Gauduchon Case in Complex Geometry

• Gauduchon metrics are Hermitian metrics on compact complex manifolds satisfying ∂∂¯(ω^(n-1))=0, ensuring a unique conformal representative in each class.
• The one-parameter family of Gauduchon connections bridges the Chern and Bismut connections, with rigidity results that force Kähler metrics outside special parameter cases.
• The theory extends to singular varieties and cohomological frameworks, underpinning stability, metric flows, and moduli construction via the Gauduchon cone.

The Gauduchon case in complex geometry refers to the structure theory, existence problem, and curvature properties of Hermitian metrics on compact complex manifolds whose ^^^^0^^^^ $(1,1)$-form $\omega$ satisfies the $(n-1)$-pluriclosed condition $\partial\bar\partial(\omega^{n-1})=0$. These Gauduchon metrics, introduced by Paul Gauduchon in the 1970s, play a central role in non-Kähler geometry. Their existence in every conformal class, deep relations to Aeppli and Bott–Chern cohomology, occurrence in special Hermitian connections, and importance in metric flows and moduli problems make the Gauduchon case a keystone of modern Hermitian theory.

1. Definition and Existence of Gauduchon Metrics

On a compact complex manifold $(X^n,J)$, a Hermitian metric $\omega$ is called a Gauduchon metric if its $(n-1)$st power satisfies

$\partial\bar\partial\omega^{n-1} = 0 .$

Gauduchon's theorem asserts that every Hermitian metric $\omega_0$ is conformally equivalent to a unique (up to scaling) Gauduchon metric in its conformal class: there exists a real function $f\in C^\infty(X)$ so that $e^f\omega_0$ is Gauduchon. This is shown by solving a linear elliptic $\partial\bar\partial$-equation for $f$, ensured by the maximum principle on compact manifolds (Ornea et al., 2024).

For varieties with singularities and even for compact normal complex spaces, similar results hold: Pan has established the existence of bounded Gauduchon metrics on arbitrary compact Hermitian varieties, appealing to log-resolution, uniform Sobolev inequalities, and Harnack-type estimates to produce the desired metric in the regular part and demonstrate its conformal uniqueness (Pan, 4 Mar 2025).

2. Gauduchon Connections: The One-Parameter Family

A defining feature of Hermitian geometry is the existence of a distinguished affine line of canonical Hermitian connections on the holomorphic tangent bundle, parametrized by $t\in\mathbb R$. For Hermitian $(X^n,g)$ with Chern connection $\nabla^c$ and Bismut connection $\nabla^b$, the $s$-Gauduchon connection is defined as

$$\nabla^s = (1-\tfrac{s}{2})\nabla^c + \tfrac{s}{2}\nabla^b,$$

or, equivalently, with $t=s/2$ as $\nabla^t = (1-t)\nabla^c + t\nabla^b$ (Yang et al., 2017, Lafuente et al., 2022).

Special values include:

• $s=0$ ($t=0$): Chern connection,
• $s=2$ ($t=1$): Bismut connection,
• $s=1$ ($t=1/2$): "associated" or first canonical connection.

These connections capture the full Hermitian holonomy range, interpolate between Chern and Bismut, and, in the Kähler case, all coincide. The curvature and torsion of $\nabla^s$ depend nonlinearly on $s$ except in the Kähler situation.

3. Rigidity, Flatness, and Kähler-Like Phenomena

A central problem is to classify Hermitian metrics whose $s$-Gauduchon connection is flat or Kähler-like (curvature satisfies the same symmetries as Kähler geometry). The breakthrough results are:

• Rigidity outside special parameters: If a compact Hermitian manifold $(M^n,g)$ has flat $s$-Gauduchon connection for $s\neq 0,2$, then $g$ is necessarily Kähler (Yang et al., 2017, Lafuente et al., 2022). Similarly, if $\nabla^s$ is Kähler-like for $s\neq0,2$, then $g$ must be Kähler, settling a conjecture of Angella–Otal–Ugarte–Villacampa.
• Classification for special cases:
  • $s=0$ (flat Chern): $g$ is complex parallelizable, a finite quotient of a complex Lie group (Yang et al., 2017).
  • $s=2$ (flat Bismut): $g$ is covered by an open subset of a "Samelson space" (compact Lie group with bi-invariant metric and left-invariant complex structure).
  • On Hermitian surfaces, if $g$ admits flat $\nabla^s$ or constant holomorphic sectional curvature for $\nabla^t$ (the $t$-Gauduchon connection), then $g$ is Kähler unless $s=2$ or $t \in \{-1,3\}$, in which case $g$ is the standard metric on an isosceles Hopf surface (Chen et al., 2022).

An interesting duality emerges: if two distinct $t$-Gauduchon connections have equal holomorphic sectional curvature, then either the metric is Kähler or the parameters $t$ and $s$ are related by $s=2-t$; this rigidity manifests on surfaces and more generally (Broder et al., 2022).

