Compactifiable Lagrangian Fibrations from Abelian Fibrations

Definitions and mathematical statement

First question: symplectic structures and compactification. Given an abelian fibration $f\colon X\to B$, determine conditions under which its total space admits a symplectic structure making $f$ a Lagrangian fibration and permitting a compactification to a possibly singular irreducible symplectic variety.

Under the usual holomorphic interpretation, a symplectic structure on a smooth complex variety $X$ is a holomorphic two-form $\sigma\in H^0(X,\Omega_X^2)$ satisfying ${\,\mathrm{d}}\sigma=0$ and nondegenerate at every point. If $\dim_{{\mathbb{C}}}X=2n$, a smooth fiber $F$ is Lagrangian when \[ \dim_{{\mathbb{C}}}F=n,\qquad \sigma|_F=0. \] An abelian variety is a connected projective algebraic group; the phrase abelian fibration indicates a fibration whose relevant smooth fibers are abelian varieties. The supplied record does not specify the precise class of fibrations, the regularity of $X$ and $B$, or whether this holomorphic interpretation is intended throughout. It also does not specify the definition of a possibly singular irreducible symplectic variety, the required behavior of the symplectic form at the compactification boundary, or whether the fibration itself must extend. These are unresolved parts of the formulation, rather than additional hypotheses imposed here.

Second question: toroidality of a relative compactified Jacobian. Let $\Delta\subset{\mathbb{C}}^k$ be a polydisk centered at $0$, and let \[ \pi\colon\mathcal C\longrightarrow\Delta,\qquad C_0=\pi^{-1}(0), \] be a family of curves that is a multiversal unfolding of the singularities of $C_0$. Index all those singularities by $i$. Each is required to have type \[ A_{\mu_i}\quad\text{with }1\leq\mu_i\leq5, \qquad\text{or}\qquad D_{\mu_i}\quad\text{with }4\leq\mu_i\leq5, \] and the total bound is \[ \sum_i\mu_i\leq5. \] In the conventional complex plane-curve notation, these types have local analytic equations \[ A_\mu:\ x^2+y^{\mu+1}=0, \qquad D_\mu:\ x^2y+y^{\mu-1}=0, \] where $\mu$ is the Milnor number. The stipulated unfolding concerns all the central fiber's singularities simultaneously. The supplied term "multiversal" is retained: the record does not give its precise deformation-theoretic definition, and no equality $k=\sum_i\mu_i$ is asserted.

The question is whether, for every family satisfying these stated conditions, its relative compactified Jacobian \[ \overline{J}(\mathcal C/\Delta) \] is toroidal. A compactified Jacobian ordinarily parametrizes rank-one torsion-free sheaves with specified numerical data and, where required, stability conditions. The record does not supply the degree, polarization, stability convention, or hypotheses on properness, flatness, connectedness, reducedness, or irreducibility of the curves needed to identify this moduli space unambiguously.

For a specified pair $(Y,D)$, toroidality conventionally describes local models by a toric variety together with its toric boundary, in a specified topology. The record does not identify a boundary on $\overline{J}(\mathcal C/\Delta)$, specify the topology for local models, or say whether "toroidal" refers to the total space, a pair, or the morphism to $\Delta$. Consequently the displayed question preserves the supplied target, but its exact toroidality predicate remains unresolved. The source explicitly presents this as a separate question; no equivalence with the first question is asserted.

Short English statement

The record asks when an abelian fibration admits a symplectic structure making its fibers Lagrangian and allowing a possibly singular irreducible symplectic compactification. Separately, it asks whether the relative compactified Jacobian of every stipulated multiversal curve family is toroidal when all central singularities are among the specified $A_\mu$ and $D_\mu$ types and their Milnor numbers sum to at most $5$. Precise compactification, moduli, unfolding, and toroidality conventions remain unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002986$, source problem code catalogue entry, title "Compactifiable Lagrangian Fibrations from Abelian Fibrations," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as $2015$.
• Supplied report citation: Mini-Workshop: Singular Curves on $K3$ Surfaces and Hyperkähler Manifolds (2016), Oberwolfach Reports, pp. 2939--2967, DOI 10.4171/owr/2015/51. https://doi.org/10.4171/owr/2015/51. The citation's displayed year $2016$ and the registry's proposed year $2015$ are retained without reconciliation.
• Supplied literature source, "Oberwolfach report question": https://oa.tib.eu/renate/server/api/core/bitstreams/205e22d5-f548-4915-b721-7e608401c35b/content.
• Supplied literature source, "Markushevich on Lagrangian fibrations": https://doi.org/10.1070/IM1995v059n01ABEH000007.
• Supplied literature source, "Hyper-Kähler compactification": https://arxiv.org/abs/1602.05534.
• Supplied dated literature curation, checked at 2026-08-21: the registry labels the problem open and assesses the exact toroidality question in the stated simple-singularity range as unresolved, with important special cases and related compactifications but no general proof for the extracted hypotheses. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. That verification claim is supplied provenance; the present draft does not independently verify the report, references, mathematical correctness, or current status.