Deformation of a smooth Deligne-Mumford stack via differential graded Lie algebra

TL;DR

This paper establishes a correspondence between the deformation theory of smooth Deligne-Mumford stacks over complex numbers and their associated Kodaira-Spencer differential graded Lie algebras, providing a new algebraic perspective.

Contribution

It introduces the Kodaira-Spencer differential graded Lie algebra for smooth Deligne-Mumford stacks and proves the isomorphism of their deformation functors in the proper case.

Findings

Deformation functor of the stack is isomorphic to that of the Kodaira-Spencer algebra.

Defines the Kodaira-Spencer differential graded Lie algebra for stacks.

Establishes the isomorphism in the proper case.

Abstract

For a smooth Deligne-Mumford stack over $\CC$, we define its associated Kodaira-Spencer differential graded Lie algebra and show that the deformation functor of the stack is isomorphic to the deformation functor of the Kodaira-Spencer algebra if the stack is proper over $\CC$.