In complex differential geometry, connections characterize the local properties of a manifold and determine its torsion, curvature, and global topological invariants. Therefore, the selection of a connection is important for studying the intrinsic geometric structure and analytic properties of the manifold. In general, non-Kähler geometry, there are two important classes of connections: the Chern connection and the Bismut connection.

If a complex manifold is endowed with a Hermitian metric, it may be viewed as a Riemannian manifold, and the Levi-Civita connection can be introduced accordingly. However, on a non-Kähler manifold, the Levi-Civita connection fails to preserve the complex structure. Therefore, on non-Kähler manifolds, the Chern connection and the Bismut connection are commonly considered.

The torsion of a connection plays an important role in geometry. It also gives several significant problems in non-Kähler geometry, such as the balanced condition, the pluriclosed condition, Gauduchon metrics, conformal invariants of the connection 1-form, and variational problems for critical metrics. In particular, the connection 1-form provides a method to determine whether a Hermitian metric is balanced or conformally balanced, and for a Gauduchon metric, the corresponding connection 1-form is co-closed [1]. Ganchev and Ivanov investigated the role of the connection 1-form on the existence and holomorphicity of harmonic 1-forms on compact balanced manifolds [2].

Prompted by problems in theoretical physics, Li and Yau provided a rigorous mathematical proof for the existence of solutions to the Strominger system, showing that certain non-Kähler manifolds admit torsion structures satisfying supersymmetry conditions [3]. In recent years, the Bismut connection and its curvature properties have attracted considerable attention. Andrada and Villacampa studied the Bismut connection on Vaisman manifolds and the related geometric structures, and characterized the corresponding holonomy group structures [4]. Barbaro established that, under appropriate conditions, a Hermitian metric with constant Bismut scalar curvature exists [5].

An n -dimensional complex manifold M is a smooth manifold of real dimension 2 n together with a holomorphic atlas { ( U k , ψ k ) } of ψ k : U k → ℂ n and ψ k ( U k ) ≃ U k . The transition map ψ j ∘ ψ k − 1 : ψ k ( U j ∩ U k ) → ψ j ( U j ∩ U k ) is holomorphic.

Let M be an n -dimensional complex manifold with tangent bundle T ℂ M . Since ℂ n ≃ ℝ 2 n and any biholomorphic map is necessarily a diffeomorphism, M can also be considered as a smooth real manifold of real dimension 2 n , which we denote by M ℝ . The complex structure on the real manifold induces a decomposition of the complexified tangent bundle

It endows T p M ℝ with the structure of a complex vector space. This decomposition induces a corresponding decomposition of complex-valued differential form and leads naturally to ( p , q ) -form.

By extending the complex structure to T ℂ M , we obtain

where T 1 , 0 M and T 0 , 1 M denote the holomorphic and anti-holomorphic tangent spaces. Let ( z 1 , ⋯ , z n ) be local holomorphic coordinates on an open set U ⊂ M , with z k = x k + i y k . Let ( x 1 , ⋯ , x n , y 1 , ⋯ , y n ) be smooth real local coordinates, and denote

Then local frames for T 1 , 0 M and T 0 , 1 M are { ∂ ∂ z k } and { ∂ ∂ z ¯ k } , respectively. The dual space of the tangent bundle is denoted by T ℂ * M , and it likewise decomposes as

Local frames for T * 1 , 0 M and T * 0 , 1 M are given by { d z k } and { d z ¯ k } .

On a complex manifold M , a k -form ∧ k T ℂ * M can be locally decomposed as

which is denoted as

We define Ω p , q ( M ) to be the space of ( p , q ) -forms on M . Any element φ ∈ Ω p , q ( M ) can be locally expressed as

The exterior differential d defined on a smooth manifold naturally extends to an n -dimensional complex manifold, where it satisfies

Consequently, we can define two differential operators on Ω p , q ( M )

Locally, they have the expressions

Since the operator ∂ ¯ satisfies ∂ ¯ 2 = 0 , it defines a cochain complex

The cohomology of the sequence is called the Dolbeault cohomology:

In particular, there exists the following isomorphism

where O M is the sheaf of holomorphic functions on M . These groups measure the obstruction to the ∂ ¯ -equation solved on ( p , q ) -forms and constitute fundamental invariants of the complex manifold M .

Consider the holomorphic tangent bundle T 1 , 0 M of a complex manifold M . For any x ∈ M , a positive definite Hermitian form h is given by

satisfying

for all Y , Z ∈ T x 1 , 0 M . In a local frame { ∂ ∂ z k } , let the components of h by

where h k l is a smooth function on M . Let the local holomorphic coordinates be ( z 1 , ⋯ , z n ) and the Hermitian metric can be expressed locally as

where h k l ¯ is a smooth section of T * ⊗ T ¯ * , satisfying

1) Hermitian symmetry, h k l ¯ = h l k ¯ ¯ ;

2) For any nonzero vector ( γ 1 , ⋯ γ n ) , h k l ¯ γ k γ ¯ l ≥ 0 .

A Hermitian metric can be defined locally, for example by pulling back the standard Hermitian metric from ℂ n . These local metrics can be patched together via a partition of unity subordinate to the atlas, thereby obtaining a global Hermitian metric.

Theorem3.1. [6] Any n -dimensional complex manifold M admits a Hermitian metric.

This fundamental fact shows that Hermitian geometry is always available on complex manifolds.

The Kähler form ω with respect to h has the following local coordinate form

Remark. If d ω = 0 , then M is called a Kähler manifold, and any submanifold of M is also a Kähler manifold.

