Obstructions to smoothing orbicurves and Orbifold Hecke algebras

Abstract

In this paper, we study obstructions to smoothing nodal orbicurves with orbighosts mapped into cyclic quotient singularities. As an application, we show, under some assumptions, that the orbifold Hecke algebra of a complex global quotient orbifold $[X/G]$ is isomorphic to the degree $0$ cohomology of the $A_\infty$-algebra of endomorphisms of a regular cotangent fiber $T_{[x]}^*[X/G]$ regarded as an object of the bulk-deformed wrapped Fukaya category of the orbifold $T^*[X/G]$ for a compact complex manifold $X$ and finite group $G \subset \operatorname{Aut}(X)$.