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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 $g = 1$.

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

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