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If $X$ and $Y$ are smooth projective varieties, $p: X \times Y \to X$ and $q: X \times Y \to Y$ are the projections, and $\mathcal{P}$ is an object in $D^b(X \times Y)$, then the Fourier--Mukai transform associated to $\mathcal{P}$ is a functor $\Phi_\mathcal{P}: D^b(X) \to D^b(Y)$ that sends $\mathcal{E}^\bullet$ to $q_*(p^*\mathcal{E}^\bullet \otimes \mathcal{P})$ (where pushforward, pullback, and tensor are the derived versions).

It is a fact that if we define $\mathcal{P}_L := \mathcal{P}^\vee \otimes q^*\omega_Y[\dim Y]$ and $\mathcal{P}_R := \mathcal{P}^\vee \otimes p^*\omega_X[\dim X]$ (where $-^\vee$ is the derived dual), then $\Phi_{\mathcal{P}_L}$ is left-adjoint to $\Phi_{\mathcal{P}}$ and $\Phi_{\mathcal{P}_R}$ is right-adjoint to $\Phi_{\mathcal{P}}$. (This is Proposition 5.9 in Huybrechts's book on Fourier--Mukai transforms, and he credits it to Mukai.)

Something that pops out to me is that the left-adjoint and the right-adjoint to $\Phi_\mathcal{P}$ are pretty similar, and that in fact they're the same if $X$ and $Y$ are Calabi--Yau and have the same dimension. My general impression is that it's a very special property for $(F, G)$ and $(G, F)$ to both be adjoint pairs (maybe it's the "next best thing" to $F$ and $G$ being inverse to one another?).

My question is:

Is this formal property that I mention in the third paragraph useful? What information about $D^b(X)$ and $D^b(Y)$ does it buy you, especially for special cases of $X$ and $Y$?

I'm not an algebraic geometer, so obvious-seeming answers are appreciated as much as non-obvious-seeming answers. The reason I'm asking is because I'm studying the symplectic version of the Fourier--Mukai transform (see this paper) and it seems to exhibit a similar formal property.

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  • $\begingroup$ It tells you that $F$ and $G$ both preserve both colimits and limits! $\endgroup$ – Qiaochu Yuan Nov 27 '13 at 21:59
  • $\begingroup$ Consider also the example of the diagonal functor $C \to C^2$. The left adjoint of the diagonal, if it exists, is the coproduct. The right adjoint, if it exists, is the product. They agree iff $C$ has biproducts, which is a very special property and in particular implies that $C$ is canonically enriched over commutative monoids. $\endgroup$ – Qiaochu Yuan Nov 27 '13 at 22:02
  • $\begingroup$ @QiaochuYuan thanks! If preservation of limits and colimits is an important consequence, then I will try and think about what that means for the derived category of coherent sheaves. I guess I failed to mention in my question that what I'm really interested in is geometric consequences for $D^b$, not "abstract nonsense consequences" that always happen when both $(F,G)$ and $(G,F)$ are adjoint pairs. $\endgroup$ – Nathaniel Bottman Nov 27 '13 at 22:16
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A pair of functors $(F,G)$ which are both left and right adjoints of each other is called a Frobenius pair. If you google that you'll find plenty of literature. The canonical example is that the induction and restriction functors in the representation theory of finite groups form a Frobenius pair. This also explains the name (it comes from Frobenius reciprocity).

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The left and the right adjoint of a functor $F$ are always conjugate to each other with respect to the Serre functors of the categories you consider (you can easily prove it by using the adjunction and the Serre duality). So, they coincide if and only if the original functor commutes with the Serre functors. Of course, for Calabi--Yau varieties of the same dimension it always holds since the Serre functors are the same twists.

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