Consider the set of complex lines through the origin on ℂ n + 1 . The complex projective space is defined as

where z , w ∈ ℂ n + 1 are equivalent, z ~ w , if there exists γ ∈ ℂ * such that w = γ z . Elements of ℙ n are denoted by homogeneous coordinates [ z 0 , z 1 , ⋯ , z n ] . The projective space can be covered by n + 1 standard open sets

On U k , we define affine coordinates

Consider the map on U k

The transition map is

Hence, ℙ n is a complex manifold.

Proposition3.2. [6] The complex projective space is a Kähler manifold.

The projective manifold is a class of complex manifolds, and it is shown in [7] that compact Kähler manifolds with positive bisectional curvature are projective.

Similarly, we equip Ω p , q ( M ) with a Hermitian metric. On a compact Hermitian manifold for any φ , ϕ ∈ Ω p , q ( M ) , we define an inner product at each point x ∈ M

The metric for φ and ϕ is given by

where d V = ω n n ! , we denote ( φ , φ ) = ‖ φ ‖ 2 .

A complex manifold is automatically orientable. Intuitively, this property arises from the fact that holomorphic coordinate changes preserve the natural orientation of the underlying real 2n-dimensional space. It distinguishes complex manifolds from smooth manifolds that fail to be orientable.

On an n -dimensional Hermitian manifold M , there exists a linear map ∗ : Ω k ( M ) → Ω 2 n − k ( M ) satisfying the following

for φ , ϕ ∈ Ω k ( M ) . By the ℂ -linear extension, it satisfies

for φ , ϕ ∈ Ω p , q . Let M be a compact manifold. Based on (2.6), (3.4) and (3.5), the dual operator is defined as

It is easy to see that

A special class of Hermitian manifolds is the balanced Hermitian manifolds, i.e., Hermitian manifolds with coclosed Kähler forms. Moreover, every Kähler manifold is balanced. In higher dimensions, there exist non-Kähler manifolds which admit balanced Hermitian metrics [8].

The Chern connection ∇ is the unique connection which is compatible with the complex structure and also the Hermitian metric.

Let ∇ be a Chern connection. The torsion tensor of ∇ is defined as

It is easy to see that T is a (1,2)-tensor, with components T α β γ defined by

denoted by h γ k ¯ T i j γ = T i j k ¯ .

Proposition4.1. Let M be a Hermitian manifold. The following equation between the Kähler form and the torsion coefficients holds:

Proof. By (2.7) and (3.3), we obtain

From h γ k ¯ T i j γ = T i j k ¯ , it gives

This completes the proof.

It follows from (4.2) that if ∇ is torsion-free, then M is a Kähler manifold. The connection 1-form on M is given by

where σ α = h μ k ¯ T α μ k ¯ . In particular, by choosing a unitary frame { e k } , we have

The linear map L is called the Lefschetz operator is defined by

Its dual is given by

Proposition4.2. Let M be an n -dimensional Hermitian manifold with σ ∂ -closed. Then

where ⊡ = − ∗ ∂ Λ ∗ . If M is compact, we have

Proof. we denote

By the dual operator Λ , it follows that

Thus

Substituting into (4.6), we can easily obtain

Using the Leibniz property of ∂ and the duality of L , it gives

which proves (4.4). From (3.6), we get

By Hermitian symmetry and the duality of L , we have

In particular, we have

The second equality of (4.8) follows from integration by parts. Therefore, (4.5) can be derived from (4.7) and (4.8).

The metric ω is called an m -Gauduchon metric for 1 ≤ m ≤ n − 1 if it satisfies

A classical result of Gauduchon states that within any conformal class of a Hermitian metric on a compact complex manifold, there exists a Gauduchon metric, which is unique up to normalization. By Proposition 4.2, we have the following corollary.

Corollary4.3. Let M be an n -dimensional Hermitian manifold with σ ∂ -closed. If ω is an ( n − 2 ) -Gauduchon metric, we have

Proof. By integration by parts, it follows that

Under the Gauduchon condition (4.5) vanishes, which proves (4.9).

In general, the fundamental 2-form ω is not necessarily closed, and the extent to which the manifold deviates from the Kähler condition can be described by the connection 1-form. Moreover, it is closely related to the topology of Hermitian manifolds, and we refer to [9].

(4.9) presents a quantitative relation between σ and ∂ ¯ ω , providing a tool to verify whether a metric satisfies the balanced or pluriclosed condition. If ‖ σ ‖ 2 = 0 , the manifold satisfies the pluriclosed condition; if ( ∂ ¯ σ , ω ) = 0 , the manifold is balanced. (4.9) holds under the ( n − 2 ) -Gauduchon condition, but the validity of (4.9) does not necessarily imply the ( n − 2 ) -Gauduchon condition. Consider the left-hand side of (4.5). If ∂ ω n − 2 ∧ ∂ ¯ ω = 0 , then ω is an ( n − 1 ) -Gauduchon metric if and only if it is a 1-Gauduchon metric.

‖ σ ‖ 2 describes the length of σ , and ‖ ∂ ¯ ω ‖ 2 is defined analogously. ( ∂ ¯ σ , ω ) measures the component of ∂ ¯ σ in the direction of ω . The integral on the left-hand side of (4.5) characterizes the extent to which the manifold deviates from the Kähler condition. Moreover, the study of higher-degree differential forms can be reduced to that of lower-degree forms via integration. Equation (4.5) shows that the global integral obtained by applying the (1, 0) and (0, 1)-directions to the Kähler form decomposes into inner products of lower-degree forms and is independent of the integral on the (1, 0)-direction.

In some cases, the σ represents Dolbeault or Bott-Chern cohomology classes. These identities can be used to relate differential forms to global invariants, for example, to compute obstructions to the existence of Kähler metrics and to study dimension inequalities between Dolbeault cohomology groups. Moreover, it is possible to apply these identities to the Chern-Ricci flow or the pluriclosed flow.