A new class of non-Kähler metrics

Cristian Ciulică

doi:10.1515/coma-2025-0014

Artikel Open Access

A new class of non-Kähler metrics

Cristian Ciulică

Veröffentlicht/Copyright: 3. Juli 2025

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Aus der Zeitschrift Complex Manifolds Band 12 Heft 1

Abstract

We study the stability at blow-ups and deformations of a class of Hermitian metrics whose fundamental two-form ω satisfies the condition ∂ ∂ ¯ ω k = 0 , for any k between 1 and n − 1 (where n is the complex dimension of the manifold). We are motivated by the existence of compact complex manifolds supporting such metrics.

Keywords: blow-up; deformation; Hermitian metric; pluriclosed; SKT

MSC 2010: 53C55; 32G07; 53C15

1 Introduction

Among all complex manifolds, those admitting Kähler structures occupy a privileged place: they lie at the intersection of Riemannian, symplectic, and complex geometry, and they often lend themselves to algebraic techniques, as they are frequently projective in the compact case. On the other hand, many interesting compact complex manifolds do not admit Kähler metrics. Therefore, various conditions have been introduced that relax Kählerianity while still maintaining enough control to derive meaningful results. Some prominent examples of non-Kähler metrics are pluriclosed (or SKT), balanced, locally conformally Kähler, and astheno-Kähler. More details about each of these types of metrics, as well as the motivation for their introduction, can be found in previous studies [1,7,11,15,19].

The question of when does a complex manifold admit metrics in several of the aforementioned classes has been intensively studied. In this note, we focus on metrics with the Hermitian 2-form ω satisfying ∂ ∂ ¯ ω k = 0 , for any k between 1 and n − 1 (where n = dim C X ). We call these metrics Endo-Pajitnov-like (EP-like). Such metrics are, in particular, both pluriclosed (for k = 1 ) and astheno-Kähler (for k = n − 2 ). Our motivation comes from the study of Endo–Pajitnov manifolds [3], which are non-Kähler compact manifolds, generalizations in higher dimensions of the Inoue surfaces of type S 0 . In the recent paper [2, Theorem 5.6], we explicitly wrote a Hermitian metric which is EP-like in the aforementioned sense on a particular class of Endo-Pajitnov manifolds (see Example 2.3). We give a second example of an EP-like metric in Example 2.4.

On the other hand, in [4], compact complex manifolds carrying both astheno-Kähler and SKT metrics are constructed. Note that, in general, metrics which belong simultaneously to two different classes of non-Kähler metrics do not exist: for example, a metric which is both astheno-Kähler and balanced, or both SKT and balanced is Kähler [6,10].

We also note that all EP-like manifolds are, in particular, k -Gauduchon for any 1 ≤ k ≤ n − 1 [7]. On the other hand, these concepts are certainly not equivalent: in [14, Theorem 2.4], the authors give an example of a nilmanifold that is k -Gauduchon for any 1 ≤ k ≤ n − 1 , but does not admit any SKT metric, hence cannot be EP-like.

In this note, we discuss the stability of EP-like metrics at deformations and with respect to blow-up along submanifolds.

Concerning the stability at deformations, we follow and generalize the work of Sferruzza for astheno-Kähler, SKT, and balanced metrics [21,23,24].

As for the blow-up, we investigate Hermitian metrics satisfying a more general property than being EP-like: for some fixed k , we ask ∂ ∂ ¯ ω i = 0 for any i = 1 , k ¯ . It turns out that this condition is stable with respect to blow-up along submanifolds. Our result then generalizes the ones about the blow-up of SKT [5] metrics and complements the ones concerning the blow-up of other non-Kähler types of metrics (e.g. [20,27] for LCK).

Note that this result contrasts the one in [25] where it is proven that a metric satisfying ∂ ∂ ¯ ω i = 0 for i = n − 2 and i = n − 3 is not preserved by blow-up.

The article is organized as follows. In Section 2, we fix the notations. The next two sections present the main results, each beginning with a subsection that provides introductory material.

2 Preliminaries

This section presents several definitions that will prove useful in the sections that follow, along with some illustrative examples.

As a first step, we will rigorously establish the necessary conventions. Throughout the articles, we shall use the conventions from [1, (2.1)] for the complex structure J acting on complex forms on a complex manifold ( M , J ) . Namely:

J α = i q − p α , for any α ∈ A C p , q M , or equivalently
J η ( X 1 , … , X p ) = ( − 1 ) p η ( J X 1 , … , J X p ) ;
the fundamental form of a Hermitian metric is given by ω ( X , Y ) ≔ g ( J X , Y ) ;
the operator d c is defined as d c ≔ − J − 1 d J , where J − 1 = ( − 1 ) deg α J .

Let π : E → M be a rank r holomorphic vector bundle and A p , q ( M , E ) ≔ A p , q ( M ) ⊗ C ∞ ( M ) E .

Definition 2.1

Let ξ = α ⊗ s ∈ A p , q ( E ) and φ = η ¯ ⊗ Z ∈ A 0,1 ( T 1,0 M ) . Following previous studies [22, Section 2.1] and [23, Section 2], we recall the definition of the contraction by φ :

ι φ : A p , q ( E ) → A p − 1 , q + 1 ( E ) ξ ↦ ι φ ( ξ ) ≔ ( η ¯ ∧ ι Z ( α ) ) ⊗ s .

Definition 2.2

Let M be a compact complex manifold of complex dimension n . A Hermitian metric g is called EP-like if its fundamental form ω satisfies ∂ ∂ ¯ ω k = 0 , for any k = 1 , n − 1 ¯ .

We end this section with some examples of manifolds that admit EP-like metrics.

Example 2.3

The first example is given by a class of Endo-Pajitnov manifolds, [3].

