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If you're taking twisted differential operators in a complex power of a line bundle, $L^c$, then you should think of them as vector bundles/sheaves on the total space of L minus its zero section, endowed with a connection that behaves specially along the fibers of the bundle projection map.

Special how? The action of $\mathbb C^*$ by fiber rotation has a differential, which is a vector field on the total space that looks like $t\frac{d}{dt}$ for any trivialization, where $t$ is the coordinate on the fiber. One should take a connection where differentiating along this vector field multiplies by c.

This follows immediately from the fact that the ring of twisted differential operators in $L^c$ are exactly $\mathbb{C}^*$-invariant differential operators on the total space minus its zero section modulo $t\frac{d}{dt}-c$.

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If you're taking twisted differential operators in a complex power of a line bundle, $L^c$, then you should think of them as vector bundles/sheaves on the total space of L minus its zero section, endowed with a connection that behaves specially along the fibers of the bundle projection map.

Special how? The action of $\mathbb C^*$ by fiber rotation has a differential, which is a vector field on the total space that looks like $t\frac{d}{dt}$ for any trivialization, where $t$ is the coordinate on the fiber. One should take a connection where differentiating along this vector field multiplies by c.

This follows immediately from the fact that the ring of twisted differential operators in $L^c$ are exactly differential operators on the total space minus its zero section modulo $t\frac{d}{dt}-c$.