A Remark on Stability Conditions on Smooth Projective Varieties

TL;DR

The paper establishes that the bounded derived category $\mathrm{D}^{b}(X)$ of any smooth projective complex variety $X$ admits Bridgeland stability conditions by a descent strategy that starts from stability on products of elliptic curves and propagates through $(\mathbf{P}^1)^n$ and $\mathbf{P}^n$ via finite maps and group actions. It develops and leverages the Bayer property and Restriction-$N$ conditions to preserve stability under pushforward/pullback along these maps, and uses double Schubert polynomials to analyze filtrations arising in the kernels of base-change morphisms. The key steps are (i) constructing $\sigma^{a,b}$ on $E^n$ with symmetry/invariance properties, (ii) descending to $\mathbf{P}^n$ through the symmetric quotient while maintaining Bayer-type control, and (iii) transferring stability to subvarieties via resolutions and Polishchuk’s families-of-t-structures. The result yields that $\mathrm{Stab}(X)$ is nonempty for all smooth projective $X$, providing a robust bridge between geometric and categorical stability frameworks with broad implications for moduli and derived-category techniques.

Abstract

Let $X$ be a smooth projective variety over $\mathbb C$. In this paper, we prove that $\mathrm{D}^b(X)$, the bounded derived category of coherent sheaves on $X$, always admits stability conditions in the sense of Bridgeland.

Theorems & Definitions (36)

• Theorem 1.1
• Definition 1.2: Bridgeland:Stab
• Definition 1.3: Bridgeland:Stab
• Definition 1.5: Kontsevich-Soibelman:stability
• Theorem 1.6: Bridgeland's Deformation Theorem,Bridgeland:StabArend:shortproof
• Definition 2.1
• Lemma 2.2: realred
• Lemma 2.3
• proof
• proof : Proof of the Claim