Jacobi complexes on the Ran space

TL;DR

This paper explores the local theory of moduli schemes through the Ran space framework, revisiting classical concepts like Jacobi complexes and deformation rings with rigorous algebraic methods.

Contribution

It provides a new rigorous algebraic treatment of Jacobi complexes, universal deformation rings, and Hitchin-type connections within the Ran space context.

Findings

Revised algebraic definitions of Jacobi complexes on the Ran space

New insights into universal deformation rings of moduli problems

Construction of Hitchin-type flat connections in an algebraic setting

Abstract

We study local theory of moduli schemes using the framework of the Ran space. With the help of the study of sheaves and complexes over the Ran space by Beilinson and Drinfeld in their theory of chiral algebras, we revisit Ran's works on the Jacobi complexes (the Chevalley complexes for sheaves of Lie algebras on the Ran space), the universal deformation rings of moduli problems, the higher Kodaira-Spencer maps, and construction of Hitchin-type flat connections. We give rigorous treatments in the algebraic setting, which seems to be new.