Non-Kähler Ricci Solitons and Broader Existence Methods

Definitions and mathematical statement

A Ricci soliton consists of a smooth Riemannian manifold $(M,g)$, a smooth vector field $X$, and a constant $\lambda\in{\mathbb{R}}$ satisfying \[ \operatorname{Ric}_g+\frac12\mathcal L_Xg=\lambda g. \] Here $\mathcal L_Xg$ is the Lie derivative of the metric. A gradient Ricci soliton additionally satisfies $X=\nabla_g f$ for a smooth function $f$; the supplied question does not explicitly require this. A metric is Einstein if $\operatorname{Ric}_g=cg$ for a constant $c$. The usual non-Einstein interpretation of "nontrivial" would exclude such metrics, but the supplied excerpt does not define that term, so this interpretation requires confirmation.

On a complex manifold $(M,J)$, a Hermitian metric satisfies $g(Ju,Jv)=g(u,v)$ for all tangent vectors $u,v$. Its fundamental two-form is $\omega(u,v)=g(Ju,v)$. It is Kähler when $\mathrm d\omega=0$, and non-Kähler when $\mathrm d\omega\not\equiv0$. Thus, under the non-Einstein interpretation of nontriviality, a candidate Hermitian non-Kähler Ricci soliton must satisfy \[ \operatorname{Ric}_g+\frac12\mathcal L_Xg=\lambda g, \qquad g(Ju,Jv)=g(u,v), \qquad \mathrm d\omega\not\equiv0, \qquad \nexists c\in{\mathbb{R}}:\ \operatorname{Ric}_g=cg. \] These displayed conditions describe candidate objects; they do not resolve the missing global hypotheses of the source question. In particular, the source excerpt does not specify compactness, completeness, boundary conditions, the sign of $\lambda$, whether $X$ must preserve $J$, or a prescribed dimension. Non-Kähler Hermitian metrics necessarily have complex dimension at least two.

For reference, a smooth Fano variety is a smooth complex projective variety whose anticanonical line bundle $K_M^{-1}$ is ample. The supplied record does not determine whether its use of "Fano varieties" is restricted to this smooth setting or also permits singular varieties, nor does it specify any singularity class or corresponding notion of soliton solution.

1. Can nontrivial Hermitian non-Kähler Ricci solitons be constructed using the techniques referred to in the original discussion, or related techniques successful in the Kähler setting?
2. Can methods from partial differential equations establish existence of nontrivial Ricci solitons on a broader class of Fano varieties than the constructions underlying that discussion?
3. Can other special geometries provide constructions of nontrivial Ricci solitons?

The first question concerns existence together with a requested construction method. The second and third are research directions whose comparison classes are not supplied. In particular, the record does not state existence on every Fano variety, does not identify which broader class is intended, and does not explicitly require the solitons in the second or third questions to be non-Kähler. A fully quantified existence conjecture cannot be extracted without resolving these qualifications.

Short English statement

The problem asks for constructions of nontrivial Hermitian non-Kähler Ricci solitons, broader existence results on Fano varieties using partial differential equations, and constructions arising from other special geometries. The source leaves the global hypotheses, the meaning of nontriviality, and the intended comparison classes unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID 30002684, exact source problem code catalogue entry, title "Non-Kähler Ricci Solitons and Broader Existence Methods," in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2014; no individual proposer is supplied.
• Supplied source citation: Mini-Workshop: Einstein Metrics, Ricci Solitons and Ricci Flow under Symmetry Assumptions (2015), Oberwolfach Reports, pp. 2529--2568, DOI 10.4171/owr/2014/45. Supplied source URL and literature-source link: https://doi.org/10.4171/owr/2014/45. The literature-source label is "Oberwolfach Report 45/2014." The supplied citation's 2015 designation differs from the report's 2014 identifier and the recorded proposed year; this discrepancy has not been resolved.
• Additional supplied literature source: Equivariant Solutions to the Prescribed Q-Curvature Equation, https://doi.org/10.1007/s12220-024-01554-4. Its relevance to the precise Ricci soliton questions is not explained in the record.
• Supplied literature assessment dated exactly 2026-08-21: the record labels the problem open and describes the targets as existence beyond known Kähler and symmetric constructions. It reports that the literature still describes compact non-Einstein Ricci soliton examples as Kähler and treats the non-Kähler and broader Fano/PDE questions as unresolved. This is supplied dated curation, not an independently verified assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, accompanied by the claim that the clean statement was checked against the official Oberwolfach report and preserves its hypotheses, notation, and open target. This verification claim is part of the supplied metadata; no source consultation or independent verification is asserted here.