We consider the Endo-Pajitnov manifold T M associated with a diagonalizable M ∈ SL ( 2 n + 1 , Z ) , with the property that there exists 1 ≤ i 0 ≤ n such that α β i 0 β ¯ i 0 = 1 and ∣ β i ∣ = 1 , for any 1 ≤ i ≤ n , i ≠ i 0 , where α is the real eigenvalue of M and the β i are the complex ones.

We take Ω ˜ = i d w ∧ d w ¯ ( Im w ) 2 + Im w d z 1 ∧ d z ¯ 1 + ∑ j n i d z j ∧ d z ¯ j on H × C n , the universal cover of T M . Then Ω ˜ is invariant under the action of the group from construction of Endo-Pajitnov manifolds and satisfies ∂ ∂ ¯ Ω ˜ k = 0 , for any k = 1 , n − 1 ¯ [2, Theorem 5.6]. In particular, Ω ˜ defines an EP-like metric on T M .

Example 2.4

We can give an example of a family of two-step complex nilmanifolds endowed with an EP-like metric [21, Example 1].

Let us consider the Lie algebra g endowed with integrable almost complex structure J such that g * is spanned by { η 1 , … , η 4 } , a set of (1, 0) complex differential forms with the structure equations:

(2.1) d η i = 0 , i = 1 , 3 ¯ , d η 4 = a 1 η 12 + a 2 η 13 + a 3 η 1 1 ¯ + a 4 η 1 2 ¯ + a 5 η 1 3 ¯ + a 6 η 23 + a 7 η 2 1 ¯ + a 8 η 2 2 ¯ + a 9 η 2 3 ¯ + a 10 η 3 1 ¯ + a 11 η 3 2 ¯ + a 12 η 3 3 ¯ ,

with a i ∈ Q [ i ] for i ∈ { 1 , … , 12 } , and η i j denotes η i ∧ η j .

Furthermore, g is the Lie algebra of a compact complex nilmanifold Γ \ G .

Let us consider

ω = i 2 ∑ i = 1 n η i i ¯ ,

where η i i ¯ = η i ∧ η i ¯ .

From [21, Example 1], we know that if a 1 = a 4 = a 6 = a 7 = a 8 = a 9 = a 11 = 0 and

∣ a 2 ∣ 2 + ∣ a 5 ∣ 2 + ∣ a 10 ∣ 2 = 2 Re ( a 3 a ¯ 12 ) ,

we have that ∂ ∂ ¯ ω = ∂ ∂ ¯ ω 2 = 0 .

The structure equations become:

(2.2) d η i = 0 , i = 1 , 3 ¯ , d η 4 = a 2 η 13 + a 3 η 1 1 ¯ + a 5 η 1 3 ¯ + a 10 η 3 1 ¯ + a 12 η 3 3 ¯ .

Then,

ω 3 = − i 8 6 ∑ 1 ≤ i 1 < i 2 < i 3 < 4 η i 1 i 1 ¯ i 2 i 2 ¯ i 3 i 3 ¯ + 2 ∑ 1 ≤ i 1 < i 2 < 4 η i 1 i 1 ¯ i 2 i 2 ¯ ∧ η 4 4 ¯ .

From (2.2), we have that ∂ η i 1 i 1 ¯ i 2 i 2 ¯ i 3 i 3 ¯ = 0 , for any i 1 , i 2 , i 3 ∈ { 1 , 2 , 3 } . Hence,

∂ ω 3 = − i 4 ∑ 1 ≤ i 1 < i 2 < 4 η i 1 i 1 ¯ i 2 i 2 ¯ ∧ ( a 2 η 12 4 ¯ + a 3 ¯ η 41 1 ¯ − a 5 ¯ η 41 3 ¯ − a 10 ¯ η 43 1 ¯ + a 12 ¯ η 43 3 ¯ ) = − i 4 ( a 12 ¯ + a 3 ¯ ) η 1 1 ¯ 2 2 ¯ 3 3 ¯ 4 .

Thus, ∂ ∂ ¯ ω 3 = − ∂ ¯ ∂ ω 3 = 0 . Therefore, ∂ ∂ ¯ ω k = 0 , for any k = 1 , 3 ¯ . Consequently, ω is an EP-like metric.

If we take a 2 = a 5 = a 10 = a 12 = 0 , we obtain [18, Example 4.5], for n = 4 .

3 Deformations of EP-like Hermitian metrics

3.1 Review of deformation theory

In this section, we will recall some definitions and properties related to the deformation of complex structures. We will review the formulas for differential operators on each element of a differentiable family of deformations, and these formulas will be expressed in terms of the operators on ( M , J ) .

Definition 3.1

([12, Definition 4.1], [22, Definition 2.1]) Let X be a smooth manifold, B ⊂ R k an open, connected set and π : X → B a differentiable surjection.

The triple ( π , X , B ) is called a differentiable family of compact complex manifolds of dimension n if:

the rank of the Jacobian of π is equal to k at every point of X ;
for any t ∈ B , X t ≔ π − 1 ( t ) is a compact connected submanifold, endowed with a complex structure;
there exists a locally finite cover { U j ∣ j ∈ I } of X and differentiable complex-valued functions ξ j 1 , … , ξ j n defined on U j such that for any t ∈ B , the atlas
{ ( U j ∩ π − 1 ( t ) , p ↦ ( ξ j 1 ( p ) , … , ξ j n ( p ) ) ) ∣ j ∈ I , U j ∩ π − 1 ( t ) ≠ ∅ }
forms a system of holomorphic coordinates on X t .

We also recall from [22, Section 2.1] the following definition of the Lie bracket of T 1,0 M -valued forms.

