Abstract
Abstract We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category that does not have a full exceptional collection consisting of line bundles. This provides a counterexample to a conjecture of Lekili and Ueda.
Highlights
We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category that does not have a full exceptional collection consisting of line bundles
Has a tilting object, which is a direct sum of line bundles
The result above is analogous to the case of toric varieties. It was asked by King if the derived category of a smooth projective toric variety admits a tilting object that is a direct sum of line bundles
Summary
It was asked by King if the derived category of a smooth projective toric variety admits a tilting object that is a direct sum of line bundles This later became known as King’s conjecture. It is known to have a full strong exceptional collection in certain cases: for example, when ≤ 3 [18] or when can be written as the Thom-Sebastiani sum of Fermat and chain polynomials [12] This has been desirable in order to establish homological mirror symmetry for mirror pairs of (gauged) Landau-Ginzburg models [4, 5, 8, 13, 14, 20, 21]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
