Gauduchon Conjecture in Complex Geometry

• Gauduchon Conjecture is a statement about the existence of Hermitian metrics with prescribed Ricci curvature on compact complex manifolds, extending the Calabi–Yau theorem to non-Kähler settings.
• Its resolution employs nonlinear Monge–Ampère equations along with continuity and parabolic flow methods to establish existence and uniqueness of the desired metrics.
• These advances have led to broader applications in addressing prescribed scalar curvature problems, Gauduchon connections, and rigidity phenomena in both singular and quaternionic geometries.

The Gauduchon Conjecture, posed in 1984 by Paul Gauduchon, concerns the existence of Hermitian metrics with prescribed Ricci curvature on compact complex manifolds. This conjecture, which generalizes the Calabi–Yau theorem to the non-Kähler setting, was resolved affirmatively by Székelyhidi, Tosatti, and Weinkove. The conjecture has influenced a broad spectrum of analytic, geometric, and cohomological questions in complex geometry and inspired a variety of elliptic and parabolic partial differential equation (PDE) approaches, as well as generalizations to singular and quaternionic settings.

1. Formulation and Mathematical Background

Let $M$ be a compact complex manifold of dimension $n$. A Hermitian metric $g$ is encoded via its real $(1,1)$-form $\omega$, defined by $\omega(X,Y) = g(JX,Y)$, where $J$ is the complex structure. The metric is called Gauduchon if $\partial\bar\partial\,\omega^{n-1}=0$. Such metrics always exist in every conformal class of Hermitian metrics on $M$.

Gauduchon's conjecture, as formally established, asserts: given a compact complex manifold $(M, \alpha)$ with a Gauduchon metric $n$0 and a real, positive $n$1-form $n$2 in the same Bott–Chern cohomology class as $n$3, there exists a unique Gauduchon metric $n$4 such that $n$5 and $n$6. In particular, for any prescribed volume form in the given Bott–Chern class, such a Gauduchon metric exists and is unique up to normalization (Zheng, 2020, Székelyhidi et al., 2015).

2. Analytic Reformulation and Elliptic Approach

The conjecture reduces to the solution of a fully nonlinear Monge–Ampère type equation involving $n$7-plurisubharmonic functions with possible gradient terms. The analytic core, as realized by Székelyhidi–Tosatti–Weinkove, is the following PDE:

$n$8

for a given real function $n$9 with $g$0. In eigenvalue form, the problem takes the shape $g$1, where the $g$2 are the eigenvalues of an appropriately defined endomorphism.

The proof employs the continuity method and a priori estimates: zero-order ($g$3), first-order (gradient), and second-order (Hessian) bounds. Key techniques include the maximum principle, subsolution theory, Evans–Krylov regularity, and a Liouville-type argument, ensuring full $g$4 regularity (Székelyhidi et al., 2015).

3. Parabolic and Continuity Equation Methods

Beyond the classical elliptic approach, the Gauduchon conjecture admits parabolic and continuity equation realizations.

• Continuity Equation: Motivated by parallels with Kähler geometry (La Nave–Tian) and Hermitian cases (Sherman–Weinkove), a continuity equation is studied:

$g$5

where $g$6 is the Chern–Ricci form, and $g$7 is chosen for closure. Existence and uniqueness are established up to a maximal parameter $g$8, characterized by explicit positivity conditions (Zheng, 2020).

• Parabolic Flow: A Monge–Ampère flow is considered:

$g$9

where $(1,1)$0 denotes a nonlinear deformation with torsion term. The flow admits global solutions, uniform a priori estimates, and smooth exponential convergence to the stationary Gauduchon metric solving the original conjecture (Zheng, 2016).

These approaches leverage a reduction to (n−1)-psh Monge–Ampère equations and Schauder theory, yielding structural analogies to Yau's solution of the Calabi conjecture.

4. Singular and Quaternionic Generalizations

Recent work extends the conjecture to singular varieties and quaternionic geometries.

• Singular Gauduchon Conjecture: On smoothable Hermitian varieties, existence and regularity are proven for metrics solving

$(1,1)$1

with a crucial $(1,1)$2-estimate and uniform bounds in families. These results generalize the theory to canonical singularities and deformation spaces, enabling applications to Hull–Strominger systems and non–Kähler Calabi–Yau metrics (Cerqueira-Gonçalves, 22 Dec 2025).

• Quaternionic Gauduchon Metrics: On compact SL(n,ℍ)-manifolds with hypercomplex structure, the quaternionic analogue seeks quaternionic Gauduchon forms Ω solving

$(1,1)$3

reduced to a fully nonlinear elliptic PDE for the potential. Key are L^\infty a priori estimates and existence in various geometric contexts (Zhang, 2023).

5. Prescribed Gauduchon Scalar Curvature Problem

The prescribed scalar curvature question is treated via conformal deformation of a background Gauduchon metric. Under conformal change $(1,1)$4, the Gauduchon scalar curvature transforms according to

$(1,1)$5

This gives rise to a Kazdan–Warner type PDE with exponential nonlinearity, where existence and uniqueness depend on the "Gauduchon degree." For negative degree, solvability is guaranteed under explicit sign conditions. Special cases recover the Chern– and Bismut–Yamabe problems (Li et al., 2022).

6. Gauduchon Connections and Rigidity Phenomena

The affine family of Gauduchon connections $(1,1)$6 interpolates between notable cases: Chern (r=1), Strominger (r=−1), and Lichnerowicz (r=0). The "Kähler-like" property for such a connection implies strong rigidity. It has been proved that if two distinct Gauduchon connections are both Kähler-like, or if a single such connection is Kähler-like for exceptional parameter values, then the metric is necessarily Kähler. This rigidity is partially explained by an involutive duality $(1,1)$7 pairing Gauduchon connections, with deep implications and open geometric questions (Zhao et al., 2021).

7. Impact and Outlook

The resolution and generalizations of the Gauduchon conjecture have elevated the role of nonlinear Monge–Ampère and Hessian equations in non-Kähler geometry, opening analytic and geometric methods for minimal model theory, moduli of Hermitian metrics, and the study of singularities in complex and quaternionic contexts. Uniform a priori estimates and continuity/parabolic techniques developed herein now underpin analytic approaches to a spectrum of Hermitian—especially non-Kähler—geometric problems and influence the landscape of canonical metric construction throughout higher-dimensional complex geometry.