# What techniques are available for constructing D-modules over smooth projective varieties?

I'm trying to learn about D-modules for computing intersection cohomology but I'm having trouble coming up with explicit constructions of D-modules on projective varieties. Since this is an involved process, I'll split this up into multiple questions:

1. How do I construct D-modules over complex projective space
2. How do I construct D-modules over smooth projective varieties
3. How can I use the constructions from (1) to find D-modules with
geometric support on singular varieties?


My goal is to start looking at D-modules on Fermat curves $$\text{Proj}\left( \frac{\mathbb{C}[x,y,z]}{x^n + y^n - z^n} \right)$$ and answer the question I raised here

• For smooth toric varieties (e.g. complex affine and projective space), you might want to take a look at sciencedirect.com/science/article/pii/S0021869301987319 Jul 25 '16 at 2:40
• If $M$ is a $D$-module (e.g. $M = R$ a ring) then the local cohomology modules $H_I^j(M)$ are $D_R$-modules. This should give you some examples I guess. Jul 25 '16 at 6:41
• You might find Vilonen's thesis useful, available here: gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002101815 Jul 25 '16 at 18:48
• @AxelStäbler Do you know of any nice introductions to local cohomology accessible to undergraduates? Aug 3 '16 at 17:13
• Eisenbud's Geometry of Syzygies has a short appendix on local cohomology that may give you a good feel for what's going on. If you know some basic dimension theory then 24 hours of local cohomology'' is a decent source if you are willing to take certain things for granted or delve into the literature to solve some of the harder exercises. In fact, I ran a seminar based on this not too long ago with students that had taken a second course in algebra. The material covered in Bruns&Herzog's Cohen-Macaulay rings is also very well presented (but they do not cover as much as other sources). Aug 8 '16 at 13:28