Problem set 4, due November 17.

(1) For algebraic varieties X, Y and an object B ∈ D^{b}(D − mod_{X×Y}) define^{1} a functor F_{B} : D^{b}(D − mod(X)) → D^{b}(D − mod(Y)) by

F_{B} : M ↦ pr_{2+}(pr_{1}^{+}(M)⊗^{L}_{O_{X×Y}} B).

Assume that X, Y are smooth and affine. Find an object B ∈ D^{b}(D_{X}^{op} ⊗

D(Y)) modules such that the functor M ↦ B⊗^{L}_{D(X)}M is isomorphic to F_{B}.

(2) Let f : X → Y be a finite map^{2} of smooth varieties and M be a holonomic D-module on X.

(a) Show that f_{+}(M) is a D-module (rather than a complex of such), i.e. it is concentrated in homological degree zero.

(b) Show that there exist an open U ⊂ im(f) and a vector bundle with a flat connection E on U we have f_{+}(O_{X}) = j_{!+}(E) (where j : U → Y is the embedding).

(3) Let f : 𝔸^{1} → 𝔸^{1} be given by f(z) = z^{n}. Show that f_{+}(O) (where O is equipped with the trivial connection) is a direct sum of n indecomposable summands and describe each summand explicitly (as an O module with an explicit connection).

(Hint: a possible way to argue is as follows. Use the action of automorphisms of the ramified covering to get the decomposition. Restriction of a summand to 𝔸^{1} \ {0} should be not hard to describe. Use base change to show that each summand is a minimal extension from 𝔸^{1} \ {0} which is clean for all the summands but one).

(4) Let Y be a smooth surface, and X → Y be a blow up of a point y ∈ Y. Show that f_{+}(O_{X}) ≅ O_{Y} ⊕ δ_{y}.

(5) (∗) Let X be the total space of the rank 2 bundle O(−1)⊕O(−1) on ℙ^{1}. It is easy to see that the space of functions on X which are linear on each fiber is four. Let f_{1}, f_{2}, f_{3}, f_{4} be a basis in this space and let f = (f_{1}, f_{2}, f_{3}, f_{4}) be the map X → 𝔸^{4}. Show that f_{+}(O) is a sum of two irreducible D-modules and support of each one of them equals im(f).

[Hint: start by observing that f is finite away from the preimage of zero and use problem 2. Then use adjointness to analyze how f_{+}(O) behaves at zero.]

(6) (Radon transform for ℙ^{1}) Let Z ⊂ ℙ^{1} × ℙ^{1} be the complement to diagonal. Show that the functors R_{!} : M ↦ pr_{2∗}pr_{1}^{+}(M) and R_{∗} : pr_{2!}pr_{1}^{+}(M) are inverse auto-equivalences of D_{hol}^{b}(D − mod(ℙ^{1})).

(7) We use notations from the last problem in problem set 3.

Let Z ⊂ V × V^{∗} be the closed subset given by the equation κ = 1, let i : Z → V × V^{∗} be the embedding. Set π′_{1} = π_{1} ∘ i, π′_{2} = π_{2} ∘ i.

The Radon transform R is a functor D−mod(V) → D−mod(V^{∗}) defined by R(M) = π′_{2+}(π′_{1})^{+}(M).

Assume that M ∈ D − mod(V) is weakly equivariant with respect to dilations but M has no non-zero submodule which is (strongly) equivariant.

Construct an isomorphism Φ(M) ≅ R(M).

1Notice the analogy with maps between spaces of functions given by an integral kernel.

2Recall that a finite morphism is both affine and proper.

[One way to do it is to use the next part of the problem].

(8) Set R = i_{+}(O_{Z}). Let e ∈ D(V), e′ ∈ D(V^{∗}) be the Euler vector fields. Show that D(V)_{(e)} = D(V)⊗_{k[e]}k[e, (e+n)^{−1}], n ∈ ℤ is a ring. Construct

an isomorphism of D(V)_{(e)} ⊗ D(V)_{(e′)}-modules:

R ⊗_{D(V ×V^{∗})} D(V)_{(e)} ⊗ D(V)_{(e′)} ≅ E_{κ} ⊗_{D(V ×V^{∗})} D(V)_{(e)} ⊗ D(V)_{(e′)}.