---
title: Gauduchon Metrics in Complex Geometry
url: https://www.emergentmind.com/topics/gauduchon-metrics
type: topic
---

# Gauduchon Metrics in Complex Geometry

A Gauduchon metric is a Hermitian metric $\omega$ on a complex manifold $X$ of dimension $n$ such that the $(n-1,n-1)$-form $\omega^{n-1}$ is $\partial\bar\partial$-closed, i.e.,
\[
\partial\bar\partial(\omega^{n-1})=0.
\]
This structure is strictly weaker than Kähler or balanced metrics and is central to non-Kähler Hermitian geometry. Gauduchon metrics play foundational roles in the study of complex manifolds, especially in the presence of torsion, curvature, and Aeppli or Bott–Chern cohomology. Their existence, canonical properties, and associated complex Monge–Ampère theory have led to significant developments, including the solution of the classical Gauduchon conjecture and a rich deformation theory.

## 1. Core Definition and Existence Theory

Given a compact complex manifold $(X,J)$ of complex dimension $n \geq 2$ and a Hermitian metric $\omega$ (a smooth, positive definite $(1,1)$-form), $\omega$ is called a Gauduchon metric if
\[
\partial\bar\partial(\omega^{n-1})=0.
\]
This is equivalent to asserting that the Lee form $\theta$ (defined by $d(\omega^{n-1})=-\theta\wedge\omega^{n-1}$) is coclosed, $d^*\theta=0$ [1912.13147].

**Key facts:**
- On any compact complex manifold, every Hermitian metric is conformally equivalent to a unique (up to scale) Gauduchon metric [2411.05595, 1912.13147, 1407.5070].
- For $n\geq2$, the equation $\partial\bar\partial(e^{(n-1)f}\omega_0^{n-1})=0$ for a real function $f$ always has a smooth solution, producing a canonical Gauduchon representative in the conformal class [2411.05595, 1407.5070].

## 2. Cohomological Cones: Gauduchon, Aeppli, and Lee–Gauduchon

Gauduchon metrics are closely tied to Aeppli and Bott–Chern cohomology:
\[
H_A^{n-1,n-1}(X,\mathbb{R}) = \frac{\{\text{real }(n-1,n-1)\text{-forms } \alpha \mid \partial\bar\partial\alpha = 0\}}{\text{Im}\,\partial + \text{Im}\,\bar\partial}.
\]
The **Gauduchon cone** $G_X$ is defined as the set
\[
G_X = \{[\omega^{n-1}]_A \in H_A^{n-1,n-1}(X,\mathbb{R}) \mid \omega>0,\ \partial\bar\partial(\omega^{n-1})=0\},
\]
which is an open convex cone. Its closure is dual (under the intersection pairing) to the pseudo-effective cone of Bott–Chern positive $(1,1)$-classes [1407.5070].

The **Lee–Gauduchon cone** is defined via the closed real $(2n-1)$-form $d^c(\omega^{n-1})$, capturing the de Rham classes of $d^c(\omega^{n-1})$ for all Gauduchon metrics. This cone is bimeromorphically invariant and plays a role analogous to the Kähler cone in non-Kähler geometry [2411.05595].

## 3. Strongly Gauduchon, Generalized Gauduchon, and Related Metrics

**Strongly Gauduchon (sG) metrics** require that $\partial(\omega^{n-1})$ is $\bar\partial$-exact. The cohomological image of $[\omega^{n-1}]_A$ vanishes under a canonical map to $H_{BC}^{n,n-1}(X,\mathbb{C})$; the set of such classes is the strongly Gauduchon cone $\mathcal{SG}_X$ [1407.5070]. 

sGG manifolds are those where every Gauduchon metric is strongly Gauduchon, characterized by numerical criteria: $b_1=2h^{0,1}$ and $h_{BC}^{0,1}=h^{0,1}$ [1407.5070].

**Generalized $k$-th Gauduchon metrics** satisfy
\[
\partial\bar\partial(\omega^k)\wedge\omega^{n-k-1}=0
\]
for $1\leq k\leq n-1$. The classical Gauduchon case is $k=n-1$ [1010.2013, 1103.1033]. For each $k$, a conformal deformation theorem holds:
- For any compact Hermitian manifold, there exists a unique (up to scale) $k$-th Gauduchon representative in each conformal class if a certain conformal invariant $y_k(\omega)$ vanishes.
- The theory interpolates between SKT ($k=1$), Gauduchon ($k=n-1$), and astheno-Kähler ($k=n-2$) [1010.2013, 1103.1033].

Balanced ($d(\omega^{n-1})=0$) and SKT ($\partial\bar\partial\omega=0$) metrics are strictly stronger than Gauduchon; all such metrics are automatically Gauduchon [2503.02759].

## 4. Nonlinear Equations: Monge–Ampère, Parabolic Flows, and the Gauduchon Conjecture

### Monge–Ampère Equations and the Calabi–Yau–Gauduchon Theorem

The classical Gauduchon conjecture postulated the existence of Gauduchon metrics with prescribed volume forms (or Chern–Ricci forms). Székelyhidi–Tosatti–Weinkove proved this by solving a nonlinear elliptic equation of Monge–Ampère type:
\[
\omega^{n-1} = \omega_0^{n-1} + i\partial\bar\partial u \wedge \omega_0^{n-2} + \Re(i\partial u \wedge \bar\partial \omega_0^{n-2}),
\]
with $\omega^n = e^{F+b}\Omega$ and $\partial\bar\partial(\omega^{n-1})=0$; $u$ is unique (up to additive constants), and the metric $\omega$ is Gauduchon [1503.04491, 1310.6326, 2512.19830].

