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The Thurston compactification of the stability manifold of a generic analytic K3 surface

Anand Deopurkar

TL;DR

The paper constructs a Thurston-like compactification of the Bridgeland stability manifold for a generic analytic K3 surface with $\operatorname{Pic}(X)=0$ by identifying the space of masses of semi-rigid objects with a 2D disk. Central to the approach is the complete classification of semi-rigid objects (up to $\mathcal{O}_X$-twists and shifts) as skyscraper sheaves, and an explicit description of their Harder–Narasimhan filtrations under standard and off-wall stability conditions. The mass embedding $m$ provides a homeomorphism from the projectivized stability space to a tiled disk $D$, whose boundary encodes lax stability phenomena, including the distinguished red endpoint $\hom(\mathcal{O}_X,-)$ and two lax-pre-stability boundary points; a $q$-analogue $m_q$ yields a closely related disk $D_q$ with an extra boundary interval and the $q$-hom endpoint $\hom_q(\mathcal{O}_X,-)$. The construction generalizes Thurston-type compactifications to a geometric setting tied to Mukai lattices, spherical twists, and the Farey-type tilings induced by the spherical twist action, and it provides concrete combinatorial and geometric descriptions of the boundary behavior of stability spaces. The results advance understanding of stability manifolds for non-algebraic K3s and offer a framework for further $q$-deformations and connections to lax stability.

Abstract

Let X be an analytic K3 surface with Pic X = 0. We describe the closure of the Bridgeland stability manifold of X obtained using the masses of semi-rigid objects.

The Thurston compactification of the stability manifold of a generic analytic K3 surface

TL;DR

The paper constructs a Thurston-like compactification of the Bridgeland stability manifold for a generic analytic K3 surface with by identifying the space of masses of semi-rigid objects with a 2D disk. Central to the approach is the complete classification of semi-rigid objects (up to -twists and shifts) as skyscraper sheaves, and an explicit description of their Harder–Narasimhan filtrations under standard and off-wall stability conditions. The mass embedding provides a homeomorphism from the projectivized stability space to a tiled disk , whose boundary encodes lax stability phenomena, including the distinguished red endpoint and two lax-pre-stability boundary points; a -analogue yields a closely related disk with an extra boundary interval and the -hom endpoint . The construction generalizes Thurston-type compactifications to a geometric setting tied to Mukai lattices, spherical twists, and the Farey-type tilings induced by the spherical twist action, and it provides concrete combinatorial and geometric descriptions of the boundary behavior of stability spaces. The results advance understanding of stability manifolds for non-algebraic K3s and offer a framework for further -deformations and connections to lax stability.

Abstract

Let X be an analytic K3 surface with Pic X = 0. We describe the closure of the Bridgeland stability manifold of X obtained using the masses of semi-rigid objects.

Paper Structure

This paper contains 15 sections, 21 theorems, 81 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Conventions
  3. Outline
  4. Acknowledgements
  5. Stability conditions on generic K3 surfaces
  6. The Mukai lattice
  7. Standard stability conditions
  8. All stability conditions
  9. Semi-rigid objects
  10. The mass embedding
  11. HN filtration of semi-rigid objects
  12. The mass map
  13. Identifying the image and its closure
  14. Points of the boundary
  15. The $q$-mass embedding

Key Result

theorem 1

Let $X$ be an analytic K3 surface with $\operatorname{Pic}(X) = 0$. Let $S \subset D^b\operatorname{Coh} (X)$ be the set of semi-rigid objects. The map $m \colon \mathbf{P} \operatorname{Stab}(D^b \operatorname{Coh}(X)) \to \mathbf{P}^S$ is a homeomorphism onto its image. The image is a 2-dimensiona

Figures (9)

  • Figure 1: For an analytic K3 surface $X$ with $\operatorname{Pic}(X)= 0$, the compactified $\mathbf{P} \operatorname{Stab}(X)$ is a closed disk, tiled by the translates of a triangle by the action of the spherical twist in $\mathcal{O}_X$. A distinguished point (red) in the boundary corresponds to the function $\hom(\mathcal{O}_X, -)$.
  • Figure 2: The closure of $m_q(\mathbf{P} \operatorname{Stab}(X))$ is also a closed disk. The boundary has an additional interval, whose blue end-point is the $q$-hom functional $\hom_q(\mathcal{O}_X,-)$.
  • Figure 3: The tiling of the disk by triangles in the $q = 1$ case (left) versus the $q \neq 1$ case (right).
  • Figure 4: For $w \in -\mathbf{H}$ (red), a central charge $Z_1$ as above defines a stability condition with heart $\operatorname{Coh}(X)$. For $z \in \mathbf{H}$ (green) and $z \in \mathbf{R}_{<0}$ (blue), a central charge $Z_2$ as above defines a stability condition whose heart is the tilt of $\operatorname{Coh}(X)$ with respect to torsion and torsion-free sheaves.
  • Figure 5: We can use the cosine rule to reconstruct the central charge of a standard $\sigma \in W_-$ from the masses $a,b,c$ of $\mathbf{k}_x, T\mathbf{k}_x, \mathcal{O}_X$ (left) and of $\sigma \in W_+$ from the masses $a,b,c$ of $T^{-1} \mathbf{k}_x, \mathbf{k}_x, \mathcal{O}_X$ (right).
  • ...and 4 more figures

Theorems & Definitions (38)

  • theorem 1
  • remark 2
  • proposition 3
  • proof
  • proposition 4
  • proof
  • proposition 5
  • proof
  • proposition 6
  • proof
  • ...and 28 more