Definition 3.2

Let φ ∈ A 0 , p ( T 1,0 M ) , Ψ ∈ A 0 , q ( T 1,0 M ) . Locally, we can write:

φ = ∑ j 1 ¯ , … , j p ¯ , i φ j 1 ¯ , … , j p ¯ i d z ¯ j 1 ∧ ⋯ ∧ d z ¯ j p ⊗ ∂ ∂ z i , Ψ = ∑ k 1 ¯ , … , k q ¯ , i Ψ k 1 ¯ , … , k q ¯ i d z ¯ k 1 ∧ ⋯ ∧ d z ¯ k q ⊗ ∂ ∂ z i .

We define:

[ φ , Ψ ] ≔ ∑ i , j = 1 n ( φ i ∧ ∂ i Ψ j − ( − 1 ) p q Ψ i ∧ ∂ i φ j ) ⊗ ∂ ∂ z j ,

where

φ j = ∑ j 1 ¯ , … , j p ¯ φ j 1 ¯ , … , j p ¯ j d z ¯ j 1 ∧ ⋯ ∧ d z ¯ j p , ∂ i φ j = ∑ j 1 ¯ , … , j p ¯ ∂ ∂ z i ( φ j ¯ 1 , … , j ¯ p j ) d z ¯ j 1 ∧ ⋯ ∧ d z ¯ j p .

and Ψ j and ∂ i Ψ j are defined analogously.

In particular, if φ , Ψ ∈ A 0,1 ( T 1,0 M ) , then

[ φ , Ψ ] = ∑ i , j = 1 n ( φ i ∧ ∂ i Ψ j + Ψ i ∧ ∂ i φ j ) ⊗ ∂ ∂ z j .

Definition 3.3

For ϕ ∈ A 0 , q ( T 1,0 M ) , we define the twisted Lie derivative:

ℒ ϕ ≔ ( − 1 ) q d ∘ ι ϕ + ι ϕ ∘ d .

Remark 3.4

We can decompose ℒ ϕ as follows:

ℒ ϕ ≔ ℒ ϕ 1,0 + ℒ ϕ 0,1 ,

where

ℒ ϕ 1,0 = ( − 1 ) q ∂ ∘ ι ϕ + ι ϕ ∘ ∂ , ℒ ϕ 0,1 = ( − 1 ) q ∂ ¯ ∘ ι ϕ + ι ϕ ∘ ∂ ¯ .

The following definition is of central importance.

Definition 3.5

[22, Section 2.1] Let ϕ ∈ A 0,1 ( T 1,0 M ) . We define e ι ϕ : A * ( M ) ⟶ A * ( M ) by:

e ι ϕ = ∑ k = 0 ∞ 1 k ! ι ϕ k .

Proposition 3.6

[13, Theorem 3.4] Let ϕ ∈ A 0,1 ( T 1,0 M ) . We have

e − ι ϕ ∘ d ∘ e ι ϕ = d − ℒ ϕ − ι 1 2 [ ϕ , ϕ ] = d − ℒ ϕ 1,0 + ι ∂ ¯ ϕ − 1 2 [ ϕ , ϕ ] ,

i.e.

e ι ϕ ∘ ∂ ¯ ∘ e ι ϕ = ∂ ¯ − ℒ ϕ 0,1 , e − ι ϕ ∘ ∂ ∘ e ι ϕ = ∂ − ℒ ϕ 1,0 − ι 1 2 [ ϕ , ϕ ] .

In what follows, we consider π : ℳ → B r a differentiable family of compact complex manifolds of complex dimension n , where B r is the unit ball in R k . More details about this description can be found in [16, pp. 147–155].

For simplification, we assume from now on that k = 1 . We denote by M 0 ≔ π − 1 ( 0 ) , M t ≔ π − 1 ( t ) .

We fix an open cover { U j } j ∈ I such that U j = { ( ξ j , t ) = ( ξ j 1 , … , ξ j n , t ) ∣ ∣ ∣ ξ j ∣ ∣ < 1 , ∣ t ∣ < r } and π ( ξ j , t ) = t and ξ j α = f j , k α ( ξ k , t ) on U j ∩ U k where f j , k is holomorphic in ξ k and differentiable in t .

Since π is a proper submersion and B r is contractible, then by a classical result of Ehresmann [26, Theorem 9.3], ℳ ≃ diff M × B r , where M is the differentiable manifold underlying the complex manifold M 0 . Via this diffeomorphism, we have U j ≃ diff U j × B r , where U j = { ξ j ∣ ∣ ∣ ξ j ∣ ∣ < 1 } .

For x ∈ M , ξ j α = ξ j α ( x , t ) is a differentiable function of ( x , t ) . If z is a holomorphic coordinate on M 0 = M , then for t = 0 , ξ j α ( z , 0 ) is holomorphic in z , while for t ≠ 0 , ξ j α ( z , t ) is merely differentiable, since the holomorphic coordinate z on M 0 is no longer necessarily holomorphic on M t when t ≠ 0 .

Starting from a compact complex manifold ( M , J ) , it is known [16, p. 150] that we can construct families of deformations ( M t ) t ∈ B of ( M , J ) , where M t = ( M , J t ) , for every t ∈ B . Here, J t is an integrable complex structure on the differentiable manifold M parameterized by a (0, 1)-vector form φ ( t ) on ( M , J ) . We proceed to explain how to characterise the complex structures on each M t . First, we remark:

Remark 3.7

We have the following formulae:

(3.1) ∂ ¯ ξ j α ( z , t ) = ∂ ¯ ( f j , k α ( ξ k ( z , t ) , t ) ) = ∑ β ∂ ∂ z ¯ β ( f j , k α ( ξ k ( z , t ) , t ) ) d z ¯ β = ∑ β ∑ γ ∂ f j , k α ∂ ξ k γ ⋅ ∂ ξ k γ ∂ z ¯ β + ∂ f j , k α ∂ ξ ¯ k γ ⋅ ∂ ξ ¯ k γ ∂ z ¯ β d z ¯ β = ∑ γ ∂ f j , k α ∂ ξ k γ ∑ β ∂ ξ k γ ∂ z ¯ β d z ¯ β = ∑ γ ∂ f j , k α ∂ ξ k γ ∂ ¯ ξ k γ ( z , t ) = ∑ β ∂ f j , k α ∂ ξ k β ∂ ¯ ξ k β .

