Gauduchon Cone in Complex Geometry

• Gauduchon cone is an open convex cone in H^(n-1,n-1)_A(X,ℝ) defined by the Aeppli classes of Gauduchon metrics on compact complex manifolds.
• It is dual to the pseudo-effective Bott–Chern cone, establishing a critical bridge between positive currents and the analytic theory of Hermitian metrics.
• The Gauduchon cone plays a pivotal role in deformation theory, bimeromorphic invariance, and constructing canonical metrics within non-Kähler geometry.

The Gauduchon cone is a central object in non-Kähler complex geometry, parameterizing Aeppli cohomology classes of bidegree $(n-1,n-1)$ on a compact complex $n$-manifold that admit positive, $\partial\bar\partial$-closed representatives; that is, representatives arising from Gauduchon metrics. The study of the Gauduchon cone and its variants provides a cohomological framework for understanding Hermitian metrics lying between the Kähler and general Hermitian settings. It also plays a crucial role in deformation theory, bimeromorphic geometry, and the analytic theory of canonical metrics.

1. Definitions and Basic Structures

Let $X$ be a compact complex manifold of dimension $n$. The Aeppli cohomology group of bidegree $(n-1,n-1)$ is

$$H^{n-1,n-1}_A(X,\R) = \frac{\ker\bigl(\partial\bar\partial\colon C^\infty_{n-1,n-1}(X)\to C^\infty_{n,n}(X)\bigr)} {\Im(\partial) + \Im(\bar\partial)}.$$

A Hermitian metric $\omega>0$ is Gauduchon if

$\partial\bar\partial\bigl(\omega^{n-1}\bigr) = 0,$

so the power $\omega^{n-1}$ determines a well-defined Aeppli class $n$0. The Gauduchon cone is then defined by

$n$1

Geometrically, $n$2 is the image in cohomology of all real, positive, $n$3-closed $n$4-forms arising from Gauduchon metrics (Popovici, 2013, Popovici et al., 2014, Ornea et al., 2024).

2. Convexity, Openness, and Duality Properties

The Gauduchon cone $n$5 is an open convex cone in the real vector space $n$6. Convexity follows since any positive linear combination of Gauduchon $n$7-forms remains positive and $n$8-closed; openness follows from the openness of positivity in the relevant topology (Popovici, 2013, Ornea et al., 2024).

There is a canonical, nondegenerate duality pairing

$n$9

where $\partial\bar\partial$0 is the Bott–Chern cohomology group. Under this pairing, the pseudo-effective cone of Bott–Chern classes of positive closed $\partial\bar\partial$1-currents,

$\partial\bar\partial$2

has as its dual the Gauduchon cone: $\partial\bar\partial$3 (Popovici, 2013, Ornea et al., 2024). The Bott–Chern Kähler cone, consisting of $\partial\bar\partial$4 with $\partial\bar\partial$5 and $\partial\bar\partial$6, embeds in the dual cone. These dualities tie the Gauduchon cone structurally to both positive current theory and the geometry of special Hermitian classes.

3. Deformation, Bimeromorphic, and Semicontinuity Aspects

The Gauduchon cone behaves in a controlled, lower-semicontinuous fashion under holomorphic deformations. If $\partial\bar\partial$7 is a proper holomorphic submersion and the central fibre $\partial\bar\partial$8 satisfies the $\partial\bar\partial$9-lemma, the Aeppli groups vary smoothly and

$X$0

where $X$1 is the canonical identification of Aeppli groups (Popovici, 2013, Popovici et al., 2014). The cone can only "shrink" in the limit. Closely related, the property of being "sGG" (manifolds where every Gauduchon metric is strongly Gauduchon) is stable under small deformations, but not necessarily closed in the parameter.

Moreover, the Gauduchon cone enjoys bimeromorphic invariance in multiple guises. In particular, it is stable under proper modifications and bimeromorphic maps—such as blowups—preserving the codimension conditions on exceptional loci (Popovici et al., 2014). Analogous statements hold for the Lee–Gauduchon cone in high-degree de Rham cohomology (Ornea et al., 2024).

