18
$\begingroup$

I'm familiar with some examples of pairs of derived equivalent varieties, for example an abelian variety and its dual, a K3 surface and certain moduli schemes on it, or the Pfaffian-Grassmannian derived equivalence.

However, when I looked for other known examples, I could only find papers constructing single examples of Fourier-Mukai partners, and no comprehensive survey.

Can anyone provide references where I can find out more about known Fourier-Mukai partners, giving me a broad overview?

$\endgroup$
  • 1
    $\begingroup$ If nobody knows an actual survey with a list, maybe we can start compiling one in the answers? $\endgroup$ – user47305 Dec 11 '15 at 23:17
6
$\begingroup$

There are actually several known Fourier Mukai partners.

  1. Standard flop/Atiyah flop. See Chapter 11 of Fourier-Mukai Transforms in Algebraic Geometry by Huybrechts.
  2. Mukai flops (Chapter 11 of Fourier-Mukai Transforms in Algebraic Geometry by Huybrechts), stratified Mukai flops https://arxiv.org/abs/1111.0688 and Grassmannian flops https://arxiv.org/abs/1206.0219.
  3. Abouf flops https://arxiv.org/abs/1706.04417 and Ueda flops https://arxiv.org/abs/1812.10688.
  4. Any birational but non-isomorphic projective Calabi-Yau 3-folds. See http://www.tom-bridgeland.staff.shef.ac.uk/publications/pub6.pdf
  5. Examples from Homological Projective Duality (HPD), e.g. intersection of two $G(2,5)$ and the intersection of the dual Grassmannian (https://arxiv.org/abs/1707.00534), the intersection of two spinor varieties of $Spin(10)$ and the intersection of their duals(https://arxiv.org/abs/1709.07736). Both of these pairs are non-birtional Calabi-Yau's. The general theorem about the derived equivalences and the kernel functors can refer to https://arxiv.org/abs/1804.00144 and https://arxiv.org/abs/1704.01050
  6. Pair of Calabi-Yau 3 folds in $G_2$ Grassmannian. https://arxiv.org/pdf/1611.08386.pdf It should be noted that these CY3 are non-birational.

I think there are many examples of FM partners from birational geometry, moduli spaces, and HPD theory.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.