Invariant stability conditions on local $\mathbb{P}^1\times \mathbb{P}^1$ (after Del Monte-Longhi)

TL;DR

This work studies stability conditions on the local Calabi–Yau threefold $X=\mathrm{Tot}\,\omega_{\mathbb{P}^1\times\mathbb{P}^1}$ that are invariant under a derived autoequivalence. Using a tilting construction to a finite-dimensional algebra $A$ and the induced quiver, the authors identify an autoequivalence $\Psi$ with $\Φ=\Psi^2$ that cyclically permutes simples; they classify all $\Φ$-invariant stability conditions and describe the complete set of $\sigma$-stable objects for $\sigma\in\mathrm{Stab}(X)^{\Φ}$. The main result shows that, up to shifts, stable objects fall into two families of special Kronecker types (I/II) determined by the central charges, plus explicit $\mathbb{P}^1$-families on certain classes, and that on the ray $\phi=\tfrac{1}{2}$ only objects of the form $s_*\mathcal{O}_{\{x\}\times\mathbb{P}^1}$ or its twist appear. Furthermore, the space $\mathrm{Stab}(X)^{\Φ}$ maps to a hyperplane-complement $\mathcal{H}^{\mathrm{reg}}$ as a covering, giving a precise topological picture of the invariant stability landscape and connecting to the DL22 framework for invariant stability. Overall, the paper provides a complete algebraic–geometric classification of invariant stability data and stable objects, clarifying how autoequivalences constrain stability in local Calabi–Yau geometries.

Abstract

Let $X$ be the total space of canonical bundle of $\pp$, we study an invariant subspace of stability conditions on $X$ under an autoequivalence of $D^b(X)$. We describe the complete set of stable objects with respect to the invariant stability conditions and characterize the space of invariant stability conditions.

Figures (5)

• Figure 1: Real slice of $\mathcal{H}^{\mathop{\mathrm{reg}}\nolimits}$
• Figure 2: Central charges of $\mathcal{U}(\mathcal{A})^{\Phi}_+$
• Figure 3: Central charges of $\mathcal{U}(\mathcal{A})^{\Phi}_-$
• Figure 4: Central charges of stable objects and $\mathcal{O}_x$ for $\sigma \in \mathcal{U}(\mathcal{A})^{\Phi}_+$
• Figure 5: Central charges of simple objects and $E$, $\delta$ for $\sigma'$

Theorems & Definitions (109)

• Lemma 1.1: =Lemma \ref{['lem:restr_stab']}
• Theorem 1.2: =Theorem \ref{['thm:stab_obj']}
• Theorem 1.3: =Theorem \ref{['thm:local_homeo']} and \ref{['thm:stab_cover']}
• Lemma 2.1: Bri07
• Definition 2.2: Torsion pair
• Theorem 2.3: Happel-Reiten-Smalø
• Remark 2.4
• Lemma 2.5
• proof
• Lemma 2.6