Resolution for Sheaf of Differential Operators on Smooth Free Geometric Quotient of Linear Action of Algebraic Group

In [R], Z. Ran gave a canonical construction for the universal deformation of a simple vector bundle using the Jacobi complex of an appropriate differential graded Lie algebra. Independently, H. Esnault and E. Viehweg made a similar construction. Using these tools, we obtain a resolution for the sheaf of differential operators on smooth geometric quotients of free linear actions of algebraic groups. Unlike previous applications of these methods, we are able to obtain global results. The terms of our resolution involve symmetric and alternating powers of vector bundles easily constructed geometrically from the algebraic group and the vector space on which it acts. Our resolution is particularly simple for projective space. For P(V ), we obtain the following sequence—