Mathematische Zeitschrift

, Volume 287, Issue 1–2, pp 155–179 | Cite as

On the existence of Fourier–Mukai functors

  • Alice RizzardoEmail author


A theorem by Orlov states that any equivalence \(F:D^{b}_{\mathrm {Coh}}(X) \rightarrow D^{b}_{\mathrm {Coh}}(Y)\) between the bounded derived categories of coherent sheaves of two smooth projective varieties X and Y is isomorphic to a Fourier–Mukai transform \(\Phi _{E}(-)=R\pi _{2*}(E\mathop {\otimes }\limits ^{L} L\pi _1^{*}(-))\), where the kernel E is in \(D^{b}_{\mathrm {Coh}}(X\times Y)\). In the case of an exact functor which is not necessarily fully faithful, we compute some sheaves that play the role of the cohomology sheaves of the kernel, and that are isomorphic to the latter whenever an isomorphism \(F\cong \Phi _{E}\) exists. We then exhibit a class of functors that are not full or faithful and still satisfy the above result.


Fourier–Mukai functor Orlov’s theorem 

Mathematics Subject Classification

13D09 18E30 



This paper is derived from part of the author’s PhD thesis. The author thanks her thesis advisor Aise Johan de Jong for suggesting the problem as well as providing invaluable guidance over the years. The last draft of this paper was written while at SISSA, in Trieste.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.SISSATriesteItaly

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