4. Aeppli Cohomology and the Gauduchon Cone

Gauduchon metrics naturally define classes in Aeppli cohomology, $H^{n-1,n-1}_A(X,\mathbb R)$. The Gauduchon cone $\mathcal G_X$ is the set

$$\mathcal G_X := \{\, [\omega^{n-1}]_A : \omega \text{ Gauduchon metric} \,\} .$$

$\mathcal G_X$ is an open, convex cone in $H^{n-1,n-1}_A(X,\mathbb R)$, nonempty and, on $\partial\bar\partial$-manifolds, filling the space (Popovici, 2013). If $[\omega^{n-1}]_A = 0$, strong vanishing results ensue for holomorphic forms (Piovani et al., 2019). The Monge–Ampère–type equation in bidegree $(n-1, n-1)$ admits, under positivity and normalization, a unique solution realizing a distinguished Gauduchon metric in any Aeppli class in the cone, providing potential for moduli constructions and deformation theory (Popovici, 2013). The deformation behavior of this cone is lower semicontinuous under smooth deformations.

5. Analytic and Flow Approaches: The Gauduchon Conjecture

The Gauduchon conjecture, inspired by Yau's resolution of the Calabi conjecture, posited the existence of Gauduchon metrics with prescribed volume or Chern–Ricci curvature within a given Aeppli class. The solution, completed by Székelyhidi–Tosatti–Weinkove, involves solving the non-Kähler complex Monge–Ampère equation

$$(\omega + i\partial\bar\partial u)^{n} = e^{F+b}\omega^{n}$$

with $\omega + i\partial\bar\partial u$ Gauduchon and appropriate normalization (Székelyhidi et al., 2015). The proof combines the continuity method with new a priori $C^0$, $C^2$, and Evans–Krylov–type estimates adapted to non-Kähler settings.

A parabolic proof via a Gauduchon-Monge–Ampère flow was established by Zheng: starting with an initial metric, the flow

$$\partial_t u = \log \frac{\omega_u^n}{\alpha^n} - \psi$$

exists for all time and converges (modulo normalization) to the unique solution of the elliptic problem, paralleling Ricci flow methods for Calabi–Yau metrics (Zheng, 2016).

A related dynamical perspective involves the Gauduchon continuity equation, evolving the $(n-1,n-1)$-form class to maximize positivity. The scalar reduction is again a fully nonlinear Monge–Ampère–type PDE, and the interval of existence is governed by positivity of a Bott–Chern–Aeppli class (Zheng, 2020).

6. Singular Varieties, Slope Stability, and Strongly Gauduchon Metrics

Recent advances have extended Gauduchon's theory to singular Hermitian varieties. Every compact irreducible normal Hermitian variety admits a bounded Gauduchon metric conformally related to any background Hermitian form (Pan, 4 Mar 2025). These metrics enable the definition of degrees and slopes for torsion-free sheaves, underpinning a full Donaldson–Uhlenbeck–Yau theory in the non-Kähler, singular context: existence and uniqueness of singular Hermite–Einstein metrics on slope-stable reflexive sheaves (Pan, 4 Mar 2025).

A stricter notion, strongly Gauduchon (sG) metrics, requires that $\partial\omega^{n-1}$ be $d$-exact. Mirroring the balanced case, sG metrics have desirable stability properties under modifications, are birationally invariant, and exist in great generality, even on singular spaces with appropriate cohomological conditions (Popovici, 2010, Meng et al., 2016).

7. Cohomological Cones and Further Curvature Geometry

The Lee–Gauduchon cone is the set of cohomology classes of closed $(2n-1)$-forms $d^c(\omega^{n-1})$ for Gauduchon metrics $\omega$ on a compact complex $n$-manifold. This cone is open, convex, bimeromorphic invariant, and dual (in a suitable sense) to the pseudo-effective cone of closed positive $(1,1)$-currents (Ornea et al., 2024). Its computation in classes of non-Kähler manifolds (e.g., Oeljeklaus–Toma or LCK manifolds) reveals the diversity and subtlety of the Gauduchon case in the context of Hermitian and non-Kähler geometry.

A key analytic feature of the Gauduchon family is the monotonicity of holomorphic sectional curvature: for any Hermitian metric, the $t$-Gauduchon holomorphic sectional curvature is maximized at the Chern case ($t=1$) and strictly lower otherwise, with rigidity phenomena enforcing Kählerness if two such holomorphic sectional curvatures are equal (Broder et al., 2022). This leads to sharp classification results for special geometric structures on complex manifolds.

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Table: Milestones in the Gauduchon Theory

Theme	Key Result(s)	References
Existence	Unique Gauduchon metric in each conformal class (smooth and singular)	(Ornea et al., 2024, Pan, 4 Mar 2025)
Flatness and Rigidity	Flat Gauduchon connection $\implies$ Kähler except for Chern/Bismut cases	(Yang et al., 2017, Lafuente et al., 2022)
Monge–Ampère Methods	Solving Gauduchon metric with prescribed volume/Chern–Ricci	(Székelyhidi et al., 2015, Zheng, 2016)
Singular Varieties	Bounded Gauduchon metrics on normal complex spaces, Hermite–Einstein metrics	(Pan, 4 Mar 2025)
Cohomological Structure	Open convex Gauduchon cone in Aeppli cohomology, invariance under deformations	(Popovici, 2013, Piovani et al., 2019)
Metric Flows	Existence, convergence, and continuity equation for Gauduchon metrics	(Zheng, 2016, Zheng, 2020)
Curvature Phenomena	Monotonicity and rigidity in holomorphic sectional curvature	(Broder et al., 2022, Chen et al., 2022)

The Gauduchon case thus provides a canonical and unifying framework for non-Kähler Hermitian geometry, cohomological theory, moduli, flow approaches, and stability problems, with deep ties to both analytic and algebro-geometric structures.