# Geometric natural transformations between Fourier-Mukai transforms

Given two schemes $X$ and $Y$ one can consider additive functors (eventually with some nice additional property) between the categories of $\mathcal{O}_X$-modules and of $\mathcal{O}_Y$ modules. Among these one has those "of a geometric nature", in particular Furier-Mukai-type functors, i.e., those of the form $\Phi_{Z,\mathcal{P}}:=g_*(\mathcal{P}\otimes f^*-)$, where $(f,g):Z\to X\times Y$ and $\mathcal{P}$ is some "twisting" sheaf on $Z$. With suitable finiteness assumptions (e.g., if $Z$ is Noetherian), $\Phi_{Z,\mathcal{P}}$ is a functor $QC(X)\to QC(Y)$. Also, one can consider a derived variant of this, and look at $\Phi_{Z,\mathcal{P}}$ as to a functor $D^b(Coh(X))\to D^b(Coh(Y))$ (all this is very well known and studied, I'm writing it only to provide a setting for my question).

Given two Furier-Mukai-type functors $\Phi_{Z,\mathcal{P}}$ and $\Phi_{W,\mathcal{Q}}$ between categories of $\mathcal{O}_X$-modules and of $\mathcal{O}_Y$-modules, are there natural "geometric" transformations between these functors? which are the main examples of these? (here "geoemtric" is somehow vague: it pretends to mean "something with the same flavor of Fourier-Mukai transform, e.g., maybe something involving a morphism $T\to Z\times W$ and some twisting kernel over $T$)

First a quick comment: if you work not with triangulated categories but with differential graded or $A_\infty$ categories, then ALL (continuous) functors are given by "Fourier-Mukai" or integral transforms (a theorem of Toen - an analogue of the Schwartz kernel theorem in analysis). Moreover the (dg) category of functors is equivalent to the category of sheaves on $X\times Y$, in other words natural transformations are the same as sheaf maps on the product.

In any case, one way to realize your question is to look eg at functors given by correspondences $X\leftarrow Z \rightarrow Y$ - i.e. at integral kernels of the form $\pi_*{\mathcal O}_Z$ for the evident map $Z\to X\times Y$. Then a map of correspondences $Z\to W$ gives a pullback map between the corresponding integral kernels, i.e., a natural transformation between the functors. You can generalize this to twisted kernels (i.e. equip $Z$ with a sheaf as well) but at that point you're very close to just looking at morphisms of integral kernels on $X \times Y$..

• Hi David, enlightening and informative as usual, thanks a lot! Just one more thing: do I also have a pushforward map between integral kernels induced by $W\to Z$? (eventually under suitable hypothesis?). This way one would associate a natural transformation to a correspondence $Z\leftarrow T \to W$, too. (as you will have surely read between the lines, my latest MO questions are secretly related to an attempt of mine to get familiar with the higher tqft constructions by Francis, Nadler and yourself) – domenico fiorenza Apr 18 '12 at 20:07