(3.2) ∂ ∂ z λ ( ξ j α ( z , t ) ) = ∑ β ∂ f j , k α ∂ ξ k β ∂ ∂ z λ ( ξ k β ( z , t ) ) .

On M 0 , ( z 1 , … , z n ) are local holomorphic coordinates, and so are ( ξ j 1 ( z , 0 ) , … , ξ j n ( z , 0 ) ) . Hence,

det ∂ ξ j α ( z , 0 ) ∂ z λ j , λ ≠ 0 ,

which implies that for small ∣ t ∣ :

det ∂ ξ j α ( z , t ) ∂ z λ j , λ ≠ 0 .

We denote by:

A j , α λ ≔ ∂ ξ j α ∂ z λ j , λ − 1

and consider the local 1-form

φ j λ ( z , t ) ≔ ∑ α A j , α j ∂ ¯ ξ j α ( z , t ) .

Claim 3.8

[16, pp. 150–151] φ j λ is well defined (i.e. does not depend on ξ j ) and is thus a global T 1,0 M -valued (0, 1) form, denoted by:

φ ( t ) ∈ A 0,1 ( T 1,0 M ) .

Thus, the explicit form of φ ( t ) is

φ ( t ) = ∂ ∂ z T ∂ ξ ∂ z − 1 ∂ ¯ ξ .

Locally, φ ( t ) can be described as follows:

φ ( t ) = φ j ¯ i d z ¯ j ⊗ ∂ ∂ z i .

Claim 3.9

[22, Section 2.1, p. 9] We have

φ j ¯ i = ∂ ξ ∂ z − 1 ∂ ξ ∂ z T j ¯ i .

Also, one can see that φ ( 0 ) = 0 and, in order for each J t to define an integrable complex structure on M , φ ( t ) must satisfy the Maurer-Cartan equation, i.e.

∂ ¯ φ ( t ) = 1 2 [ φ ( t ) , φ ( t ) ] .

Claim 3.10

Let φ = φ j ¯ i d z ¯ j ⊗ ∂ ∂ z i and φ ¯ = φ ¯ l k ¯ d z l ⊗ ∂ ∂ z ¯ k . We define φ φ ¯ ≔ φ k ¯ i φ ¯ j ¯ k d z j ⊗ ∂ ∂ z i . Then φ ¯ ⌞ φ = φ φ ¯ .

Proof

We have that φ ∈ A 0,1 ( T 1,0 M ) . Hence, φ ¯ ∈ A 1,0 ( T 0,1 M ) . We compute the right-hand side and obtain

□ ι φ ¯ φ = φ ¯ ˩ φ = ∑ i , j , k , l d z l ∧ ∂ ∂ z ¯ k d z ¯ j ⊗ ∂ ∂ z i φ ¯ l ¯ k φ j ¯ i = ∑ i , j , k , l d z l δ j k ⊗ ∂ ∂ z i φ ¯ l ¯ k φ j ¯ i = ∑ i , k , l φ ¯ l ¯ k φ k ¯ i d z l ⊗ ∂ ∂ z i .

Remark 3.11

We have φ ∈ A 0,1 ( T 1,0 M ) . Hence, φ = η ¯ ⊗ Z , where η ¯ ∈ A 0,1 ( M ) , Z ∈ T 1,0 ( M ) .

Then,

ι φ d z i = η ¯ ∧ i Z d z i = Z ( z i ) η ¯ ∈ A 0,1 ( M ) .

Moreover,

ι φ 2 d z i = ι φ ( i φ ( d z i ) ) = ι φ ( Z ( z i ) η ¯ ) = Z ( z i ) η ¯ ( Z ) η ¯ = 0 .

Thus,

ι φ 2 d z i = ⋯ = ι φ k d z i = 0 ,

for any k ∈ N .

By using the definition of e ι φ ( t ) , we obtain e ι φ ( t ) ( d z i ) = d z i + ι φ d z i .

Proposition 3.12

[22, Lemma 2.5] We have the following identities:

d ξ α = ∂ ξ α ∂ z i ( e ι φ ( d z i ) ) ;
∂ ξ α ∂ z i ∂ ∂ ξ α = ( ( I − φ φ ¯ ) − 1 ) i j ∂ ∂ z j − ( ( I − φ ¯ φ ) − 1 φ ¯ ) i j ¯ ∂ ∂ z ¯ j .

Definition 3.13

Let ( π , ℳ , I ) be a differentiable family of compact complex manifolds parametrized by φ ( t ) , for I = ( − ε , ε ) . For each t , we have

d : A p , q ( M t ) ⟶ A p + 1 , q ( M t ) ⊕ A p , q + 1 ( M t ) .

Then, we define ∂ t ≔ π t p + 1 , q ∘ d and ∂ ¯ t ≔ π t p , q + 1 ∘ d .

