# Splitting of higher order jet sequence

Let $$X$$ be a smooth variety. Because $$\mathcal{O}_X$$ admits a canonical connection $$\mathrm{d} : \mathcal{O}_X \to \Omega_X$$ the sequence, $$0 \to \Omega_X \to J^1(\mathcal{O}_X) \to \mathcal{O}_X \to 0$$ splits canonically.

I say that $$X$$ has the jet splitting property at level $$n$$ if the sequence, $$0 \to \mathrm{Sym}^n(\Omega_X) \to J^{n}(\mathcal{O}_X) \to J^{n-1}(\mathcal{O}_X) \to 0$$ splits (we could say $$\mathcal{O}_X$$ admits a $$n^{\mathrm{th}}$$-order connection in this case).

Question. What is known about varieties with jet splitting at level $$n > 1$$?

On the affine space, these sequences are always split. For a complete curve, I do not believe there can ever be splitting at any level $$n > 1$$ unless $$g = 1$$.

• @DonuArapura edited Feb 27, 2021 at 18:28
• @BenC You have two occurrences of the same bundle in your second sequence; I think the middle one is supposed to be $J^n$? Feb 27, 2021 at 19:10
• Thank you for the correction Feb 27, 2021 at 19:12

Look at the paper P. Jahnke and I. Radloff, Splitting jet sequences. They classify such splittings on compact Kaehler manifolds. Those which admit a vector bundle with splitting jet sequence are precisely projective spaces, compact complex manifolds covered by a complex torus, and those compact Kaehler manifolds whose universal covering space is the unit ball in complex Euclidean space. In particular, every complete curve has such a sequence. On the other hand, for splitting of jet sequences where the vector bundle is $$\mathcal{O}$$, the compact Kaehler manifold is covered by a complex torus. In particular, the only complete curves with such splittings are genus one curves.