Non-Kähler Analogues of Yau's Monge–Ampère Theorem

Definitions and mathematical statement

A Hermitian metric on a complex manifold $X$ is represented by a smooth positive real $(1,1)$-form $\omega$. It is Kähler if ${\,\mathrm{d}}\omega=0$, and non-Kähler otherwise. A non-Kähler metric need not be on a manifold that admits no Kähler metric; the supplied question does not distinguish these settings.

For reference, the classical Kähler problem mentioned in the source has the following formulation. Let $X$ be a compact connected complex manifold of complex dimension $n\geq 1$, and let $\omega$ be a Kähler form. For every $F\in C^\infty(X,{\mathbb{R}})$ satisfying \[ \int_X e^F\omega^n=\int_X\omega^n, \] Yau's theorem gives a unique $\varphi\in C^\infty(X,{\mathbb{R}})$, normalized by $\sup_X\varphi=0$, such that \[ \omega_\varphi:=\omega+\sqrt{-1}\,\partial\bar\partial\varphi>0, \qquad \omega_\varphi^n=e^F\omega^n. \] The associated Calabi-conjecture statement prescribes the Ricci form within a fixed Kähler class: for every smooth real closed $(1,1)$-form $\rho$ representing $2\pi c_1(X)$ in real de Rham cohomology, there is a unique Kähler form $\widetilde\omega$ in $[\omega]$ with $\operatorname{Ric}(\widetilde\omega)=\rho$. Here $c_1(X)$ is the first Chern class of the holomorphic tangent bundle, and, locally for a Hermitian metric matrix $(g_{j\bar k})$, the convention is \[ \operatorname{Ric}(\omega)=-\sqrt{-1}\,\partial\bar\partial\log\det(g_{j\bar k}). \] These are reference formulations of the classical results, not additional hypotheses supplied for the requested generalization.

Source question. Are there analogues for non-Kähler metrics of Yau's solution of the complex Monge--Ampère equation and of the resulting Calabi-conjecture statement?

Unresolved scope. The source does not specify the class of manifolds or Hermitian metrics, the equation to be solved, admissible prescribed data, normalization or compatibility conditions, an equivalence class replacing the fixed Kähler class, or the intended existence and uniqueness conclusions. Consequently, it supplies a research question rather than a single fully quantified conjecture. In particular, the displayed Kähler normalization and conclusions cannot simply be imposed on the unspecified non-Kähler analogue.

Short English statement

The question asks for non-Kähler counterparts of Yau's prescribed-volume theorem and its prescribed-Ricci-form consequence. The source leaves the geometric setting and the desired conclusions unspecified, so no single precise non-Kähler conjecture is determined. Its supplied status is partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30002604$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The project registry locations are data/unsolvedmath/problems.json and catalog/problems.json; this draft uses the supplied record without independently inspecting those files. The record gives proposed year 2014 and identifies no proposer.
• Supplied report citation: Geometrie (2015), Oberwolfach Reports, DOI 10.4171/owr/2014/29, pp. 1569--1623. https://doi.org/10.4171/owr/2014/29. The supplied report title year differs from the year in the DOI and the proposed-year field; this discrepancy is unresolved.
• Supplied literature source: Székelyhidi--Tosatti--Weinkove, "prescribed volume form." https://arxiv.org/abs/1503.04491.
• Supplied literature source: Tosatti--Weinkove, "Hermitian Monge--Ampère." https://arxiv.org/abs/1310.6326.
• Supplied literature source: Oberwolfach Report 29/2014. https://ems.press/content/serial-article-files/46519.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and states that analogues are available for Gauduchon metrics on all compact complex manifolds and for balanced or astheno-Kähler settings, while no single full analogue of all Kähler results is known. This is supplied dated curation, not an independently verified literature assessment. The background explicitly calls for independent verification of time-sensitive status and cited claims.