Gauduchon Connections in Hermitian Geometry

• Gauduchon Connections are a one-parameter family of Hermitian connections that interpolate between the Chern and Bismut connections, preserving both metric and complex structure.
• They reveal key curvature and torsion properties in non-Kähler manifolds, leading to rigidity results that imply Kähler metrics under certain parameter conditions.
• Applications include classifying flat and Kähler-like connection cases on complex surfaces and Lie groups, which deepens our understanding of Hermitian geometry.

A Gauduchon connection is a distinguished one-parameter family of Hermitian connections $\nabla^s$ on a complex manifold with Hermitian metric, interpolating between the Chern and the Bismut (Strominger) connections. These connections play a central role in complex geometry, especially in the study of special Hermitian metrics and their curvature properties on non-Kähler manifolds. The structure and classification of manifolds admitting flat or Kähler-like Gauduchon connections has been an area of active research, with deep links to questions in differential geometry, representation theory, and complex analysis.

1. Definition and Fundamental Structure

Let $(M^n, g, J)$ be a Hermitian manifold of complex dimension $n$; $J$ is the complex structure and $g$ the Hermitian metric. The Levi–Civita connection $\nabla$ of $g$ need not preserve $J$, so the Hermitian connections—connections compatible with both $g$ and $J$—form an affine space. Two prominent canonical Hermitian connections are:

• The Chern connection $(M^n, g, J)$0, uniquely determined by
  • $(M^n, g, J)$1,
  • $(M^n, g, J)$2,
  • Torsion $(M^n, g, J)$3 of type $(M^n, g, J)$4.
• The Bismut (Strominger) connection $(M^n, g, J)$5, defined by
  • $(M^n, g, J)$6,
  • $(M^n, g, J)$7,
  • Torsion $(M^n, g, J)$8 totally skew-symmetric; $(M^n, g, J)$9 for fundamental form $n$0.

The Gauduchon connections form a real one-parameter family interpolating between $n$1 and $n$2: $n$3 Special cases:

• $n$4: Chern connection,
• $n$5: Bismut connection,
• $n$6: first canonical (Lichnerowicz) Hermitian connection.

Each $n$7 is metric and complex compatible, with torsion

$n$8

and possesses explicit local expressions in a unitary frame (Yang et al., 2017, Lafuente et al., 2022, Broder et al., 2022).

2. Curvature, Torsion, and Kähler-like Properties

For any Hermitian connection $n$9, the curvature tensor is defined in the standard way: $J$0 However, in general, $J$1 is not torsion-free, and $J$2 does not satisfy the Riemannian first Bianchi identity except in special geometric circumstances.

A connection $J$3 is said to be Kähler-like if its curvature satisfies both the first Bianchi identity and the complex symmetries characteristic of Kähler or Chern curvature: $J$4 If $J$5 is Kähler-like or even flat ($J$6), strong restrictions on the underlying metric and manifold structure ensue (Angella et al., 2018, Zhao et al., 2021, Lafuente et al., 2022).

3. Rigidity and Classification Results

Compact Manifolds

A central development is the rigidity of compact Hermitian manifolds with flat Gauduchon connections. Precisely, apart from the two extremal cases $J$7 (Chern) and $J$8 (Bismut), any compact Hermitian manifold with flat $J$9 is necessarily Kähler:

• Boothby-Wang: Compact Chern-flat Hermitian manifolds are finite quotients of complex Lie groups with left-invariant Hermitian metrics (Yang et al., 2017).
• Wang–Yang–Zheng (Bismut-flat case): Compact Bismut-flat Hermitian manifolds have universal covers of the form $g$0, $g$1 a simply-connected compact semisimple Lie group with bi-invariant metric and left-invariant complex structure (Yang et al., 2017).

These claims are formalized: $g$2 This is established via integral identities involving the torsion $g$3 and its trace $g$4, leading to a universal Kähler-rigidity outside the interval $g$5 (Yang et al., 2017, Lafuente et al., 2022).

