A remark on the jet bundles over the projective line

Abstract

This is a footnote of a recent interesting work of Cohen, Manin and Zagier, where they, among other things, produce a natural isomorphism between the sheaf of (n-1)-th order jets of the n-th tensor power of the tangent bundle of a Riemann surface equipped with a projective structure and the sheaf of differential operators of order n (on the trivial bundle) with vanishing 0-th order part. We give a different proof of this result without using the coordinates, and following the idea of this proof we prove: Take a line bundle L with $L^2 = T$ on a Riemann surface equipped with a projective structure. Then the jet bundle $J^n(L^n)$ has a natural flat connection with $J^n(L^n) = S^n(J^1(L))$. For any $m >n$ the obvious surjection $J^m(L^n) \rightarrow J^n(L^n)$ has a canonical splitting. In particular, taking $m = n+1$, one gets a natural differential operator of order $n+1$ from $L^n$ to $L^{-n-2}$.