Problem set 1, due September 21

(1) Let ∇ be a connection on a trivial bundle over 𝔸^{n} with fiber V. Let ω ∈ Ω^{1}(𝔸^{n}) ⊗ End(V) be the connection form. Prove that ∇ is flat if and only if

dω + [ω, ω] = 0

(this is called the Maurer-Cartan equation).

(2) 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 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).

(3) In this problem we work over ℂ. Let G denote the symplectic group Sp(2n, ℂ). Note that G acts on D = D(𝔸^{n}) by automorphisms (namely, it acts on the 2n-dimensional vector space span(x_{i}, \frac{∂}{∂x_{j}}) which generates D and symplectic transformations (with respect to the standard symplectic form) extend to automorphisms of D). Thus if M is a D-module and g ∈ G we denote by M^{g} the same module but with D-action twisted by g.

Let M be a holonomic D-module on 𝔸^{n} and let c = c(M).

a) Show that for 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 ∈ ℂ).

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 Stonevon 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).

(4) (For those familiar with commutative and homological algebra)

Let X be an algebraic variety. Recall that for a closed subset Z ∈ X the local cohomology functor RΓ_{Z} on QCoh(X) is the derived functor of the functor of sections supported (set-theoretically) on Z.

Let X be an n-dimensional smooth algebraic variety over a field, x ∈ X be a (closed) point. Show that R^{i}Γ_{x}(O_{X}) = 0 for i ≠ n, and R^{n}Γ_{z}(O_{X}) is the D_{X}-module of distributions supported at x.