Hyperkähler Compactification of Projective Fibrations

Definitions and mathematical statement

Write $\pi_U:M_U\to U$ for the given smooth projective fibration and $i:U\hookrightarrow\mathbb{P}^n$ for the supplied inclusion. A compactification of $M_U$ means a compact space $M$ with an open immersion $j:M_U\hookrightarrow M$ having dense image, in the geometric category intended by the source.

Compactification question. Characterize the given fibrations $\pi_U$ for which there exist a smooth hyper-Kähler compactification $(M,j)$ and an everywhere-defined regular morphism $\pi:M\to\mathbb{P}^n$ satisfying \[ \pi\circ j=i\circ\pi_U. \] The requested characterization is a necessary-and-sufficient condition on the input fibration; the record does not supply such a condition.

Unresolved conventions. The source does not specify the ground field, the geometric structure or openness of $U$, the allowed values of $n$, or whether "fibration" requires surjectivity and connected fibres. The usual complex-geometric interpretation would take the ground field to be ${\mathbb{C}}$, but this is an interpretation requiring confirmation. For a compact complex manifold, an irreducible holomorphic symplectic interpretation of "hyper-Kähler" requires that $M$ be Kähler and simply connected, with \[ H^0(M,\Omega_M^2)={\mathbb{C}}\omega, \qquad \omega\text{ everywhere nondegenerate}. \] The record does not specify whether this irreducible convention or a broader hyper-Kähler convention is intended. It also does not explicitly require $M$ to be projective. If $U$ is open, the potentially stronger requirement $j(M_U)=\pi^{-1}(U)$ is not explicitly stated and must be clarified before it is imposed. Thus the displayed extension condition records the supplied target, while these missing conventions prevent a unique fully specified formulation.

Short English statement

Determine exactly when a given smooth projective fibration over a subset of projective space admits a smooth hyper-Kähler compactification whose projection extends regularly to all of projective space. The intended geometric category, hyper-Kähler convention, and precise relation over $U$ require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004712$, source problem code catalogue entry, entitled "Hyperkähler Compactification of Projective Fibrations." The project identifies data/unsolvedmath/problems.json and the supplementary catalog/problems.json as local registries; their contents have not been independently inspected for this draft.
• Supplied primary report citation: Classical Algebraic Geometry (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/29, pp. 1519--1577. https://doi.org/10.4171/owr/2021/29. The background describes this collection as workshop reports documenting mathematical discussions and open problems. The record gives proposed year 2021 and supplies no named proposer.
• Supplied literature reference, attributed to Saccà: https://arxiv.org/abs/2411.06505. No title or more specific bibliographic attribution is supplied.
• Supplied literature reference, attributed to Saccà: https://arxiv.org/abs/2002.01420. No title or more specific bibliographic attribution is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and states that compactification criteria and broad classes of torsors and group schemes are known, including smooth hyper-Kähler examples, while no necessary-and-sufficient criterion for the general question is established. This is supplied dated literature triage, not an independently verified assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified; its accompanying verification note describes only consistent mathematical delimiters for the projective-space inclusions and morphisms, without changing the question. This label does not establish mathematical verification of the formulation.