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Decorated sheaves and morphisms in tilted hearts

Yinbang Lin, Sz-Sheng Wang, Bingyu Xia

Abstract

We identify limit stable pairs and stable framed sheaves as epimorphisms and monomorphisms, respectively, in tilts of the standard heart, under suitable conditions. We then identify the moduli spaces with the corresponding Quot spaces, obtaining the projectivity of the Quot spaces in these cases. We also prove a formula in a motivic Hall algebra relating the Quot spaces under a tilt.

Decorated sheaves and morphisms in tilted hearts

Abstract

We identify limit stable pairs and stable framed sheaves as epimorphisms and monomorphisms, respectively, in tilts of the standard heart, under suitable conditions. We then identify the moduli spaces with the corresponding Quot spaces, obtaining the projectivity of the Quot spaces in these cases. We also prove a formula in a motivic Hall algebra relating the Quot spaces under a tilt.

Paper Structure

This paper contains 11 sections, 19 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. $t$-structures and torsion pairs
  3. Quot spaces and moduli of quotient husks
  4. Stable pairs and framed sheaves
  5. Stable pairs
  6. Framed sheaves
  7. Change of Quot space under tilting
  8. The stack of pairs
  9. Motivic Hall algebra
  10. Laurent subsets
  11. Identities in the Laurent Hall algebra

Key Result

Lemma 2.2

Let $\mathcal{A}$ be the heart of a bounded $t-$structure on a triangulated category $\mathcal{D}$ and $(\mathcal{T}, \mathcal{F})$ a torsion pair of $\mathcal{A}$. Let $\mathcal{A}^{\#}$ be the tilt in $\mathcal{A}$ at this pair.

Theorems & Definitions (39)

  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Example 3.4
  • Remark 3.5
  • ...and 29 more