# Kernels for Grassmann Flops

## TL;DR

This paper generalizes a kernel construction for Grassmann flips, creating a canonical idempotent kernel that induces semi-orthogonal decompositions and aligns with Kapranov's exceptional collections on Grassmannians.

## Contribution

It introduces a new kernel construction for Grassmann flips that provides canonical decompositions and links to known exceptional collections.

## Key findings

- Constructs a canonical idempotent kernel on the derived category.
- Establishes a semi-orthogonal decomposition comparing flipped varieties.
- Identifies a window in the derived category matching Kapranov's exceptional collection.

## Abstract

We develop a generalization of the $Q$-construction of the first author, Diemer, and the third author for Grassmann flips. This generalization provides a canonical idempotent kernel on the derived category of the associated global quotient stack. The idempotent kernel, after restriction, induces a semi-orthogonal decomposition which compares the flipped varieties. Furthermore its image, after restriction to the geometric invariant theory semistable locus, "opens" a canonical "window" in the derived category of the quotient stack. We check this window coincides with the set of representations used by Kapranov to form a full exceptional collection on Grassmannians.