Beltrami differential. R. S. Hamilton, in [3], and discussed by C. Earle in [1], found an elegant proof of the formula using only first principles and not requiring uniformization theory. His proof uses a small amount of Hodge theory, the Riemann bilinear period relations, and a simple operator construction. In this article, we find an analogue of Rauch’s formula for the Prym differentials using some of Hamilton’s techniques, the Hodge theorem for vector bundles, and the “Prym version” of the Riemann bilinear relations. We discover a complicated set of formulas for the variation of the Prym differentials, with different specific solutions depending to the make-up of the Prym character. We conclude that the variation of the Prym periods with a given character depends on the differentials for the character and the differentials for its inverse. This explains the simplicity of the classical case, where the character is its inverse.

On the period classes of Prym differentials. I.

On the period classes of Prym differentials. I.

A property of the periods of a Prym differential

The periods of Prym differentials can be used to prove the invariance of Picard bundles on Jacobian varieties.

A Property of the Periods of a Prym Differential

The periods of Prym differentials can be used to prove the invariance of Picard bundles on Jacobian varieties.