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New examples of non-Fourier–Mukai functors
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- Journal:
- Compositio Mathematica / Volume 158 / Issue 6 / June 2022
- Published online by Cambridge University Press:
- 12 August 2022, pp. 1254-1267
- Print publication:
- June 2022
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A celebrated result by Orlov states that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of geometric origin, i.e. it is a Fourier–Mukai functor. In this paper we prove that any smooth projective variety of dimension
$\ge 3$ equipped with a tilting bundle can serve as the source variety of a non-Fourier–Mukai functor between smooth projective schemes.
Stability and the Fourier–Mukai transform II
- Part of
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- Journal:
- Compositio Mathematica / Volume 145 / Issue 1 / January 2009
- Published online by Cambridge University Press:
- 01 January 2009, pp. 112-142
- Print publication:
- January 2009
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We consider the problem of preservation of stability under the Fourier–Mukai transform ℱℰ:D(X)→D(Y ) on an abelian surface and a K3 surface. If Y is the moduli space of μ-stable sheaves on X with respect to a polarization H, we have a canonical polarization
on Y and we have a correspondence between (X,H) and 
. We show that the stability with respect to these polarizations is preserved under ℱℰ, if the degree of stable sheaves on X is sufficiently large.
on 
. We show that the stability with respect to these polarizations is preserved under ℱ