On Derived Categories of Generalized Grassmannian Flips

TL;DR

This paper introduces a new class of flips called generalized Grassmannian flips, extending standard flip constructions to complex algebraic groups, and verifies the DK flip conjecture for a specific 9-fold case.

Contribution

It constructs and classifies generalized Grassmannian flips and proves the DK flip conjecture for a particular 9-fold flip in symplectic groups.

Findings

Constructed a new family of flips for generalized Grassmannians.

Classified these flips within algebraic geometry.

Verified the DK flip conjecture for a 9-fold flip in Sp(6,C).

Abstract

In this paper, we construct and classify a new family of flips, called generalized Grassmannian flips, by generalizing the construction of standard flips for $\mathbb{P}^m\times \mathbb{P}^n$ to any generalized Grassmannian $G/P$, where $P$ is a maximal parabolic subgroup of a complex semi-simple algebraic group. In addition, we show that a 9-fold generalized Grassmannian flip for $Sp(6, \mathbb{C})$ satisfies the DK flip conjecture by Bondal-Orlov and Kawamata via mutation techniques by Kuznetsov and Thomas' chess game method.