Problem set 1, due September 24

(1) Let V be a finite dimensional vector space.

a) Check that Fourier transform gives an isomorphism D(V) ≅ D(V^{∗}) where V^{∗} is the dual space.

b) Let W ⊂ V a vector subspace and U ⊂ V be a complement to W. Consider the D-module M = O(W) ⊠ δ_{0}(U). Show that M depends

only on W, not on U. Describe the D-module on V^{∗} which is the Fourier transform of M.

(2) In this problem we work over a field of characteristic zero, and D(X) stands

for the ring of Grothendieck differential operators on X.

a) Let X be given by equations xy = 0 in 𝔸^{2}. What is D(X)? What is grD(X)? Is it noetherian?

b)^{∗} Do the same for the quadratic cone xy = z^{2}.

c)^{∗∗} Let X be the cubic cone x^{3} + y^{3} + z^{3} = 0. Show that D(X) is not noetherian.

(3) Consider the (left) D-module δ_{x} on a smooth variety X. Show that any vector field v vanishing at x multiplies the canonical generator of δ_{x} by a scalar. For X = 𝔸^{n}, x = 0, v = ∑ a_{I,J}x^{J}∂^{J} give an explicit formula for that scalar.

(4) (For those familiar with basic homological algebra). In this problem we consider D-modules on 𝔸^{1} with coordinate x. For every a ∈ A^{1} we let δ_{a} denote the corresponding D-module of δ-functions.

Let O = k[x] with the standard action of differential operators. Show that Ext^{1}(δ_{a}, O) and Ext^{1}(O, δ_{a}) (in the category of D-modules) are both isomorphic to k (hint: construct explicitly the correspondind extensions 0 → O → M → δ_{a} → 0, 0 → δ_{a} → M → O → 0).

(5) Let G denote the symplectic group Sp(2n). Recall that G acts on D = D(𝔸^{n}) by automorphisms. If M is a D-module and g ∈ G we denote by M^{g} the same module with the D-action twisted by g. Let M be a (arithmetically) holonomic D-module on 𝔸^{n}.

Let c = c(M) (notations of the lecture of 9/10), i.e. the Hilbert polynomial for M has the form h_{M}(t) = \frac{c}{n!}t^{n}+ lower order terms.

a) Show that for a generic g ∈ G the module M^{g} is O-coherent of rank c (hint: think what the transformation M ↦ M^{g} does to the arithmetic singular support of M and then formulate the O-coherence condition in terms of the arithmetic singular support).

Note that this gives another definition of c.

b) Explain what a) says explictly for M = δ_{a} where a ∈ k^{n}, i.e. determine which g are generic.

c) Assume that the base field is k = ℂ. Let S(ℝ^{n}) denote the Schwartz space of ℝ^{n}, i.e. the space of functions all of whose partial derivatives are rapidly decreasing at ∞. Let S^{∗}(ℝ^{n}) denote the appropriate topological dual space. It has a natural structure of D-module. This is the space of tempered distributions on ℝ^{n}. The Stone-von Neumann theorem says (in particular) that (S^{∗}(ℝ^{n}))^{g} is isomorphic to S^{∗}(ℝ^{n}) for every g ∈ Sp(2n, ℝ) (the corresponding isomorphisms were explictly written by Weil). Show that for every holonomic D-module M we have

dim Hom_{D}(M, S^{∗}(ℝ^{n})) ≤ c(M).

In other words, the space of solutions of M in the space of tempered distributions has dimension ≤ c(M) (hint: use a).