Over $\mathbb{A}^n$, it is easy to construct D-modules by writing down an explicit linear system of PDE's and then writing a presentation of the associated D-module $$ \mathcal{D}^n \xrightarrow{} \mathcal{D}^m \to \mathcal{M} \to 0 $$ Is there an analogous process for constructing D-modules over projective space as an explicit system of differential equations?
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You can construct $\mathcal{D}$ modules on projective space (or actually any space) by precisely the same process: write down a system of PDE's (using differential operators on the whole of $\mathbb{P}^n$) and consider the corresponding quotient. The key caveat is that on a general space, not every $\mathcal{D}$-module will be of this form: some of them can be defined using systems on different coordinate patches which are consistent, but not globally (I think a degree 0 line bundle on an elliptic curve should give a counterexample). However, $\mathbb{P}^n$ is special, and every $\mathcal{D}$-module can be defined this way (by the Beilinson-Bernstein theorem).
answered Jan 3, 2017 at 0:58
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$\begingroup$ Will $\mathcal{D}_{\mathbb{P}^n}(\mathbb{P}^n) = \mathbb{C}\left[\frac{\partial}{\partial x_0}, \ldots, \frac{\partial}{\partial x_n}\right]$? $\endgroup$ Commented Jan 3, 2017 at 1:42
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2$\begingroup$ @user251222 Those expressions don't even make sense as differential operators, since they have poles at $\infty$. The polynomial differential operators are generated by $x_i\frac{\partial}{\partial x_j}$. $\endgroup$– Ben Webster ♦Commented Jan 3, 2017 at 2:24
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$\begingroup$ Oh, why is that true? Do you have a reference? $\endgroup$ Commented Jan 3, 2017 at 4:38
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1$\begingroup$ Look at Section 11.3 of Hotta, Takeuchi and Tanisaki (math.columbia.edu/~scautis/dmodules/hottaetal.pdf). The essential point is that these vector fields (thought of as functions on $T^*\mathbb{P}^n$) generate all the polynomial functions on the cotangent bundle, since the induced map from $T^*\mathbb{P}^n$ to $(n+1)\times (n+1)$-matrices is a resolution of singularities of the rank $\leq 1$ matrices. $\endgroup$– Ben Webster ♦Commented Jan 3, 2017 at 13:33
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1$\begingroup$ Also take a look at "D-modules on Smooth Toric Varieties" (arxiv.org/abs/math/0007099) $\endgroup$ Commented Jan 17, 2017 at 15:30