We need to recall the following fundamental proposition [16,17,22]:

Proposition 3.14

[22, Proposition 2.7] The holomorphic structure on M t is determined by φ ( t ) . Specifically, a differentiable function f defined on any open subset of M 0 is holomorphic with respect to the holomorphic structure of M t if and only if

∂ ¯ − ∑ i φ i ( t ) ∂ ∂ z i f ( z ) = 0 ,

where φ i ( t ) = ∑ j φ ( t ) j ¯ i d z ¯ j , or equivalently,

( ∂ ¯ − φ ( t ) ˩ ∂ ) f ( z ) = 0 .

Remark 3.15

From the proof of the previous proposition, we can write the explicit form of ∂ ¯ t :

∂ ¯ t f = e ι φ ¯ ( ( I − φ ¯ φ ) − 1 ˩ ( ∂ ¯ − φ ˩ ∂ ) f ) .

We need to understand the decomposition of each cotangent bundle T * M t , and we want to link ( p , q ) -forms form M 0 to ( p , q ) -forms from M t . For this, we must use the next definition.

Definition 3.16

[22, Definition 2.8] Let α ∈ A p , q ( M 0 ) . We recall the definition of the extension map

e ι φ ( t ) ∣ ι φ ¯ ( t ) ( α ) = α i 1 ⋯ i p j ¯ 1 ⋯ j ¯ q ( z ) ( e ι φ ( t ) ( d z i 1 ∧ ⋯ d z i p ) ) ∧ ( e ι φ ¯ ( t ) ( d z ¯ j 1 ∧ ⋯ ∧ d z ¯ j q ) ) ,

where, locally, α = α i 1 ⋯ i p j ¯ 1 ⋯ j ¯ q d z i 1 ∧ ⋯ d z i p ∧ d z ¯ j 1 ∧ ⋯ ∧ d z ¯ j q .

We have the following result:

Proposition 3.17

[22, Lemma 2.9] The extension map e ι φ ( t ) ∣ ι φ ¯ ( t ) : A p , q ( M 0 ) ⟶ A p , q ( M t ) is a linear isomorphism as t is arbitrarily small.

Definition 3.18

[22, Section 2.2] We recall the definition of the simultaneous contraction of a ( p , q ) form by a (0, 1) vector form φ as follows:

φ ╛ α ≔ α i 1 ⋯ i p j ¯ 1 ⋯ j ¯ q φ ˩ d z i 1 ∧ ⋯ ∧ φ ˩ d z i p ∧ φ ˩ d z ¯ j 1 ∧ ⋯ φ ˩ d z ¯ j q ,

where, φ ∈ A 0,1 ( T 1,0 M t ) and α ∈ A p , q M t is written, locally, as α = α i 1 ⋯ i p j ¯ 1 ⋯ j ¯ q d z i 1 ∧ ⋯ ∧ d z i p ∧ d z ¯ j 1 ∧ ⋯ ∧ d z ¯ j q .

We will prove a useful computational fact about the relation between the e i φ operator and the wedge product.

Claim 3.19

e ι φ d z i 1 ∧ ⋯ ∧ e ι φ d z i p = e ι φ ( d z i 1 ∧ ⋯ ∧ d z i p ) .

Proof

Let I ′ ≔ I − { i 1 } . Then

ι φ ( d z i 1 ∧ d z I ′ ) = ( ι φ d z i 1 ) ∧ d z I ′ + d z i 1 ∧ ι φ ( d z I ′ ) .

By induction, we can prove that

ι φ k ( d z i 1 ∧ d z I ′ ) = k ι φ d z i 1 ∧ ι φ k − 1 d z I ′ + d z i 1 ∧ ι φ k ( d z I ′ ) ,

for any k .

Hence,

e ι φ ( d z i 1 ∧ d z I ′ ) = ∑ k = 0 ∞ 1 k ! e φ k ( d z i 1 ∧ d z I ′ ) = ∑ k = 0 ∞ 1 k ! ( k ι φ d z i 1 ∧ ι φ k − 1 d z I ′ + d z i 1 ∧ ι φ k ( d z I ′ ) ) = ι φ d z i 1 ∧ ∑ k = 1 ∞ k k ! ι φ k − 1 d z I ′ + d z i 1 ∧ ∑ k = 0 ∞ 1 k ! ι φ k ( d z I ′ ) .

Also, we have

∑ k = 1 ∞ k k ! ι φ k − 1 d z I ′ = ∑ k = 1 ∞ 1 ( k − 1 ) ! ι φ k − 1 d z I ′ = ∑ k = 0 ∞ 1 k ! ι φ k d z I ′ = e ι φ d z I ′ .

Then,

e ι φ ( d z i 1 ∧ d z I ′ ) = e ι φ ( d z i 1 ) ∧ e ι φ ( d z I ′ ) + d z i 1 ∧ e ι φ ( d z I ′ ) = ( e ι φ ( d z i 1 ) + d z i 1 ) ∧ e ι φ ( d z I ′ ) = e ι φ ( d z i 1 ) ∧ e ι φ ( d z I ′ ) .

By repeating the process n times, we obtain the conclusion.□

With the previous claim, we can write the extension map using simultaneous contraction.

Lemma 3.20

[22, Section 2.2]

e ι φ ∣ ι φ ¯ = ( I + φ + φ ¯ ) ╛ .

Proof

Let α = ∑ I , J α I , J d z I ∧ d z ¯ J .

Then,

( I + φ + φ ¯ ) ╛ α = ∑ I , J α I , J ( ( I + φ + φ ¯ ) ˩ d z i 1 ∧ ⋯ ∧ ( I + φ + φ ¯ ) ˩ d z i p ∧ ( I + φ + φ ¯ ) ˩ d z ¯ j 1 ∧ ⋯ ∧ ( I + φ + φ ¯ ) ˩ d z ¯ j q ) = ∑ I , J α I , J ( d z i 1 + ι φ d z i 1 ) ∧ ⋯ ∧ ( d z i p + ι φ d z i p ) ∧ ( d z ¯ j 1 + ι φ ¯ d z ¯ j 1 ) ∧ ⋯ ∧ ( d z ¯ j q + ι φ ¯ d z ¯ j q ) = ∑ I , J α I , J e ι φ ( d z i 1 ) ∧ ⋯ ∧ e ι φ ( d z i p ) ∧ e ι φ ¯ ( d z ¯ j 1 ) ∧ ⋯ ∧ e ι φ ¯ ( d z ¯ j q ) .

