Zero-Free One-Forms on Compact Kähler Varieties

Definitions and mathematical statement

Let $X$ be a smooth complex projective variety. Write $\Omega_X^1$ for its holomorphic cotangent bundle and $\kappa(X)$ for its Kodaira dimension, defined as the maximum dimension of the images of its pluricanonical rational maps, with value $-\infty$ if no positive tensor power of the canonical bundle has a nonzero global section. A holomorphic one-form $\omega\in H^0(X,\Omega_X^1)$ is nowhere vanishing if $\omega_x\ne 0$ in $T_x^*X$ for every $x\in X$. A birational model of $X$ is a variety birational to $X$. A complex abelian variety is a connected projective complex algebraic group.

1. Compact Kähler threefold extension question. A compact Kähler threefold is a compact complex manifold of complex dimension three admitting a Kähler metric. Its underlying differentiable manifold fibers over the circle if it admits a smooth locally trivial fiber bundle map to $S^1=\{z\in{\mathbb{C}}:|z|=1\}$. Does the classification of smooth projective threefolds whose underlying differentiable manifolds fiber over $S^1$ extend to compact Kähler threefolds with this property?

   The supplied record does not state that projective classification or the conclusion intended by its extension. Consequently, this part identifies the intended question but does not determine a precise classification conjecture. No equivalence between fibering over the circle and possessing a nowhere-vanishing holomorphic one-form is asserted here.

2. Birational splitting question. For every smooth complex projective variety $X$ satisfying \[ \kappa(X)\ge 0 \quad\text{and}\quad \exists\,\omega\in H^0(X,\Omega_X^1)\ \forall x\in X,\ \omega_x\ne 0, \] does there exist a birational model $Y$ of $X$ admitting a finite $\acute{e}$tale cover with a positive-dimensional abelian variety as a factor?

   Interpreting "as a factor" as a global product decomposition, the requested data have the form \[ Y\dashrightarrow X\ \text{birational},\qquad p:\widetilde Y\longrightarrow Y\ \text{finite, surjective, and $\acute{e}$tale}, \qquad \widetilde Y\cong A\times Z, \qquad \dim_{{\mathbb{C}}}A>0, \] where $A$ is a complex abelian variety and $Z$ is the remaining factor. Here an $\acute{e}$tale morphism is understood in the usual algebraic sense; over ${\mathbb{C}}$, its analytification is locally a biholomorphism. The supplied record does not specify whether $Y$ must be smooth or projective, or the required regularity and projectivity of $Z$. The displayed formulation therefore records the intended product-splitting target while leaving those qualifications unresolved.

Short English statement

The record asks whether a projective threefold classification involving smooth fibrations over the circle extends to compact Kähler threefolds. It also asks whether every smooth complex projective variety of nonnegative Kodaira dimension carrying a nowhere-vanishing holomorphic one-form has a birational model with a finite $\acute{e}$tale cover splitting off a positive-dimensional abelian variety. The classification to be extended and some conditions on the birational model and product factor are absent from the supplied record.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30005491$, title Zero-Free One-Forms on Compact Kähler Varieties, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives the proposed year as 2023 and names no individual proposer.
• Supplied report citation: Komplexe Analysis -- Differential and Algebraic methods in Kähler spaces (2023), Oberwolfach Reports, pp. 965--1030, DOI 10.4171/owr/2023/18. https://doi.org/10.4171/owr/2023/18. The supplied original-statement fragment contains only the assertion that both conjectures remain open and a page marker for p. 995; it does not reproduce either conjecture.
• Supplied literature source, labeled "OWR 2023 report": https://ems.press/content/serial-article-files/47013.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels both the compact Kähler extension conjecture and the birational or $\acute{e}$tale-cover expectation as open, and attributes this assessment to the report. This is supplied dated curation, not an independently verified assessment of the literature or current status.
• The registry labels the statement corrected_verified and supplies a verification note claiming comparison with the official report. That label and note are source metadata; no independent source consultation or mathematical verification is claimed in this draft.