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Jacobi Cohomology, Local Geometry of Moduli Spaces, and Hitchin Connections
Author(s) -
Ran Ziv
Publication year - 2006
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1017/s0024611505015704
Subject(s) - mathematics , moduli space , moduli of algebraic curves , cohomology , geometric invariant theory , pure mathematics , moduli , modular equation , differential geometry , algebraic geometry , connection (principal bundle) , geometry , geometry and topology , algebra over a field , mathematical analysis , differential equation , ordinary differential equation , differential algebraic equation , physics , quantum mechanics
We develop some cohomological tools for the study of the local geometry of moduli and parameter spaces in complex Algebraic Geometry. Notably, we develop canonical formulae for the differential operators 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 the Lie bracket of vector fields on a moduli space. As an application, we obtain another construction and proof of flatness for the familiar KZW or Hitchin connection on moduli spaces of curves. 2000 Mathematics Subject Classification 14D15, 32G05.