By using Claim 3.19, we obtain

□ ∑ I , J α I , J e ι φ ( d z i 1 ∧ ⋯ ∧ d z i p ) ∧ e ι φ ¯ ( d z ¯ j 1 ∧ ⋯ ∧ d z ¯ j q ) = e ι φ ∣ ι φ ¯ ( α ) .

Thus, using the proof of [22, Proposition 2.13], we can write explicitly the action of ∂ t and ∂ ¯ t operator on e i φ ∣ i φ ¯ ( α ) , where α ∈ A p , q ( M 0 ) . More precisely,

(3.3) ∂ t ( e ι φ ∣ ι φ ¯ α ) = e ι φ ∣ ι φ ¯ ( ( I − φ φ ¯ ) − 1 ╛ ( [ ∂ ¯ , ι φ ¯ ] + ∂ ) ( I − φ φ ¯ ) ╛ α ) ,

(3.4) ∂ ¯ t ( e ι φ ∣ ι φ ¯ α ) = e ι φ ∣ ι φ ¯ ( ( I − φ ¯ φ ) − 1 ╛ ( [ ∂ , ι φ ] + ∂ ¯ ) ( I − φ ¯ φ ) ╛ α ) .

3.2 Stability of EP-like Hermitian metrics at deformations

We derive a necessary condition that the deformation must satisfy in order to preserve the property of a Hermitian metric to be EP-like.

Let ε > 0 and I ≔ ( − ε , ε ) . Suppose { M t } t ∈ I is a small complex deformation (Definition 3.1 with M 0 = M . As explained in Section 3.1, the complex structures on M t can be described by a family of tensors φ ( t ) ∈ A 0,1 ( T 1,0 ( M ) ) , for t ∈ I . Let { ω t } t ∈ I be a smooth family of Hermitian metrics on { M t } t ∈ I . By Proposition 3.17, for each t ∈ I , there exists ω ( t ) ∈ A 1,1 ( M ) such that:

ω t = e ι φ ∣ ι φ ¯ ( ω ( t ) ) ,

where ω ( 0 ) = ω 0 = ω .

Locally, for each k and index families I = { i 1 , … , i k , j 1 , … , j k ∣ i l < i s , j l < j s , ∀ l < s } , we find ω I ∈ C ∞ ( I × M ) such that

ω k ( t ) = ∑ ω I ( t , ⋅ ) d z i 1 ∧ d z ¯ j 1 ∧ ⋯ ∧ d z i k ∧ d z ¯ j k ,

where the sum is taken over all index families as mentioned earlier. We will also write ω I ( t ) ≔ ω I ( t , ⋅ ) .

Then ω t k has local expression e ι φ ∣ ι φ ¯ ( ω k ( t ) ) = e ι φ ∣ ι φ ¯ ( ω I ( t ) d z i 1 ∧ d z ¯ j 1 ∧ ⋯ ∧ d z i k ∧ d z ¯ j k ) , set

( ω k ( t ) ) ′ ≔ ∂ ∂ t ( ω I ( t ) ) d z i 1 ∧ d z ¯ j 1 ∧ … d z i k ∧ d z ¯ j k ∈ A k , k ( M ) .

Theorem 3.21

Let { M t } t ∈ I be a small complex deformation of a compact complex n-manifold and { ω t } t ∈ I be a smooth family of Hermitian metrics along { M t } t ∈ I . If every metric, ω t satisfies ∂ t ∂ ¯ t ω t k = 0 , for t ∈ I and k fixed between 1 and n − 1 , it must hold that

(3.5) 2 i Im ( ∂ ∘ ι φ ′ ( 0 ) ∘ ∂ ) ( ω k ) = ∂ ∂ ¯ ( ω k ( 0 ) ) ′ .

Proof

From hypothesis, we have that ∂ t ∂ ¯ t ω t k = 0 , for any t ∈ I . Then,

∂ ∂ t ( ∂ t ∂ ¯ t ω t k ) = 0 .

By definition of extension map, we have

(3.6) ∂ t ∂ ¯ t ( ω t k ) = ∂ t ∂ ¯ t ( e i φ ∣ i φ ¯ ( ω k ( t ) ) ) .

Using relations (3.3) and (3.4), the last relation becomes

∂ t ∂ ¯ t ( e ι φ ∣ ι φ ¯ ( ω k ( t ) ) ) = ∂ t ( e ι φ ∣ ι φ ¯ ( ( I − φ ¯ ϕ ) − 1 ╛ ( [ ∂ , ι φ ] + ∂ ¯ ) ( I − φ φ ¯ ) ╛ ω k ( t ) ) ) = e ι φ ∣ ι φ ¯ ( ( ( I − φ φ ¯ ) − 1 ╛ ( [ ∂ ¯ , ι φ ¯ ] + ∂ ) ( I − φ φ ¯ ) ) ╛ ( ( I − φ ¯ φ ) − 1 ╛ ( [ ∂ , ι φ ] + ∂ ¯ ) ( I − φ ¯ φ ) ╛ ω k ( t ) ) ) .

We will use the Taylor series expansion around 0 for φ and ω k , and we obtain

φ ( t ) = t φ ′ ( 0 ) + o ( t ) , ω k ( t ) = ω k ( 0 ) + t ω k ( 0 ) ′ + o ( t ) .

