A new 5-fold flop and derived equivalence

We describe a new example of a flop in 5-dimensions, due to Roland Abuaf, with the nice feature that the contracting loci on either side are not isomorphic. We prove that the two sides are derived equivalent.Comment: v1. It may well be that this example has appeared before - references welcome! v2. Minor changes. Final version, to appear in Bull. London Math. So

Hori-mological projective duality

Kuznetsov has conjectured that Pfaffian varieties should admit non-commutative crepant resolutions which satisfy his Homological Projective Duality. We prove half the cases of this conjecture, by interpreting and proving a duality of non-abelian gauged linear sigma models proposed by Hori.Comment: 55 pages. V2: slightly rewritten to take advantage of the `non-commutative Bertini theorem' recently proved by the authors and Van den Bergh. V3: lots of changes in exposition following referees' comments. Section 5 has been mostly cut because it was boring. To appear in Duke Math. J. V3: added funder acknowledgemen

Quintic threefolds and Fano elevenfolds

The derived category of coherent sheaves on a general quintic threefold is a central object in mirror symmetry. We show that it can be embedded into the derived category of a certain Fano elevenfold. Our proof also generates related examples in different dimensions.Comment: V1: 12 pages. V2: added reference to work of Iliev and Manivel. V3: persistent sign error corrected. Other minor changes following referee's suggestions. To appear in Crell

The Pfaffian-Grassmannian equivalence revisited

We give a new proof of the 'Pfaffian-Grassmannian' derived equivalence between certain pairs of non-birational Calabi-Yau threefolds. Our proof follows the physical constructions of Hori and Tong, and we factor the equivalence into three steps by passing through some intermediate categories of (global) matrix factorizations. The first step is global Knoerrer periodicity, the second comes from a birational map between Landau-Ginzburg B-models, and for the third we develop some new techniques.Comment: Improved exposition, minor corrections. 32 page

Window shifts, flop equivalences and Grassmannian twists

We introduce a new class of autoequivalences that act on the derived categories of certain vector bundles over Grassmannians. These autoequivalences arise from Grassmannian flops: they generalize Seidel-Thomas spherical twists, which can be seen as arising from standard flops. We first give a simple algebraic construction, which is well-suited to explicit computations. We then give a geometric construction using spherical functors which we prove is equivalent.Comment: Improved structure and formatting. Minor edits to some explanations. Added acknowledgements and addresses. 38 pages, 7 figure

A non-commutative Bertini theorem

We prove a version of the classical 'generic smoothness' theorem with smooth varieties replaced by non-commutative resolutions of singular varieties. This in particular implies a non-commutative version of the Bertini theorem.Comment: 6 pages. v2: added funder acknowledgement. Published in J. Noncommutative Geometr