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DK Conjecture for Some $K$-inequivalences from Grassmannians

Naichung Conan Leung, Ying Xie

Abstract

The DK conjecture of Bondal-Orlov and Kawamata states that there should be an embedding of bounded derived categories for any $K$-inequivalence, which is proved to be true for the toroidal case. In this paper, we construct examples of non-toroidal $K$-inequivalences from Grassmannians inspired by Kuznetsov, Kanemitsu, Ueda, and Morimura, and we show that these $K$-inequivalences satisfy the DK conjecture.

DK Conjecture for Some $K$-inequivalences from Grassmannians

Abstract

The DK conjecture of Bondal-Orlov and Kawamata states that there should be an embedding of bounded derived categories for any -inequivalence, which is proved to be true for the toroidal case. In this paper, we construct examples of non-toroidal -inequivalences from Grassmannians inspired by Kuznetsov, Kanemitsu, Ueda, and Morimura, and we show that these -inequivalences satisfy the DK conjecture.

Paper Structure

This paper contains 13 sections, 9 theorems, 51 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Vanishing of Cohomology and Mutations
  3. Mutations on Derived category of $X\, (N=2n+1)$
  4. Proof of Theorem \ref{['thm']} by Chess Game Method: $N=2n+1$
  5. Mutation of the SOD (\ref{['SOD1mut']})
  6. Mutation on the SOD (\ref{['SOD2']})
  7. Proof of Theorem \ref{['thm']}
  8. Even dimensional case:
  9. Background on mutations and Borel-Weil-Bott
  10. Semiorthogonal decompositions and mutations
  11. Borel-Weil-Bott Theorem
  12. Proof of Lemma \ref{['van']}
  13. Proof of Lemma \ref{['mut']}

Key Result

Theorem 1.1

The $K$-inequivalence birational map $f: X_2\dashrightarrow X_1$ in (aflip) satisfies the DK Conjecture, i.e., there is a fully-faithful embedding of triangulated categories:

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1: Kapranov kapranov1988derived and Kuznetsov kuznetsov2008exceptional
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • Remark 3.7
  • Proposition 4.1
  • ...and 15 more