Equivariant Derived Categories of Symplectic Surfaces

Definitions and mathematical statement

The supplied record leaves the meaning of "symplectic surface" and the condition that the action be "suitable" unspecified. In particular, it does not specify the ground field, smoothness or properness hypotheses, or whether the surfaces are algebraic or complex analytic. Consequently, the following is a question schema whose admissible objects and actions still require specification.

Write $\mathcal D(S)$ for the intended derived category of a symplectic surface $S$. A coherent action of a finite group $G$ consists of autoequivalences $F_g$ together with natural isomorphisms $c_{g,h}\colon F_gF_h\xrightarrow{\sim}F_{gh}$, satisfying the associativity and unit coherence conditions. An equivariant object is an object $E$ with isomorphisms $\phi_g\colon F_g(E)\xrightarrow{\sim}E$ satisfying the unit condition and \[ \phi_{gh}\circ c_{g,h,E}=\phi_g\circ F_g(\phi_h). \] Morphisms commute with these isomorphisms. Denote the intended equivariant derived category by $\mathcal D(S)^G$. The categorical enhancement or construction needed to give this category its intended derived structure is not specified in the record.

The question is whether, for every symplectic surface $S$ in the intended class and every finite group $G$ with a suitable coherent action on $\mathcal D(S)$, there exist a symplectic surface $S'$ and a Brauer twisting class $\alpha$ on $S'$ such that \[ \mathcal D(S)^G\simeq\mathcal D(S',\alpha). \] Here $\mathcal D(S',\alpha)$ denotes the intended derived category of $\alpha$-twisted sheaves; the trivial class is allowed and gives the untwisted case. Under the usual bounded coherent-sheaf interpretation, the displayed target would read \[ \left(D^b(\operatorname{Coh}(S))\right)^G \simeq D^b(\operatorname{Coh}(S',\alpha)). \] That interpretation, the notion of Brauer class, and the required structure preserved by the equivalence must be confirmed against the source. No precise admissibility condition replacing "suitable" is supplied, so this schema does not assert the equivalence for arbitrary finite-group actions.

Short English statement

The question asks whether taking equivariant objects for a suitable finite-group action on the derived category of a symplectic surface produces the derived category of another symplectic surface, possibly with a Brauer twist. The admissible surfaces, actions, and categorical constructions remain unspecified. The supplied dated curation labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004469$, exact source problem code catalogue entry, title: Equivariant Derived Categories of Symplectic Surfaces. Collection: Oberwolfach Reports Open Problems. The record gives proposed year $2020$ and names no proposer.
• Supplied source citation: Algebraic Geometry: Moduli Spaces, Birational Geometry and Derived Aspects (2021), Oberwolfach Reports, pp. 977--1021, DOI 10.4171/owr/2020/19. Original OWR report: https://doi.org/10.4171/owr/2020/19. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. No specific page or speaker for this question is supplied.
• Supplied literature reference, attributed to Beckmann--Oberdieck: https://arxiv.org/abs/2006.13899. No title or theorem locator is supplied in the record.
• Literature assessment dated exactly 2026-08-22: the supplied curation states that Beckmann and Oberdieck provide a criterion and examples yielding an equivalence with a symplectic surface, while the equivalence under general reasonable conditions is not proved. The supplied status is partially_solved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; the status and cited claims require independent checking.