Jet Curvature and the First Chern Class

Definitions and mathematical statement

Let $X$ be a complex manifold, and let $T_X$ denote its holomorphic tangent bundle. Define its real first Chern class by \[ c_1(X)_{{\mathbb{R}}}:=\iota\left(c_1(T_X)\right)\in H^2(X;{\mathbb{R}}), \] where $H^2$ denotes singular cohomology and $\iota:H^2(X;{\mathbb{Z}})\to H^2(X;{\mathbb{R}})$ is induced by the inclusion of coefficients.

The supplied question asks whether the implication \[ X\text{ has nondegenerate negative }k\text{-jet curvature} \quad\Longrightarrow\quad c_1(X)_{{\mathbb{R}}}\ne 0 \] holds for complex manifolds $X$. Here $k$ denotes a jet order, conventionally a positive integer; the record does not explicitly specify its range or quantification.

The curvature hypothesis is retained verbatim as a source-dependent condition. The supplied record does not define the relevant jet space, metric, curvature inequality, or nondegeneracy condition, so a precise geometric definition of this hypothesis cannot be recovered from the supplied text alone. Compactness, projectivity, dimension restrictions, and any further hypotheses are likewise unspecified. None are added here. Consequently, this is a statement-only transcription with unresolved scope and terminology, requiring comparison with the cited report.

Short English statement

Does nondegenerate negative $k$-jet curvature on a complex manifold force the first Chern class of its holomorphic tangent bundle to be nonzero in real cohomology? The curvature definition, the quantification over $k$, and any additional hypotheses require source review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002180$, source problem code catalogue entry, title Jet Curvature and the First Chern Class; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The record gives proposed year 2012.
• Supplied report citation: Komplexe Analysis (2013), Oberwolfach Reports, pp. 2597--2656, DOI 10.4171/owr/2012/43. https://doi.org/10.4171/owr/2012/43. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's 2013 designation and the 2012 provenance are retained without reconciliation.
• Supplied literature source: Oberwolfach Report 43, jet-curvature question. https://ems.press/content/serial-article-files/46414.
• Supplied literature source: Demailly, jet metrics and curvature. https://www-fourier.univ-grenoble-alpes.fr/~demailly/source_files/takagi16/JJM1566_final_page_proof.pdf.
• Supplied literature assessment, checked 2026-08-21: the record labels the problem open, describes the implication as unresolved, and says that later work retains a surrounding jet-curvature equivalence as a difficult conjectural problem. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. These are supplied registry assertions, not independent verification of the mathematical formulation or source fidelity.