Canonical tilting bundles on cotangent bundles of Grassmannians of two-planes
File(s)
Author(s)
Zhou, Wei
Type
Thesis
Abstract
Pioneering work of Beilinson (1979) on projective spaces and of Kapranov (1985) on Grassmannians studied certain collections of natural vector bundles, which have proven extremely useful in understanding the derived category of bounded complexes of coherent sheaves. These collections have no higher extensions, yet are able to generate all coherent sheaves by taking shifts and cones. The direct sum of the bundles in such a collection is known as a tilting bundle. A remarkable work of Kaledin (2008) proved the existence of tilting bundles on a large class of smooth symplectic varieties via the quantization theory. However, it has been a nontrivial question to describe the bundles explicitly. Among the simplest such symplectic varieties are the cotangent bundles of projective spaces and Grassmannians. It is well-known that the pullback of Beilinson's line bundles on a projective space to its cotangent bundle still defines a tilting bundle, but for a Grassmannian of rank higher than one, the pullback of Kapranov's collection to the cotangent bundle no longer has vanishing higher extensions. This thesis addresses this problem for Grassmannians of rank two: a new collection of vector bundles is constructed by replacing some of the vector bundles in Kapranov's collection by others using filtrations on certain natural vector bundles. This ensures that the new collection still generates the derived category. Then, by iteratively taking specific extensions of each of the resulting vector bundles, we are able to eliminate their higher extensions. Furthermore, as a motivation for this work, the tilting bundle is shown to be stable under the derived equivalence for the stratified Mukai flop of the cotangent bundles of Grassmannians via the geometric categorical sl(2) action of Cautis, Kamnitzer and Licata (2013). Consequently, the new collection provides a categorical lift of the canonical basis in the equivariant K-theory.
Version
Open Access
Date Issued
2024-10-27
Date Awarded
2025-06-01
Copyright Statement
Attribution-NonCommercial 4.0 International Licence (CC BY-NC)
License URL
Advisor
Schedler, Travis
Sponsor
Engineering and Physical Sciences Research Council (Great Britain)
Grant Number
EP/S021590/1
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)