Stability Conditions from Surface Degenerations

Definitions and mathematical statement

Let $\mathcal D$ be a triangulated category intended to be associated with a surface, and let $K_0(\mathcal D)$ be its Grothendieck group. A locally finite Bridgeland stability condition on $\mathcal D$ consists of a group homomorphism $Z:K_0(\mathcal D)\to{\mathbb{C}}$, called the central charge, and a slicing $\mathcal P(\phi)\subseteq\mathcal D$, indexed by $\phi\in{\mathbb{R}}$, satisfying the following conditions:

1. Every nonzero $E\in\mathcal P(\phi)$ satisfies $Z([E])=m(E)e^{i\pi\phi}$ for some $m(E)>0$.
2. $\mathcal P(\phi+1)=\mathcal P(\phi)[1]$ for every $\phi\in{\mathbb{R}}$.
3. If $\phi_1>\phi_2$, then $\operatorname{Hom}_{\mathcal D}(E_1,E_2)=0$ whenever $E_j\in\mathcal P(\phi_j)$.
4. Every nonzero object admits a finite Harder--Narasimhan filtration by distinguished triangles, with nonzero factors in $\mathcal P(\phi_1),\ldots,\mathcal P(\phi_n)$ for strictly decreasing phases $\phi_1>\cdots>\phi_n$.
5. There exists $\varepsilon>0$ such that, for every $\phi\in{\mathbb{R}}$, the extension-closed category generated by the $\mathcal P(\psi)$ with $\phi-\varepsilon<\psi<\phi+\varepsilon$ has finite length in its natural quasi-abelian structure.

Write $\operatorname{Stab}(\mathcal D)$ for the space of these stability conditions. This definition supplies terminology; the record does not specify whether the intended conditions must additionally be numerical or satisfy a support property.

Source-qualified construction problem. Produce further Bridgeland stability conditions for some surfaces by studying degenerations of geometric stability conditions associated with ample classes. The supplied reconstructed statement particularly highlights degenerations involving the contraction of several rational curves or passage to mildly singular surfaces. The request is existential and exploratory; the record does not assert a construction for every surface or every such degeneration.

For a fixed-category interpretation, the intended task would require specifying a surface $X$, an associated category $\mathcal D_X$, a family $\sigma_t\in\operatorname{Stab}(\mathcal D_X)$ of the relevant geometric stability conditions associated with ample classes, and a degeneration procedure yielding further stability conditions. If the degeneration changes the surface to a singular surface $Y$, the category on which the resulting stability condition is to live, and its relationship with $\mathcal D_X$, must also be specified. These are requirements for making a construction precise, not hypotheses supplied by the record.

Unresolved qualifications. The verbatim extract refers to degenerations of "the examples above," but those examples are absent from the supplied record. Consequently, the intended surfaces, ground field, derived categories, geometric stability conditions, degeneration parameters, and meaning of "new" cannot be recovered precisely. Neither the configurations of rational curves that may be contracted nor the permitted singularities are specified. The record also does not say whether degeneration means convergence in a fixed stability space, a limiting construction outside that space, or a construction on a different category. Thus the supplied evidence determines a research direction rather than a fully quantified existence assertion.

Short English statement

The problem asks for further Bridgeland stability conditions on some surfaces by degenerating known geometric conditions associated with ample classes, especially in situations involving several contracted rational curves or mildly singular surfaces. The referenced starting examples and essential hypotheses are missing, so the precise class of constructions remains unresolved. The supplied record labels this direction open; this draft does not independently verify that label.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005960$, title Stability Conditions from Surface Degenerations, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The project registry locations are data/unsolvedmath/problems.json and the supplementary catalog/problems.json; this draft uses the supplied complete record.
• Supplied report citation: Algebraic Geometry: Wall Crossing and Moduli Spaces, Varieties and Derived Categories (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/32, pp. 1787--1854. Supplied source URL and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2024/32. No individual proposer is supplied; the record gives proposed year 2024.
• The supplied original extract asks for more stability conditions for some surfaces through degenerations of preceding examples. The record marks its expanded statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims, not independent verification performed for this draft.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and describes producing further stability conditions through surface degenerations as an unresolved future direction. This is supplied dated curation, not an independently verified literature assessment. The background expressly requires independent verification of time-sensitive status and cited claims.