Small Resolutions Detected by Symplectic Cohomology

Definitions and mathematical statement

Under the conventional complex interpretation, an isolated compound Du Val (cDV) singularity is a three-dimensional hypersurface germ $(X,0)=\{f=0\}\subset({\mathbb{C}}^4,0)$, singular only at $0$, whose general hyperplane section through $0$ has a Du Val (ADE) surface singularity. Its Milnor fibre is $F=f^{-1}(t)\cap B_\varepsilon$, for sufficiently small regular $t\ne0$ and suitable Milnor ball $B_\varepsilon$, with its usual exact symplectic structure. This is the conventional setting; the supplied statement does not explicitly specify the ground field or geometric category.

A small resolution is a proper birational resolution $\pi:Y\to X$, with $Y$ smooth and $\pi$ an isomorphism over $X\setminus\{0\}$, whose exceptional locus $E=\pi^{-1}(0)$ has no divisorial components. In this three-dimensional setting, its irreducible exceptional components are curves. The source does not specify whether the resolution is required to be algebraic, complex analytic, or projective, so that qualification remains unresolved.

Write $SH^k(F)$ for degree-$k$ symplectic cohomology. The coefficient ring or field, the meaning of rank over it, and the integer-grading conventions are not supplied. Subject to fixing those conventions and the category of resolution, the intended Evans--Lekili conjecture is the following: for every isolated cDV singularity and every component count $\ell\in{\mathbb{Z}}_{\ge0}$, \[ \begin{gathered} \exists\text{ a small resolution }\pi:Y\to X \text{ whose exceptional locus has exactly }\ell \text{ irreducible components} \\[2pt] \Longleftrightarrow \quad \forall k\in{\mathbb{Z}}\text{ with }k<0,\qquad \operatorname{rank}SH^k(F)=\ell. \end{gathered} \] Here nonnegative integrality of $\ell$ expresses its role as a component count; the source gives no explicit range or separate convention for $\ell=0$. The statement does not assert validity for every possible choice of coefficients or grading.

The supplied excerpt describes isolated cDV singularities as terminal threefold hypersurface singularities and as the only class of Gorenstein threefold singularities that can admit a small resolution. It also states that their links are index positive by its reference [6], and consequently that negative-degree symplectic cohomology of their Milnor fibres is a contact invariant by its reference [2]. These are attributed background claims; their precise hypotheses and the identities of those numbered references are not supplied here.

The excerpt reports explicit rank formulas for Milnor fibres of all quasihomogeneous $cA_n$ singularities, including those without small resolutions, using an algorithm attributed to its reference [2]. Its discussion ends with an incomplete sentence. These contextual claims are retained as supplied evidence, not as a proof of the displayed equivalence.

Short English statement

The conjecture attributed to Evans and Lekili asks whether an isolated cDV singularity has a small resolution with exactly $\ell$ irreducible exceptional curves precisely when every negative-degree symplectic cohomology group of its Milnor fibre has rank $\ell$. Coefficients, grading conventions, and the category of resolution need clarification. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30006448$, title Small Resolutions Detected by Symplectic Cohomology, exact source problem code catalogue entry. This draft uses the supplied complete registry record; the project registry locations are data/unsolvedmath/problems.json and catalog/problems.json. No file lookup or external source verification was performed.
• Supplied report citation: Singularities (2026), Oberwolfach Reports, pp. 2413--2490, DOI https://doi.org/10.4171/owr/2025/45. The supplied literature list calls this Oberwolfach Report 45/2025, and the proposed year is 2025; the relationship between these year labels remains unverified.
• The excerpt attributes the conjecture to Evans and Lekili. It identifies the contribution as A glimpse into the birational geometry of quasi-homogeneous $cA_n$ singularities, by Aline Zanardini, joint work with Nikolas Adaloglou and Federica Pasquotto. No separate proposer is supplied. The excerpt labels the displayed conjecture "Conjecture 1" and begins with an incomplete reference to "Conjecture 1.4]".
• Supplied literature source: Symplectic cohomology of cDV singularities, https://doi.org/10.5802/ahl.177.
• Supplied literature source: Symplectic cohomology of quasihomogeneous cA singularities, https://doi.org/10.1007/s00209-026-03943-7.
• Supplied literature assessment dated 2026-08-22: the record labels the problem partially solved and states that the Evans--Lekili equivalence remains conjectural in general, while explicit calculations provide evidence and verify it for broad quasihomogeneous $cA$ subfamilies. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The registry's statement-status label is corrected_verified; its accompanying explanation says that only formula delimiters, notation, grammar, and typography were repaired, preserving mathematical content and scope. This supplied label does not establish mathematical correctness or independent verification of the conjecture.