Holomorphic-Curve Structure of Calabi–Yau Blow-Up Sets

Definitions and mathematical statement

An almost-complex structure on a smooth manifold $X$ is a smooth bundle endomorphism $J:TX\to TX$ satisfying $J^2=-\mathrm{id}_{TX}$. In the usual parametrized terminology, a $J$-holomorphic curve is represented by a nonconstant map $u:(\Sigma,j)\to(X,J)$ from a Riemann surface satisfying \[ \mathrm{d}u\circ j=J\circ\mathrm{d}u. \] Its image is a subset of $X$. The supplied record does not specify which class of such curves is intended.

The proposed conditional assertion concerns Donaldson's almost-complex Calabi--Yau equation: assume the intended $\varepsilon$-regularity estimate holds. If the a priori estimates in Donaldson's conjecture fail, must the resulting blow-up set $B$ admit a representation \[ B=\bigcup_{\alpha\in I}C_\alpha, \] where each $C_\alpha$ is a $J$-holomorphic curve?

This is a schematic formulation of the supplied speculative assertion. The record does not state the equation, the ambient geometric hypotheses, the regularity estimate, the a priori estimates, the degenerating sequence, or the definition of $B$. Consequently, the relevant domains, boundary conditions, convergence conditions, and quantification over blow-up sequences cannot be made precise from this record. It also does not specify whether the union is finite or locally finite, whether curves are closed or compact, or whether singularities or multiplicities are intended. These qualifications remain unresolved. The proposed curve structure is described as potentially leading toward Donaldson's conjecture; no implication establishing that conjecture is supplied.

Short English statement

Assuming an unspecified $\varepsilon$-regularity estimate for Donaldson's almost-complex Calabi--Yau equation, the question asks whether failure of the conjectured a priori estimates produces a blow-up set made of $J$-holomorphic curves. The supplied formulation is speculative and lacks the geometric and analytic hypotheses needed for a fully quantified statement.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30001492$, title "Holomorphic-Curve Structure of Calabi--Yau Blow-Up Sets," exact source problem code catalogue entry; Oberwolfach Reports Open Problems collection, proposed year 2010. No individual proposer is supplied.
• Supplied report citation: Geometrie (2010), Oberwolfach Reports, DOI 10.4171/owr/2010/26, pp. 1535--1572. https://doi.org/10.4171/owr/2010/26. The supplied literature-source label is "original OWR Report 26/2010" and points to this same URL. No problem-specific page or named author is supplied.
• Supplied literature assessment dated exactly 2026-08-21: status "open"; the assertion is described as a speculative blow-up consequence intended to lead toward the preceding conjecture, with no general proof identified by that assessment. 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 metadata; the report has not been independently consulted for this draft.