So,

( I − φ φ ¯ ) − 1 = ( I − φ ¯ φ ) − 1 = ( I − φ φ ¯ ) = ( I − φ ¯ φ ) = I + o ( t ) .

By using Lemma 3.20, we have

∂ t ∂ ¯ t ω t k = ( I + t φ ′ ( 0 ) + t φ ¯ ′ ( 0 ) ) ╛ ( [ ∂ ¯ , t φ ′ ( 0 ) ¯ ˩ ] + ∂ ) ( [ ∂ , t φ ′ ( 0 ) ˩ ] + ∂ ¯ ) ( ω k ( 0 ) + t ( ω k ( 0 ) ) ′ ) + o ( t ) .

But we see that

( [ ∂ , t φ ′ ( 0 ) ˩ ] + ∂ ¯ ) ( ω k ( 0 ) + t ( ω k ( 0 ) ) ′ ) + o ( t ) = [ ∂ , t φ ′ ( 0 ) ˩ ] ω k ( 0 ) + ∂ ¯ ω k ( 0 ) + t ∂ ¯ ( ω k ( 0 ) ′ ) + o ( t ) .

Then,

(3.7) ∂ t ∂ ¯ t ω t k = ( I + t φ ′ ( 0 ) + t φ ′ ( 0 ) ¯ ) ╛ ( [ ∂ ¯ , t φ ′ ( 0 ) ¯ ˩ ] + ∂ ) ( [ ∂ , t φ ′ ( 0 ) ˩ ] ω k ( 0 ) + ∂ ¯ ω k ( 0 ) + t ∂ ¯ ( ω k ( 0 ) ′ ) ) + o ( t ) .

We denote by A ≔ [ ∂ , t φ ′ ( 0 ) ˩ ] ω k ( 0 ) + ∂ ¯ ω k ( 0 ) + t ∂ ¯ ( ω k ( 0 ) ′ ) .

Hence,

∂ A = ∂ ( ∂ ( t φ ( 0 ) ′ ˩ ω ( 0 ) k ) − t φ ( 0 ) ′ ˩ ∂ ω k ( 0 ) ) + ∂ ∂ ¯ ω k ( 0 ) + t ∂ ∂ ¯ ( ω k ( 0 ) ′ ) = − t ∂ ( φ ( 0 ) ′ ˩ ∂ ω k ( 0 ) ) + + t ∂ ∂ ¯ ( ω k ( 0 ) ′ ) .

Thus,

[ ∂ ¯ , t φ ′ ( 0 ) ¯ ˩ ] + ∂ ( [ ∂ , t φ ( 0 ) ′ ˩ ] ω k ( 0 ) + ∂ ¯ ω k ( 0 ) + t ∂ ¯ ( ω k ( 0 ) ′ ) ) + o ( t ) = [ ∂ ¯ , t φ ′ ( 0 ) ¯ ˩ ] ( [ ∂ , t φ ( 0 ) ′ ˩ ] ω k ( 0 ) ) + [ ∂ ¯ , t φ ′ ( 0 ) ¯ ˩ ] ( ∂ ¯ ω k ( 0 ) ) + [ ∂ ¯ , t φ ′ ( 0 ) ¯ ˩ ] ( t ∂ ¯ ω k ( 0 ) ′ ) + ∂ A + o ( t ) .

We observe that in the brackets of the first and third terms, there are terms containing t, so they will be absorbed into o ( t ) . The last equality becomes

[ ∂ ¯ , t φ ′ ( 0 ) ¯ ˩ ] ( ∂ ¯ ω k ( 0 ) ) + ∂ A + o ( t ) = ∂ ¯ ( t φ ′ ( 0 ) ¯ ˩ ∂ ¯ ω k ( 0 ) ) − t φ ( 0 ) ′ ˩ ∂ ¯ ∂ ¯ ω k ( 0 ) + ∂ A + o ( t ) = t ∂ ¯ ( φ ′ ( 0 ) ¯ ˩ ∂ ¯ ω k ( 0 ) ) − t ∂ ( φ ′ ( 0 ) ˩ ω k ( 0 ) ) + t ∂ ∂ ¯ ( ω k ( 0 ) ′ ) + o ( t ) .

Hence, (3.7) becomes

∂ t ∂ ¯ t ω t k = ( I + t φ ′ ( 0 ) + t φ ′ ( 0 ) ¯ ) ╛ ( − t ∂ ( φ ′ ( 0 ) ˩ ω k ( 0 ) ) + t ∂ ¯ ( φ ′ ( 0 ) ¯ ˩ ∂ ¯ ω k ( 0 ) ) + t ∂ ∂ ¯ ( ω k ( 0 ) ′ ) ) + o ( t ) .

The relevant part is only the simultaneous contraction with the identity; the other two terms contain t , and in the simultaneous contraction with forms that contain t , they will be absorbed in o ( t ) .

Thus,

∂ t ∂ ¯ t ω t k = − t ∂ ( φ ′ ( 0 ) ˩ ω k ( 0 ) ) + t ∂ ¯ ( φ ′ ( 0 ) ¯ ˩ ∂ ¯ ω k ( 0 ) ) + t ∂ ∂ ¯ ( ω k ( 0 ) ′ ) + o ( t ) .

From the hypothesis, ∂ ∂ t t = 0 ( ∂ t ∂ ¯ t ω t k ) = 0 . Hence,

− ∂ ( φ ′ ( 0 ) ˩ ∂ ω k ( 0 ) ) + ∂ ¯ ( φ ′ ( 0 ) ¯ ˩ ∂ ¯ ω k ( 0 ) ) + ∂ ∂ ¯ ( ω k ( 0 ) ) ′ = 0 .

