Topological recursive relations in $H^{2g}(M_{g,n})$

TL;DR

This paper proves a vanishing result for high-degree polynomials in tautological classes on moduli spaces of curves, generalizing previous results and confirming a version of Getzler's conjecture using Gromov-Witten invariants.

Contribution

It establishes a new vanishing theorem for tautological classes in high degree, extending prior work and providing a novel proof approach via relative Gromov-Witten invariants.

Findings

Degree at least g polynomial in tautological classes vanishes on M_{g,n} for g ≥ 2

Generalizes Looijenga's result and proves a version of Getzler's conjecture

Includes a quick proof of a recent conjecture by Vakil

Abstract

We show that any degree at least $g$ polynomial in descendant or tautological classes vanishes on $M_{g,n}$ when $g\ge 2$. This generalizes a result of Looijenga and proves a version of Getzler's conjecture. The method we use is the study of the relative Gromov-Witten invariants of $P^1$ relative 2 points combined with the degeneration formulas of [IP1]. At the end of the paper, we also included a quick proof of a very recent conjecture made by Vakil.