Canonical tilting bundles: the first nonabelian examples

TL;DR

The paper constructs an explicit tilting bundle on the nonabelian symplectic resolution $T^{*}\mathrm{Gr}(2,\mathbb{C}^{N})$ by combining Kapranov’s exceptional collection with iterated GL$(\mathbb{C}^{N})$-equivariant extensions. The resulting bundle $\mathcal{E}$, comprising $\mathcal{E}_{\lambda}=\mathbb{S}^{\lambda}V_{2}$ and iterated cones $\mathcal{E}_{k}$, is a classical generator with Ext-vanishing and a Koszul endomorphism algebra, and its image under the stratified Mukai flop is explicitly controlled via a geometric $\mathfrak{sl}(2)$ action. This yields a canonical-basis-compatible tilting object: the K-theory classes of the summands are invariant under the bar involution and sit inside the canonical basis, up to shifts, providing a categorical lift of canonical bases for Nakajima quiver varieties. The work bridges noncommutative resolutions, derived equivalences, and symplectic duality, with tools including Lascoux resolutions and Borel–Weil–Bott computations, and it extends the abelian Beilinson-type picture to a substantial nonabelian setting. The constructions and invariance results have potential implications for understanding categorical representations and canonical bases in more general Nakajima quiver-variety contexts.

Abstract

We present an explicit construction of tilting bundles on cotangent bundles of Grassmannians of 2-planes. This construction is based on Kapranov's exceptional collection for the underlying Grassmannians, and utilizes specific iterative extensions. The resulting tilting bundle exhibits invariance under the derived equivalence for the stratified Mukai flop through the geometric categorical sl(2) action, providing a categorical lift of the K-theoretic canonical basis up to shifts.

Theorems & Definitions (70)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1: Kr
• Theorem 2.2: Wey
• Theorem 2.3: Wey
• Remark 2.4
• Definition 2.5
• Theorem 2.6: CKL-sl2