Jacobi Cohomology, Local Geometry of Moduli Spaces, and Hitchin Connections

We develop some cohomological tools for the study of the localgeometry of moduli and parameter spaces in complex AlgebraicGeometry. Notably, we develop canonical formulae for the differentialoperators of arbitrary order and their natural action on suitable‘natural’ modules (for example, functions); in particular,we obtain a formula, in terms of the moduli problem, for theLie bracket of vector fields on a moduli space. As an application,we obtain another construction and proof of flatness for thefamiliar KZW or Hitchin connection on moduli spaces of curves.2000 Mathematics Subject Classification 14D15, 32G05.