Noncommutative rigidity of the moduli stack of stable pointed curves

TL;DR

The authors address the noncommutative deformation problem for the moduli stack of stable pointed curves by proving that $HH^2(ar{rak M}_{g,n})$ vanishes for almost all $(g,n)$ in characteristic zero, i.e., noncommutative deformations are obstructed except in a finite list of exceptional cases. The key strategy combines the Hochschild–Kostant–Rosenberg decomposition for DM stacks with a logarithmic Kodaira–Katzik–Nakano vanishing theorem on normal crossing DM pairs, plus a detailed classification of twisted sectors in the inertia stack up to codimension two. They show $HH^2$ vanishes in all non-exceptional cases, while identifying eight exceptional pairs $(g,n)$ where obstructions may persist, and for genus zero with five marked points a six-dimensional family of noncommutative deformations exists. The results connect noncommutative rigidity with the geometry of twisted sectors and the positivity properties of psi-classes and the log-canonical divisor, suggesting that noncommutative deformations of curves are largely governed by the underlying modular geometry. The paper thus provides a near-complete rigidity statement for the moduli of stable pointed curves in characteristic zero and clarifies the precise loci where noncommutative deformations may occur.

Abstract

We prove that the second Hochschild cohomology group of the moduli stack of stable $n$-pointed genus $g$ curves vanishes for all but finitely many $(g,n)$.

Theorems & Definitions (76)

• Theorem 1.1: Main Theorem
• Remark 1
• Theorem 1.2: $=$\ref{['theorem:logarithmic_KAN']} in characteristic $0$
• Theorem 1.3
• Theorem 1.4
• Lemma 1
• proof
• Proposition 1
• proof
• Lemma 2