Famous Fourier-Mukai Results I

A while back I (tried to) give a talk about some of the uses of the Fourier-Mukai transform.  I’ve only learned about the Fourier-Mukai transform in the context of dealing with smooth projective varieties, so everything will be at least that.  For F \in D^b(X \times Y), the Fourier-Mukai transform is \Phi_F \colon D^b(X) \to D^b(Y) via G \mapsto Rp_{2*}\bigl(p_1^*G\otimes^L F\bigr).

In my early readings these were the most prominent results I came across:

  1. If A is an Abelian variety then the Poincare bundle \mathcal{P} \in Pic(A \times \widehat{A}) gives an isomorphism \Phi_\mathcal{P} \colon D^b(A) \to D^b(\widehat{A}).
  2. (Orlov’s Result) Any functor F \colon D^b(M) \to D^b(X) which is full, faithful and exact is represented by an object on the product.
  3. If \omega_X \in Pic(X) or its inverse is ample then D^b(X) determines X.
  4. For Abelian varieties A,B, D^b(A) \cong D^b(B) \Leftrightarrow A \times \widehat{A} \cong B \times \widehat{B}

Some Details for 1

The proof I plan to outline depends on the following results

1. If L \ne O_A and L \in Pic^0(A) then H^*(A,L) = 0.  See cohom of pic zero in this post.

2. ( Adjunction )For \Phi_P \colon D^b(X) \to D^b(Y) and P^\vee = R\hom(P,O_{X\times Y}) set

P_L = P^\vee\otimes p_2^*\omega_Y[\dim Y]

P_R = P^\vee\otimes p_1^*\omega_X[\dim X].

Then P_L,P_R represent left and right adjoints to \Phi_P respectively.

pf:

\hom_Y(\Phi_P F, E) \cong \hom_{X\times Y}(p_1^*F \otimes^L P, p_2^*E\otimes p_1^*\omega_X[\dim X])

this isomorphism comes from the fact that \omega_{X\times Y} \cong p_1^*\omega_X \otimes p_2^*\omega_Y and Grothendieck-Verdier duality:

Let f \colon W \to Z be a morphism of smooth schemes over a field k (lets say algebraically closed).  There is an isomorphism

hom_Z(Rf_*E, F) \cong \hom_W(E, Lf^*F \otimes \omega_X \otimes \omega_Y^{-1}[\dim W - \dim Z]).

continuing the hom isomorphisms…

\cong \hom_{X\times Y}(p_1^*F , p_2^*E\otimes^L P^\vee \otimes p_1^*\omega_X[\dim X])

\cong \hom_{X}(F ,Rp_{1*}\bigl( p_2^*E\otimes^L P^\vee \otimes p_1^*\omega_X[\dim X]\bigr))

\cong \hom_{X}(F,\Phi_{P_R}E). QED.

3. \Phi_P \colon D^b(X) \to D^b(Y) if fully faithful iff \forall x,y \in X

\hom(\Phi_P k(x),\Phi_P k(y))[i] = \begin{cases} k \mbox{ if } i = 0, x = y\\ 0 \mbox{ if } x \ne y \mbox{ or } i \not \in [0, \dim X]\end{cases}

4. (purely category theory) If F \colon D \to D' is a fully faithful, exact functor between triangulated categories and D contains objects not isomorphic to 0 and D' is indecomposable

roughly a triangulated category C is decomposable if there subcategories C_1,C_2 s.t. $\forall O \in ob(C) \exists$  a distinguished triangle O_1 \to O \to O_2 with O_i \in ob(C_i) and \hom(ob(C_i),ob(C_j)) = \delta_{ij}; also both subcateogories have to contain objects not isomorphic to 0.

Then F is an equivalence iff F has left and right adjoints $G,H$ and for O \in ob(D'), H(B) \cong 0 \Rightarrow G(B) \cong 0.

A proof of this can be found in Huybrechts book on the Fourier-Mukai transform.  This result gives away the idea of the proof: use algebraic geometry results to check the hypothesis of this assertion in the case at hand.

Putting it all together.

Using 2 it is easily checked that \mathcal{P}_L = \mathcal{P}_R = \mathcal{P}^\vee hence \Phi_\mathcal{P} is exact with left and right adjoints.

Showing that D^b(\widehat{A}) is indecomposable is more category theoretic and I’m skipping that for now.

Unwrapping the definitions,

\hom(\Phi_\mathcal{P} k(b),\Phi_\mathcal{P} k(a))[i] = \hom_{\widehat{A}}(\mathcal{P}_{a \times \widehat{A}}, \mathcal{P}_{b \times \widehat{A}})[i]

= \hom(O_{\widehat{A}}, \mathcal{P}_a \otimes \mathcal{P}_b^{-1})[i] = H^i(\widehat{A}, \mathcal{P}_{a -b})

so \Phi_\mathcal{P} is fully faithful using 3 and 1.

I’m putting some details for Orlov’s result in a whole other post.

I don’t know how to prove result 3, but I know a few important points regarding its proof.  The triangulated D^b(X) consists of all the categorical data plus the data of a Serre functor S_X \colon D^b(X) \to D^b(X) given by S_X(E) = E \otimes \omega_X[\dim X].

Central to the proof are the notions of point like objects and invertible objects.  P\in D^b(X) is point like if

  1. S_X(P) = P[\dim X]
  2. \hom(P,P[i]) = 0 for i<0
  3. \hom(P,P) =: k(P) is a field.

An object L \in D^b(X) is invertible if \forall P \in D^b(X) point like $\exists n(L,P) \in \mathbb{Z}$ such that \hom(L,P[i]) = \begin{cases} k(P) \mbox{ for } i = n(L,P)\\ 0 \mbox{ otherwise } \end{cases}.

My understanding is roughly is picking out this information allows you to pick out \omega_X \in D^b(X) and consequently the canonical ring \oplus_{i \in \mathbb{Z}} H^0(X,\omega_X^i) which determines X.

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