# 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 $X$.