Holomorphic Symplectic Geometry — Conjecture 12

Definitions and mathematical statement

The supplied excerpt does not define the ambient space $X$, the loci $X_s$, or the admissible indices $s$. Retaining these as unresolved source notation, its proposed statement is \[ X_s\neq\varnothing \quad\Longrightarrow\quad \text{there exists a component }Z\text{ of }X_s\text{ with }\dim Z>s. \] This is a conditional formulation template: the hypotheses on $X$, the definition of $X_s$, the range of $s$, and the intended meanings of component and dimension must be recovered before it becomes a fully specified mathematical conjecture. In particular, the excerpt does not justify defining $X_s$ by any particular rank condition on a Poisson tensor.

The source presents this as an "optimistic modification" after suggesting that the condition "$X$ Fano" is probably too strong. This does not specify which other hypotheses are retained or what replaces the Fano hypothesis. The preceding contextual passage describes an unspecified result as exceeding a naive dimension count and says that it implies that a Poisson field vanishing at a point must vanish along a curve. That contextual implication is preserved as a supplied source claim; its missing antecedent prevents its formulation or verification here.

Short English statement

Beauville's supplied Conjecture 12 excerpt proposes that every nonempty locus $X_s$ should have a component of dimension strictly greater than $s$. It is described as an optimistic weakening of a formulation involving a Fano hypothesis. The excerpt omits the definitions and remaining hypotheses needed to make this precise.

Sources

• Local UnsolvedMath registry, permanent ID $800203$, exact source code catalogue entry, title: Holomorphic Symplectic Geometry --- Conjecture 12. The supplied record attributes the problem to Beauville, with proposed year 2010, and labels it open; that label is supplied registry metadata, not an independently verified status assessment.
• Beauville, Holomorphic Symplectic Geometry: A Problem List (2010), Conjecture 12. Supplied source URL: https://arxiv.org/abs/1002.4321. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; no independent source visit or verification is asserted.