A Note on Commuting Reflection Functors for Calabi-Yau d-folds

TL;DR

This paper investigates commuting reflection functors in derived categories of sheaves on Calabi-Yau varieties, revealing their structure through orthogonal spherical objects and analyzing the kernels of associated transforms.

Contribution

It characterizes commuting reflection functors via mutually orthogonal spherical objects and links kernels of transforms to torsion-free sheaves with zero-dimensional singularities.

Findings

Commuting reflection functors are determined by orthogonal spherical objects.

Kernels of transforms correspond to torsion-free sheaves with zero-dimensional singularities.

Spherical twists are shown to give equivalences through detailed analysis.

Abstract

We study sets of commuting reflection functors in the derived category of sheaves on Calabi-Yau varieties. We show that such a collection is determined by a set of mutually orthogonal spherical objects. We also show that when the spherical objects are locally-free sheaves then the kernel of the composite transform parametrizes properly torsion-free with zero-dimensional singularity sets and conversely that such a kernel gives rise to a collection of mutually orthogonal spherical vector bundles. We do this using a more detailed analysis of the reason why spherical twists give equivalences.