Remarks on Collino cycles and hyperelliptic Johnson homomorphisms

TL;DR

This paper explores the relationship between Collino cycles on hyperelliptic Jacobians and the hyperelliptic Johnson homomorphism, using relative completion to construct cohomology classes and analyze their span.

Contribution

It recasts Colombo's explicit monodromy computation within the framework of relative completion and constructs new Collino classes for hyperelliptic mapping class groups.

Findings

Constructed Collino classes as cohomology classes with symplectic coefficients.

Determined the dimension of the span of these classes for level two hyperelliptic mapping class groups.

Recast Colombo's monodromy results in the context of relative completion.

Abstract

A Collino cycle is a higher cycle on the Jacobian of a hyperelliptic curve. The universal family of Collino cycles naturally gives rise to a normal function, whose induced monodromy relates to the hyperelliptic Johnson homomorphism. Colombo computed this monodromy explicitly and made this relation precise. We recast this in the perspective of relative completion. In particular, we use Colombo's result to construct Collino classes, which are cohomology classes of hyperelliptic mapping class groups with coefficients in a certain symplectic representation. We also determine the dimension of their span in the case of the level two hyperelliptic mapping class group.