### Parabolic Flows

An alternative parabolic approach (Zheng) evolves a potential $u(t)$ by a Monge–Ampère-type flow:
\[
\frac{\partial u}{\partial t} = \log\frac{\left(\alpha + \frac{1}{n-1}[(\Delta_\alpha u)\alpha - i\partial\bar\partial u] + Z(u)\right)^n}{\alpha^n},
\]
with detailed a priori $C^0$, gradient, and higher estimates. The flow preserves the Gauduchon property and converges (after normalization) to the solution of the Gauduchon conjecture, thus providing a parabolic proof paralleling Cao's approach for the Kähler–Calabi–Yau problem [1609.07854].

### Continuity Path

The continuity equation for Gauduchon metrics generalizes the Kähler–Ricci flow/continuity method:
\[
\omega(s)^{n-1} = \omega_0^{n-1} - s(n-1)\left[\mathrm{Ric}(\omega(s))\wedge\omega(s)^{n-2} - \Re(i\partial\bar\partial\log\omega(s)^{n-2})\right].
\]
Solvability is guaranteed up to the maximal $s$ such that the right side remains positive-definite, extending the techniques of La Nave–Tian and Sherman–Weinkove beyond the Kähler context [2004.06335].

### Singular Settings

Recent work extends Gauduchon theory to compact Hermitian varieties with singularities, showing that a unique bounded Gauduchon metric exists in every conformal class on the regular locus, and providing uniform estimates in smoothing families [2106.06259, 2503.02759, 2512.19830].

## 5. Curvature, Functionals, and Variational Properties

### Curvature

The Gauduchon (or $t$-Gauduchon) connections interpolate between the Chern, Lichnerowicz, Bismut, and Minimal Hermitian connections. Ricci-flat and HSC-flat metrics are explored for various $t$, with monotonicity properties showing that the Chern connection yields maximal holomorphic sectional curvature among the Gauduchon line [2211.05973]. Explicit suspensions over Sasaki–Einstein manifolds yield $t$–Gauduchon Ricci-flat metrics for $t<1$.

### Variational Problems

The $L^2$-norm of the Lee form energy functional identifies balanced metrics as the only critical points (in all dimensions), while the $L^2$-norm of the full Chern torsion admits non-Kähler critical points, including Chern-flat and Strominger-parallel metrics. The variational approach highlights the energetic optimality of balanced and related Hermitian classes [2205.12480].

## 6. Cohomology, Deformation Theory, and Moduli

Gauduchon metrics interact intricately with Aeppli, Bott–Chern, and de Rham cohomology. Strongly Gauduchon properties, sGG conditions, and the shape of the Gauduchon cone are characterized both analytically and numerically (e.g., $b_1=2h^{0,1}$, $h_{BC}^{0,1}=h^{0,1}$) [1407.5070]. 

Stability properties are notable:
- Gauduchon metrics exist on arbitrary compact complex manifolds, and their existence is stable under small deformations and bimeromorphic modifications (including blow-ups and blow-downs) [1407.5070, 2503.02759, 1009.5408, 2106.06259].
- The convexity and duality properties of the Gauduchon, sGG, and Lee–Gauduchon cones are robust under deformation and modification [2411.05595, 1407.5070].

Moduli applications include the structure of the Gauduchon cone in families of Calabi–Yau $\partial\bar\partial$-manifolds and the study of Hermite–Einstein metrics and slope stability for sheaves on non-Kähler varieties [2503.02759].

## 7. Examples and Explicit Constructions

- On nilmanifolds, balanced and SKT metrics occur only in specific algebraic types; classification results identify cases with sGG but no balanced metric, or sGG without “superstrong” Gauduchon representatives [1407.5070, 1103.1033].
- Products of Sasakian manifolds, circle bundles over quasi-Sasakian bases, and principal torus bundles via the Swann twist yield broad families of k-th Gauduchon metrics, not necessarily Kähler or SKT [1103.1033].
- On Hopf and Inoue surfaces, Gauduchon metrics exist but may fail to be strongly Gauduchon due to the presence of d-exact positive currents [1009.5408].
- Singular examples, such as conifold transitions of Calabi–Yau threefolds, exhibit bounded Gauduchon metrics on the smooth locus, crucial for the generalized Hermite–Einstein correspondence [2503.02759].

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In summary, Gauduchon metrics are a fundamental and flexible class of Hermitian metrics intermediate in strength between balanced and general Hermitian structures. They admit a canonical representative in each conformal class, possess strong cohomological and curvature-theoretic properties, and serve as a robust analytic foundation for structure, deformation, and moduli theory in non-Kähler complex geometry. Their elliptic, parabolic, and continuity formulations open perspectives for further developments in curvature flows, singular geometries, and stability phenomena.

Source: https://www.emergentmind.com/topics/gauduchon-metrics