Problem set 2, due October 13.

(1) In this problem we work over the base field ℂ. The algebra D(𝔸^{1}) of polynomial differential operators on 𝔸^{1} acts on the space C = C_{c}^{∞}(ℝ) of smooth compactly supported functions of one variable. Let Dist = C^{∗} denote the dual space, it is a right D(𝔸^{1})-module.

For each of the following distributions σ ∈ Dist show that the submodule generated by σ is of finite length. Describe the Jordan-Hoelder series of that D-module and decomposition as a sum of indecomposable modules.

(a) σ(f) = ∫_{ℝ} |x|f(x)dx.

(b) σ(f) = P V ∫_{ℝ} \frac{f(x)}{x}dx where P V ∫ is the principal value integral lim_{ϵ→0}(∫_{−∞}^{−ϵ} + ∫_{ϵ}^{∞}).

(c) σ(f) = ∫_{ℝ} \frac{f(x)}{|x|}dx when f(0) = 0. (For such f the integral converges). σ(const) = 0.

(d) σ(f) = ∫_{ℝ} log(|x|)f(x)dx.

(e) σ(f) = ∫_{ℝ} log^{2}(|x|)f(x)dx.

(2) Show that Fourier transform interchanges the D-modules from parts (c)

and (d) of the previous problem.

Find an example of a distribution that generates a D module isomorphic to Fourier transform of the D-module from part (e) of that problem

(3) Let j : 𝔸^{1} \ 0 → 𝔸^{1}.

(a) Compute the singular support of j_{∗}(O) equipped with the tautological connection.

(b) Let M be the D-module exp(1/z) on 𝔸^{1} \ {0} (thus M = (O, ∇ = d − \frac{dz}{z^{2}} ). Show that the singular support of j_{∗}(M) is the union of the zero section taken with multiplicity one and the fiber at zero taken with multiplicity two. Deduce that the singular support of a holonomic D-module does not determine the singular support of its ∗ direct image under an affine open embedding.

(4) Let X be an affine smooth variety. Consider the regular bi-module B ^{1} for D(X). Show that B⊗_{O(X)}Ω(X)^{−1} is naturally a D(X)⊗D(X) = D(X×X) module, i.e. a D-module over X^{2}.

What is the support of this D-module? Describe the D-module geometrically.

(5) Let E be a vector bundle on a smooth algebraic variety X and E^{∗} be the dual vector bundle. Generalize Fourier transform to an equivalence of categories between D-modules on the total spaces of E and E^{∗}.

Show that this equivalence comes from an isomorphism of sheaves of rings over X provided that the line bundle Λ^{top}(E) on X is a trivial.

(Hints provided upon request).

1Recall that a bimodule for a k-algebra A is a module over A ⊗_{k} A^{op}. A regular bimodule is identified with A as a vector space, and the action of A ⊗ A^{op} is given by a_{1} ⊗ a_{2} : x ↦ a_{1}xa_{2}.