Abstract
The aim of this mainly expository note is to point out that, given an Fourier-Mukai functor, the condition making it fully faithful is an instance of generic vanishing. We test this point of view on some fairly classical examples, including the strong simplicity criterion of Bondal and Orlov, the standard flip and the Mukai flop.
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Notes
Note that in loc cit condition (i) is stated in a different way, namely \(R^j\Phi ^{Y\rightarrow Z}_{{\mathcal {E}}^\vee \otimes p_Z^*\omega _Z}({\mathcal {F}}^\vee \otimes \omega _Y)=0\) for \(j>\dim Y\). This is equivalent to (i) by duality and base-change. Indeed by duality (i) is equivalent to the fact the loci \(V^j_{{\mathcal {P}}^\vee \otimes p_Z^*\omega _Z}(Y, {\mathcal {F}}^\vee \otimes \omega _Y)\) are empty for all \(j>\dim Y\), which is in turn equivalent (by an easy application of base-change) to the vanishing of the sheaves \(R^j\Phi ^{ Y\rightarrow Z}_{{\mathcal {E}}^\vee \otimes p_Z^*\omega _Z}({\mathcal {F}}^\vee \otimes \omega _Y)=0\) for all \(j>\dim Y\).
Condition (c) is more usually expressed in the dual way, namely \(H^i(Y,F\otimes \Phi _{{\mathcal {P}}[\dim Z]}^{Z\rightarrow Y}(A^\vee ))=0\) for \(i\ne 0\) (see e.e. [21] Cor. 3.11(b)). By duality and Serre vanishing it is easily seen that the two formulations are equivalent.
In fact it is well known that the sheaf \({{\widehat{\omega _Y}}}\) has the “base-change property”, namely, in the present case, the natural map \(tor_0({{\widehat{\omega _Y}}},k_{(x,x\prime )})\rightarrow H^{\dim Y}(Y,\omega _Y\otimes \Phi _{{\mathcal {E}}^\vee \boxtimes _Y{\mathcal {E}}}^{X\times X\rightarrow Y}(k_{(x,x^\prime )}))\) is an isomorphism. This is proved as follows: condition (a) can be stated as \(V^i_{{\mathcal {E}}\boxtimes _Y{\mathcal {E}}^\vee }(Y, OO_Y)=\emptyset \) for \(i>0\) or, dually, \(V^i_{{\mathcal {E}}^\vee \boxtimes _Y{\mathcal {E}}}(\omega _Y)=\emptyset \) for \(i>\dim Y\). By a well known base change argument (see e.g. [17] 1.3 or [21], proof of Lemma 3.6) this implies that \({{\widehat{\omega _Y}}}\) has the base change property.
This is also the key ingredient in the proof of the equivalence between (b) and (c) of Theorem 2.3.
Note that this implies that \(\Phi _{\mathcal {P}}^{X\rightarrow {\widehat{X}}}\) it is in fact an equivalence. Moreover, since \({\mathcal {P}}^\vee =(-id,id)^*=(id,-id)^*{\mathcal {P}}\), this proves also that
$$\begin{aligned} \Phi _{\mathcal {P}}^{X\rightarrow {\widehat{X}}}\circ \Phi _{\mathcal {P}}^{{\widehat{X}}\rightarrow X}=(-id)^*[-\dim X] \end{aligned}$$i.e. the theorem of Mukai in [14].
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In memory of Alexandru T. Lascu.
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Pareschi, G. Fully faithful Fourier-Mukai functors and generic vanishing . Ann Univ Ferrara 63, 185–199 (2017). https://doi.org/10.1007/s11565-016-0256-9
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DOI: https://doi.org/10.1007/s11565-016-0256-9