# Symbol map for sheaves of differential operators

For a complex manifold $X$, its sheaf of differential operators $\mathcal{D}_X$ is a sheaf of filtered algebras, and there is an isomorphism of sheaves of graded algebras

$$\text{gr } \mathcal{D}_X \xrightarrow{\sim}\mathcal{S}(\Omega_X)$$

(the right side is the symmetric algebra of the cotangent bundle). This is called the "symbol map".

One can also consider the sheaf of differential operators $\mathcal{E} \rightarrow \mathcal{F}$, for vector bundles $\mathcal{E}$ and $\mathcal{F}$. Is there a "symbol map" (which can be expressed in a way similar to the one above) in this situation too?

• What is your definition for the sheaf of differential operators between vector bundles (presumably over the same base)? – KReiser Aug 24 '18 at 6:30
• See, e.g., part 2 of the answer here: math.stackexchange.com/a/1089287/81996 – rj7k8 Aug 24 '18 at 16:32