Tautological classes with twisted coefficients

Abstract

Let $M_g$ be the moduli space of smooth genus $g$ curves. We define a notion of Chow groups of $M_g$ with coefficients in a representation of $Sp(2g)$, and we define a subgroup of tautological classes in these Chow groups with twisted coefficients. Studying the tautological groups of $M_g$ with twisted coefficients is equivalent to studying the tautological rings of all fibered powers $C_g^n$ of the universal curve $C_g\to M_g$ simultaneously. By taking the direct sum over all irreducible representations of the symplectic group in fixed genus, one obtains the structure of a twisted commutative algebra on the tautological classes. We obtain some structural results for this twisted commutative algebra, and we are able to calculate it explicitly when $g\le 4$. Thus we completely determine the tautological rings of all fibered powers of the universal curve over $M_g$ in these genera. We also give some applications to the Faber conjecture.

50 pages. v2, exposition clarified, Section 12 rewritten. Final version to appear in Annales de l'ENS

References in corpus (6)

• Three questions in Gromov-Witten theory
• A spectral sequence for stratified spaces and configuration spaces of points
• Representations of twisted Yangians associated with skew Young diagrams
• Relations in the tautological ring
• Relations on ${\overline{M}}_{g,n}$ via equivariant Gromov-Witten theory of ${\mathbb{P}}^1$
• A conjectural connection between R^*(C_g^n) and R^*(M_{g,n}^rt)