Holomorphic Symplectic Geometry — Conjecture 11

Definitions and mathematical statement

The supplied excerpt does not define the ambient space $X$ or the tensor $\tau$. Under the conventional holomorphic Poisson interpretation, $X$ would be a smooth complex projective variety and $\tau\in H^0(X,\bigwedge^2T_X)$ a holomorphic bivector satisfying the Poisson identity $[\tau,\tau]=0$. These ambient hypotheses are a contextual interpretation requiring confirmation, rather than hypotheses explicitly supplied by the excerpt. In this interpretation, $X$ is Fano when its anticanonical line bundle $K_X^{-1}$ is ample, and the rank at $x$ is the rank of the contraction map \[ \tau_x^{\sharp}:T_x^*X\longrightarrow T_xX. \] This rank is even. For each nonnegative even integer $s$, define \[ X_s=\{x\in X:\operatorname{rank}(\tau_x^{\sharp})=s\}, \qquad X_{\leq s}=\bigcup_{\substack{k\in 2{\mathbb{Z}}_{\geq0}\\k\leq s}}X_k. \] Thus \[ X=\bigsqcup_{s\in 2{\mathbb{Z}}_{\geq0}}X_s. \] Here $X_{\leq s}$ is the closed rank degeneracy locus, and a component means an irreducible component of its underlying closed set.

The source attributes to Bondal the conjectural assertion that, if $X$ is Fano and $s$ is even, then \[ \exists\, Z\text{ an irreducible component of }X_{\leq s} \quad\text{such that}\quad \dim_{{\mathbb{C}}}Z>s. \] The allowed range of $s$ is unresolved in the supplied record. If $n=\dim_{{\mathbb{C}}}X$, the assertion cannot hold for $s\geq n$, since $X_{\leq s}\subseteq X$. A restriction such as $0\leq s<n$ is therefore a possible intended qualification, but is not stated in the excerpt and is not adopted here as verified source text. The ambient hypotheses, the nature of $\tau$, and the range of $s$ must be confirmed before this becomes a fully specified conjecture.

Short English statement

Bondal's conjectural assertion, as recorded in Beauville's problem list, says that an even-rank degeneracy locus on a Fano variety has an irreducible component whose dimension exceeds the rank threshold. The supplied excerpt leaves the ambient Poisson hypotheses and the necessary range of the threshold unspecified, so this is a statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $800202$, exact source problem code catalogue entry; title: Holomorphic Symplectic Geometry --- Conjecture 11. The supplied registry labels the problem open; this label is not an independently verified mathematical or literature assessment.
• Beauville, Holomorphic Symplectic Geometry: A Problem List (2010), Conjecture 11. Supplied source URL: https://arxiv.org/abs/1002.4321. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.
• The excerpt explicitly attributes the conjecture to Bondal, citing the source labels [Bd] and [P] (the latter as an additional reference). The supplied record does not include the bibliographic entries corresponding to these labels. The registry lists Beauville as proposer and 2010 as the proposed year; this registry attribution is preserved alongside the excerpt's attribution to Bondal.