Problem set 7, due November 9.

Let j : 𝔸^{1} \ 0 → 𝔸^{1} and i : {0} → 𝔸^{1} be the embeddings. Let E_{n} be the local system on 𝔸^{1} \ 0 given by E_{n} = (O^{n}, d + N \frac{dz}{z}), where N is the n × n matrix N_{ij} = δ_{i,j−1}.

(1) Show that every indecomposable D-module on 𝔸^{1} \0, which has a filtration whose associated graded is isomorphic to a sum of several copies of O, is isomorphic to E_{n} for some n.

(2) Describe the Jordan-Hoelder series of j_{!∗}(E_{n}).

[Hint: its length is 2n − 1.]

(3) Let Ξ_{n} be the co-kernel of a non-zero map O_{𝔸^{1}} → j_{∗}(E_{n+1}).

Show that i^{∗}(Ξ_{n}), i^{!}(Ξ_{n}) are concentrated in homological degree zero. Find the dimensions of these vector spaces.

(4) Show that Ξ_{n} is the only extension of E_{n} to 𝔸^{1}, such that i^{∗}(Ξ_{n}), i^{!}(Ξ_{n})

are concentrated in homological degree zero.^{1}

Deduce that Ξ_{n} is self-dual.

(5) Show that Fourier transform sends Ξ_{n} to j_{!∗}(E_{n+1}).

(6) Let M be the D-module exp(1/z) on 𝔸^{1} (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.

(7) Recall the sheaf of twisted differential operators D_{h}, h ∈ H^{1}(X, Ω_{cl}^{1}) defined in problem 1, set 5. Let Ω_{X} be the line bundle of top degree forms, and set ω_{X} = \frac{1}{2}c_{1}(Ω_{X}) ∈ H^{1}(X, Ω_{cl}^{1}).

Let X be the total space of a vector bundle on a smooth variety Y, and X′ be the space of the dual vector bundle. Construct a canonical isomorphism of sheaves of rings on Y: D_{ω_{X}} ≅ D_{ω_{X′}}, which for Y being the point reduces to the Fourier transform.^{2}

1Ξ_{n} is the particular case of the so-called maximal extension defined by Beilinson. (The extension j_{!∗} is sometimes called the minimal extension). Beilinson usually calls Ξ_{1} “the two-tails beast”. Explain why.

2Hints provided upon request.