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Derived categories of quartic double fivefolds

Raymond Cheng, Alexander Perry, Xiaolei Zhao

TL;DR

The paper analyzes the Kuznetsov component of singular quartic double fivefolds and constructs a crepant categorical resolution by a twisted Calabi-Yau threefold, via a detailed quadric-bundle geometry and Clifford-algebra description. It then identifies a rational specialization in which the Brauer twist disappears, yielding a geometric Calabi-Yau threefold whose derived category matches the Kuznetsov component, and it demonstrates a conifold-type transition between twisted and geometric CY3s. These results provide evidence for a higher-dimensional Kuznetsov rationality and a noncommutative version of Reid's fantasy about connected moduli of Calabi-Yau threefolds, while outlining a framework for deforming Ku(X) through singularities and mutations. The work also extends the toolkit for relating CY categories to both twisted and untwisted geometric models via quadric bundles, Clifford algebras, and Azumaya structures.

Abstract

We construct singular quartic double fivefolds whose Kuznetsov component admits a crepant categorical resolution of singularities by a twisted Calabi--Yau threefold. We also construct rational specializations of these fivefolds where such a resolution exists without a twist. This confirms an instance of a higher-dimensional version of Kuznetsov's rationality conjecture, and of a noncommutative version of Reid's fantasy on the connectedness of the moduli of Calabi--Yau threefolds.

Derived categories of quartic double fivefolds

TL;DR

The paper analyzes the Kuznetsov component of singular quartic double fivefolds and constructs a crepant categorical resolution by a twisted Calabi-Yau threefold, via a detailed quadric-bundle geometry and Clifford-algebra description. It then identifies a rational specialization in which the Brauer twist disappears, yielding a geometric Calabi-Yau threefold whose derived category matches the Kuznetsov component, and it demonstrates a conifold-type transition between twisted and geometric CY3s. These results provide evidence for a higher-dimensional Kuznetsov rationality and a noncommutative version of Reid's fantasy about connected moduli of Calabi-Yau threefolds, while outlining a framework for deforming Ku(X) through singularities and mutations. The work also extends the toolkit for relating CY categories to both twisted and untwisted geometric models via quadric bundles, Clifford algebras, and Azumaya structures.

Abstract

We construct singular quartic double fivefolds whose Kuznetsov component admits a crepant categorical resolution of singularities by a twisted Calabi--Yau threefold. We also construct rational specializations of these fivefolds where such a resolution exists without a twist. This confirms an instance of a higher-dimensional version of Kuznetsov's rationality conjecture, and of a noncommutative version of Reid's fantasy on the connectedness of the moduli of Calabi--Yau threefolds.
Paper Structure (15 sections, 29 theorems, 78 equations, 1 figure)

This paper contains 15 sections, 29 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quartic double fivefolds singular along a line
  3. Linear projection from the line
  4. Quartic fourfolds
  5. Double quartic fivefolds
  6. Sections of the singular quadric surface bundles
  7. General situation: twisted geometric component
  8. Crepant resolution of $\mathcal{K}u(X)$
  9. Clifford algebra description of $\widetilde{\mathcal{K}u}(X)$
  10. A twisted Calabi--Yau threefold
  11. Non-projectivity
  12. Special rational examples
  13. Projection from a section
  14. Conifold transition
  15. Geometric Kuznetsov component

Key Result

Theorem 1.3

Let $X \to \mathbf{P}^5$ be a double cover branched along a quartic hypersurface $Y \subset \mathbf{P}^5$ which is singular along a line $L \subset \mathbf{P}^5$. For general such $Y$, there is a crepant categorical resolution of singularities of $\mathcal{K}u(X)$ by $\mathrm{D^b}(W^+,\mathcal{A}^+)

Figures (1)

  • Figure 1: This diagram summarizes the sequence of mutations we perform to transform the exceptional collection coming from \ref{['DbtX2']} to the exceptional collection coming from the decomposition of Lemma \ref{['lemma-DbtX']}. The symbol $(a,b)$ represents the object $\mathcal{O}_{\widetilde{X}}(aH + bh)$, and $i_*(a,b)$ represents the object $i_*\mathcal{O}_Z(aH + bh)$. An arrow indicates that the next row is obtained by mutating the object at the tail of the arrow through the object at the head of the arrow. Arrows that point off the sides indicate the application of Serre duality to flip underlined objects to the other side.

Theorems & Definitions (55)

  • Remark 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6: Higher dimensions
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 45 more