# List of known Fourier Mukai partners?

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?

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

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.