Homological projective duality for Grassmannians of lines

By Alexander Kuznetsov. 2006

119 citations recorded by the source corpus.

We show that homologically projectively dual varieties for Grassmannians Gr(2,6) and Gr(2,7) are given by certain noncommutative resolutions of singularities of the corresponding Pfaffian varieties. As an application we describe the derived categories of linear sections of these Grassmannians and Pfaffians. In particular, we show that (1) the derived category of a Pfaffian cubic 4-fold admits a semiorthogonal decompositions consisting of 3 exceptional line bundles, and of the derived category of a K3-surface; (2) mutually orthogonal Calabi-Yau linear sections of Gr(2,7) and of the corresponding Pfaffian variety are derived equivalent. We also conjecture a rationality criterion for cubic 4-folds in terms of their derived categories.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d67544cd58f77a062e44835f3279d9f92c2a7dbf

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. K. Hori
2. David Tong
3. Alexander Kuznetsov
4. C. Vafa
5. Emanuele Macrì
6. P. Stellari
7. L. Katzarkov
8. L. Borisov

In Mathematical Physics

1. K. Hori
2. David Tong
3. Alexander Kuznetsov
4. Emanuele Macrì
5. P. Stellari
6. Alexander Perry
7. I. Smith
8. T. Bridgeland

Citations in the corpus

• Homological projective duality (2005, background): “In a forthcoming paper [ K2 ] we will describe homological projective duality for all quadrics.”
• Aspects of non-abelian gauge dynamics in two-dimensional Script N = (2,2) theories (2006, methodology): “A mathematical proof of the derived equivalence has been given by A. Kuznetsov in [ 40 ] as an application of his Homological Projective Duality [41].”
• Exceptional collections for Grassmannians of isotropic lines (2005, background): “[8] also gives an answer for SGr(2, 2m).”
• Lefschetz decompositions and categorical resolutions of singularities (2006, methodology): “We illustrate the approach by constructing (conjecturally minimal) noncommutative resolutions of singularities of Pfaan varieties (see Section 8). We use the constructed resolutions in [ 16 ] to show that they are homologically projective duals to the Grassmannians of lines.”
• Duality in two-dimensional (2,2) supersymmetric non-Abelian gauge theories (2011, methodology): “We would like to thank Alexey Bondal, Mike Douglas, Simeon Hellerman, Shinobu Hosono, Minxin Huang, Johanna Knapp, Alexander Kuznetsov, Todor Milanov, Dave Morrison, Yongbin Ruan and David Tong for useful discussions and instructions.”