Tautological rings on Jacobian varieties of curves with automorphisms

TL;DR

This paper introduces new tautological rings on Jacobian varieties of curves with automorphisms, extending Beauville's work, and explores tautological rings on related abelian subvarieties.

Contribution

It develops extended tautological rings for Jacobians with automorphisms and analyzes their induced structures on complementary abelian subvarieties.

Findings

New tautological rings on Jacobians with automorphisms

Existence of tautological rings on complementary abelian subvarieties

Extension of Beauville's tautological ring framework

Abstract

Let $J$ be the Jacobian of a smooth projective complex curve $C$ which admits non-trivial automorphisms, and let $A(J)$ be the ring of algebraic cycles on $J$ with rational coefficients modulo algebraic equivalence. We present new tautological rings in $A(J)$ which extend in a natural way the tautological ring studied by Beauville (Compos Math 140(3):683-688, 2004). We then show there exist tautological rings induced on special complementary abelian subvarieties of $J$.