Symplectic Moduli Spaces of Sheaves

Definitions and mathematical statement

An irreducible holomorphic symplectic manifold is a compact, connected, simply connected Kähler manifold $X$ of complex dimension $2n$, where $n\geq 1$, such that \[ H^0(X,\Omega_X^2)={\mathbb{C}}\sigma \] for an everywhere nondegenerate holomorphic two-form $\sigma$. Here $\Omega_X^2$ denotes the sheaf of holomorphic two-forms, and nondegeneracy means that contraction with $\sigma$ induces an isomorphism $T_X\to\Omega_X^1$.

The supplied question is: for every irreducible holomorphic symplectic manifold $X$ and every moduli space $M$ of sheaves on $X$ intended by the source, must $M$ itself be an irreducible holomorphic symplectic manifold?

The phrase "moduli space of sheaves" is not specified sufficiently to determine the range of this question. The record does not specify the class of sheaves, numerical invariants, stability condition, polarization, or whether $M$ denotes a coarse moduli space, a moduli stack, or a selected component. It also supplies no nonemptiness, smoothness, compactness, or dimension assumptions on $M$. Consequently, this is a faithful statement of the supplied question with an unresolved quantifier domain, rather than a fully specified universal conjecture. None of these missing hypotheses is imposed here.

The supplied background itself says that higher-dimensional examples can be singular or require a symplectic resolution. This qualification leaves unresolved whether the intended conclusion concerns $M$ itself, a suitable component, or a resolution. A resolution cannot silently replace $M$ in the stated question.

Short English statement

The source asks whether a moduli space of sheaves on an irreducible holomorphic symplectic manifold must again have that property. The allowed sheaves and moduli spaces are unspecified, and the supplied discussion of singularities and resolutions leaves the intended scope unresolved. The supplied open-status label is dated curation, not an independently verified conclusion.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30002183$, title "Symplectic Moduli Spaces of Sheaves," Oberwolfach Reports Open Problems collection, exact source problem code catalogue entry. No individual proposer is identified; the supplied proposed year is 2012.
• Supplied report citation: Komplexe Analysis (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/43, pp. 2597--2656. https://doi.org/10.4171/owr/2012/43. The collection describes these reports as documenting mathematical workshops and their open-problem discussions. The supplied title year and proposed year differ; their relationship has not been verified.
• Supplied literature reference: "Wandel, tautological sheaves and moduli." https://arxiv.org/abs/1308.4304.
• Supplied literature reference: "Examples of smooth components of sheaf moduli." https://doi.org/10.1007/s00229-020-01223-0.
• Supplied literature reference: "Recent IHS moduli-space constructions." https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/ogrady-tenfolds-as-moduli-spaces-of-sheaves/390DF9D375D2DDED972F3F0D0FEE38F0.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the general question open, reports many positive constructions on K3 and abelian surfaces, and states that moduli spaces on higher-dimensional irreducible holomorphic symplectic manifolds can be singular or require a symplectic resolution. This is supplied dated curation derived from the supplied Oberwolfach Reports curation, not an independently verified literature assessment. Its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that the cleanup is meaning-preserving and self-contained. These are supplied editorial assertions; they do not resolve the missing mathematical hypotheses or establish mathematical correctness.