4. The Small Gauduchon Cone and sGG Manifolds

A compact complex manifold $X$2 is called sGG if every Gauduchon metric is strongly Gauduchon, i.e., the Gauduchon and strongly Gauduchon cones coincide: $X$3 where

$X$4

and the strongly Gauduchon condition is

$X$5

Several numerical characterizations of sGG manifolds are available (Popovici et al., 2014):

• $X$6
• $X$7
• The injectivity and surjectivity of canonical maps between de Rham, Dolbeault, and Aeppli cohomology

Prominent examples include the Iwasawa manifold and its small deformations, which are sGG yet not $X$8-manifolds. The sGG property is bimeromorphically invariant but may not be preserved in the limit of a family.

Property	Criterion / Example	Reference
$X$9	sGG manifolds; see Iwasawa manifold	(Popovici et al., 2014)
$n$0	Each Gauduchon $n$1 strongly Gauduchon	(Popovici et al., 2014)
$n$2	sGG classification for nilmanifolds	(Popovici et al., 2014)

5. The Gauduchon Cone in Locally Conformally Kähler and Other Settings

For a locally conformally Kähler (LCK) manifold $n$3 with $n$4, every conformal class of Hermitian metrics contains a unique Gauduchon metric (Ornea et al., 2024). The Gauduchon cone $n$5 remains an open convex cone in $n$6. It plays an essential role in the duality with pseudo-effective Bott–Chern classes and with the “Lee cone” in $n$7. For various classes of LCK manifolds (Vaisman, Oeljeklaus–Toma, Kato), the Gauduchon cone lies strictly in the interior of the dual to the cone generated by $n$8, which is pseudo-effective. This precludes the existence of balanced metrics on such manifolds—pairing with a balanced metric would force vanishing, but strict positivity instead holds (Ornea et al., 2024).

A related object, the Lee–Gauduchon cone in top-degree de Rham cohomology, arises from the class $n$9, for which strong convexity and bimeromorphic invariance hold. In particular, balanced manifolds always have a trivial Lee–Gauduchon cone (Ornea et al., 2024).

6. Analytical and Metric-Theoretic Aspects

The Gauduchon cone is closely linked to canonical metrics and PDE theory. Popovici (Popovici, 2013) proposes a Monge–Ampère–type equation in bidegree $(n-1,n-1)$0 prescribing volume forms for metrics representing a given Gauduchon class. In the Kähler case, this specializes to the balanced Monge–Ampère equation. Uniqueness (with positivity normalization) is established for solutions within an Aeppli class; existence in the non-Kähler setting remains open. This analytic perspective enables the construction of canonical Gauduchon metrics attached to given classes and underpins applications to moduli of Calabi–Yau $(n-1,n-1)$1-manifolds.

Generalized Gauduchon cones for $(n-1,n-1)$2-Gauduchon metrics, parameterized by sign conditions of associated $(n-1,n-1)$3 functionals, have been introduced. The classical ($(n-1,n-1)$4) Gauduchon cone is dual to the balanced cone (Fu et al., 2010).

7. Examples and Applications

Fundamental Examples

• Balanced manifolds: The balanced cone of $(n-1,n-1)$5st powers of balanced Hermitian forms sits inside the Gauduchon cone.
• Surfaces: On compact complex surfaces, the Gauduchon, Kähler, and balanced cones coincide due to the result of Lamari.
• Nilmanifolds: The Iwasawa manifold exemplifies an sGG manifold not admitting a $(n-1,n-1)$6-structure.

Applications

• Deformation theory of complex manifolds, via the semicontinuity and stratification of the Gauduchon cone (Popovici, 2013, Popovici et al., 2014).
• Moduli of Calabi–Yau $(n-1,n-1)$7-manifolds and canonical metric construction within Aeppli classes (Popovici, 2013).
• Bimeromorphic geometry and invariance phenomena for Hermitian and cohomological invariants (Popovici et al., 2014, Ornea et al., 2024).

A plausible implication is that a more complete understanding of the Gauduchon cone, its analytic and topological structure, and its interplay with other geometric cones, is integral for advancing the theory of non-Kähler metrics, canonical representatives in cohomology, and the birational classification of complex manifolds.