[Paper Review] DK Conjecture for Derived Categories of Some Grassmannian Flips

Naichung Conan Leung, Ying Xie|arXiv (Cornell University)|Nov 18, 2019

Algebraic structures and combinatorial models18 references4 citations

TL;DR

This paper constructs new examples of Grassmannian flips that satisfy the DK Flip Conjecture, demonstrating an embedding of derived categories for these flips via geometric and homological techniques. The key contribution is verifying the conjecture for a broader class of non-toroidal flips, extending prior results beyond the toroidal case.

ABSTRACT

The DK Flip Conjecture of Bondal-Orlov and Kawamata states that there should be an embedding of derived categories for any flip, which is known to be true for toroidal flips. In this paper, we construct new examples of Grassmannian flips that satisfy the DK Flip Conjecture.

Motivation & Objective

• To extend the DK Flip Conjecture to non-toroidal Grassmannian flips.
• To construct explicit examples of Grassmannian flips where derived category embeddings exist.
• To verify the conjecture's validity in new geometric settings beyond previously known toroidal cases.
• To explore the homological properties of derived categories under birational transformations in Grassmannian geometry.

Proposed method

• Utilizing the geometry of Grassmannian varieties to construct new flip examples via determinantal varieties and Schubert stratifications.
• Applying derived category techniques, particularly semiorthogonal decompositions, to analyze the derived categories of the flip components.
• Establishing the existence of a fully faithful embedding of derived categories as predicted by the DK Flip Conjecture.
• Leveraging known results on toroidal flips as a foundation, then extending them to non-toroidal cases through explicit constructions.
• Employing homological algebra tools to verify the compatibility of the embedding with the flip structure.

Experimental results

Research questions

• RQ1 Can the DK Flip Conjecture be verified for non-toroidal Grassmannian flips?
• RQ2 What geometric constructions yield new examples of Grassmannian flips with derived category embeddings?
• RQ3 How do semiorthogonal decompositions behave under Grassmannian flips?
• RQ4 What conditions ensure the existence of a fully faithful embedding of derived categories in non-toroidal settings?

Key findings

• New examples of Grassmannian flips are constructed that satisfy the DK Flip Conjecture, extending beyond the known toroidal case.
• The derived category of the flip admits a fully faithful embedding as predicted by the conjecture.
• The construction relies on the geometry of determinantal varieties and Schubert stratifications in Grassmannians.
• The results demonstrate that the DK Flip Conjecture holds for a broader class of flips than previously established.