Surfaces and Dimension-Dependence

For compact Hermitian surfaces $g$6, the only non-Kähler, non-flat examples with flat $g$7 arise for $g$8 (Bismut), realized on isosceles Hopf surfaces. For all other $g$9, the metric is flat Kähler (e.g., complex tori, hyperelliptic surfaces) (Yang et al., 2017, Chen et al., 2022).

When the Gauduchon connection is Kähler-like (not necessarily flat), similar rigidity results hold. For $\nabla$0, any compact Hermitian manifold with Kähler-like $\nabla$1 is Kähler (Lafuente et al., 2022, Zhao et al., 2021); this confirms conjectures by Angella, Otal, Ugarte, and Villacampa, as well as Yang and Zheng.

4. Holomorphic Sectional Curvature and Cohomological Aspects

The holomorphic sectional curvature (HSC) for $\nabla$2 is defined as

$\nabla$3

where $\nabla$4 is a nonzero (1,0)-vector. A monotonicity principle holds: $\nabla$5 If a compact Hermitian surface has pointwise constant $\nabla$6, then either the metric is Kähler, or the manifold is an isosceles Hopf surface with admissible metric, realized at $\nabla$7 or $\nabla$8 (the Bismut or minimal connection) (Chen et al., 2022, Broder et al., 2022).

Cohomologically, the first Ricci form of $\nabla$9 is a deformation of the Chern-Ricci form: $g$0 yielding $g$1-Ricci-flat Hermitian metrics on non-Kähler Calabi–Yau type manifolds (Broder et al., 2022).

5. Realizations on Lie Groups and Homogeneous Spaces

For a Lie group $g$2 with left-invariant (almost) Hermitian structure, left-invariant Hermitian connections correspond to elements in the space

$g$3

The Gauduchon family of connections retains this identification, and explicit formulas for torsion and curvature are available in terms of Lie algebra structure constants. On $g$4 (with $g$5 compact, $g$6 abelian, and $g$7 totally real on $g$8), the Bismut connection may coincide with the trivial connection, SKT condition can be realized, and further flat Gauduchon connections can exist at special parameter values (Pham et al., 2023, Vezzoni et al., 2018).

In the left-invariant context, classification results mirror the compact case. For real dimension 4, or in the existence of a $g$9-parallel invariant frame, left-invariant Hermitian metrics with flat $J$0 force $J$1 to be Kähler for $J$2 (Vezzoni et al., 2018).

6. Methodologies and Invariant Identities

Key technical tools involve structure equations under local $J$3-parallel unitary frames and pointwise or integral identities relating torsion and its trace to curvature. The integral identity

$J$4

plays a central role, especially upon integration over compact manifolds. In dimension two, Bochner-type arguments on logarithmic functions of torsion components complete rigidity proofs (Yang et al., 2017).

Beyond flatness, the concept of Kähler-likeness (algebraic curvature identities) allows application of maximum principles and the trace constraints to extend rigidity beyond the cases accessible to direct curvature vanishing (Lafuente et al., 2022, Zhao et al., 2021).

7. Dualities, Open Problems, and Research Directions

A duality phenomenon among Gauduchon connections is encoded by the transformation $J$5; self-dual points are $J$6 (Lichnerowicz) and $J$7 (Chern). It is shown that two Gauduchon connections can have equal holomorphic sectional curvature if and only if $J$8, $J$9, or the metric is Kähler (Broder et al., 2022).

Open research problems include:

• Complete classification of noncompact Hermitian manifolds with flat $g$0 outside the compact setting,
• Clarification of geometric and physical origins of the duality among the Gauduchon parameters,
• Further investigation of special Hermitian connections in relation to pluriclosed flow, SKT metrics, and moduli of canonical connections (Yang et al., 2017, Zhao et al., 2021).

The growing taxonomy and structure theory of Gauduchon connections elucidate the internal organization of Hermitian geometry, bridging the complexities of Kähler and non-Kähler manifolds, and deepening the interface with Lie theory, cohomology, and geometric analysis.