If we rewrite, we have

− ( ∂ ∘ ι φ ′ ( 0 ) ∘ ∂ ) ω k + ( ∂ ¯ ∘ ι φ ′ ( 0 ) ¯ ∘ ∂ ¯ ) ω k + ∂ ∂ ¯ ( ω k ( 0 ) ) ′ = 0 .

Hence,

( − ∂ ∘ ι φ ′ ( 0 ) ∘ ∂ + ∂ ∘ ι φ ′ ( 0 ) ∘ ∂ ¯ ) ω k = − ∂ ∂ ¯ ( ω k ( 0 ) ) ′ ,

which is equivalent to

□ 2 i Im ( ∂ ∘ ι ϕ ′ ( 0 ) ∘ ∂ ) ( ω k ) = ∂ ∂ ¯ ( ω k ( 0 ) ) ′ .

As a direct consequence, we have the following corollary.

Corollary 3.22

Let ( M , J ) be a compact Hermitian manifold endowed with a metric g and associated fundamental form ω with the property that ∂ ∂ ¯ ω k = 0 for some and k fixed between 1 and n − 1 . If along the family of deformations ( M t ) t there exists ( ω t ) t a smooth family of metrics with the property that ∂ t ∂ ¯ t ω t k = 0 , such that ω 0 = ω , then the following equation must hold

(3.8) [ Im ( ∂ ∘ ι φ ′ ( 0 ) ∘ ∂ ) ( ω k ) ] H B C k + 1 , k + 1 ( M ) = 0 .

Thus, we immediately obtain the following result about EP-like metric.

Corollary 3.23

Let ( M , J ) be a compact complex manifold of dimension n endowed with an EP-like metric g and associated fundamental form ω . Let { M t } t ∈ I be a small complex deformation of M and { ω t } t ∈ I be a smooth family of Hermitian metrics along { M t } t ∈ I . If every metric ω t is an EP-like metric, it must hold that

(3.9) 2 i Im ( ∂ ∘ ι φ ′ ( 0 ) ∘ ∂ ) ( ω k ) = ∂ ∂ ¯ ( ω k ( 0 ) ) ′ , ∀ k = 1 , n − 1 ¯ .

Also, we obtain the following corollary.

Corollary 3.24

Let ( M , J ) be a compact Hermitian manifold endowed with an EP-like metric g and associated fundamental form ω . If along the family of deformations ( M t ) t there exists ( ω t ) t a smooth family of EP-like metrics, then the following equation must hold

(3.10) [ Im ( ∂ ∘ ι φ ′ ( 0 ) ∘ ∂ ) ( ω k ) ] H B C k + 1 , k + 1 ( M ) = 0 , ∀ k = 1 , n − 1 ¯ .

4 Blow-ups of EP-like metrics

4.1 Basic facts about blow-ups

We will briefly present the construction of the blow-up of a complex manifold along a closed submanifold. This object will also be a complex manifold together with a holomorphic map between it and the initial manifold. We will follow the notations from [9, Chapter 2.5].

Let X be a complex manifold of dimension n and Y ⊂ X an arbitrary compact submanifold of dimension m . We choose an atlas ( U i , φ i ) , φ i : U i ⟶ V i ≔ φ i ( U i ) such that φ i ( U i ∩ Y ) = φ i ( U i ) ∩ C m .

Then φ i ∘ φ j − 1 : φ j ( U i ∩ U j ) ⟶ φ i ( U i ∩ U j ) ⊂ V i and (see [9, proof of Proposition 2.4.7])

( φ i ∘ φ j − 1 ) k ( z 1 , … , z n ) = ∑ s = m + 1 n z s φ k s i j ( z 1 , … , z n ) , for k ≥ m + 1 , where φ k s i j are power series in z 1 , … , z n .

We define

B l V i ∩ C m ( V i ) ≔ { ( z , l ) ∣ z i l j = z j l i , i , j ≥ m + 1 } ⊂ V i × P n − m − 1 .

Let

Ψ i j : B l V i ∩ φ i ( U i ∩ U j ) ∩ C m ( V i ∩ φ i ( U i ∩ U j ) ) ⟶ B l V j ∩ φ j ( U i ∩ U j ) ∩ C m ( V j ∩ φ j ( U i ∩ U j ) ) ( z , l ) ↦ ( φ j ∘ φ i − 1 ( z ) , ( φ s r i j ( z ) ) s , r = m + 1 n ⋅ l ) .

Then Ψ i j is a well-defined biholomorphism and all the blow-ups on the various charts φ i ( U i ) naturally glue. We have the following proposition:

Proposition 4.1

[9, Proposition 2.5.3] Let Y be a complex submanifold of X. Then there exists a complex manifold X ˆ = B l Y ( X ) , the blow-up of X along Y, together with a holomorphic map σ : X ˆ ⟶ X such that σ : X ˆ \ σ − 1 ( Y ) ≃ X \ Y and σ : σ − 1 ( Y ) ⟶ Y is isomorphic with P ( N Y ∣ X ) ⟶ Y .

Let ( U α , φ α ) be an atlas for X with the previous properties, and let us denote V α ≔ φ α ( U α ) and Ψ α : B l V α ∩ C m ( V α ) ↪ B l Y ( X ) . Let p r 2 : B l V α ∩ C m ( V α ) ⟶ P n − m − 1 be the projection onto the second factor. We consider O P n − m − 1 ( − 1 ) as the tautological line bundle, and we denote by E α ≔ p r 2 * O P n − m − 1 ( − 1 ) . Let h be the natural metric on O P n − m − 1 ( − 1 ) (i.e. the metric such that, if we consider e ∈ O P n − m − 1 ( − 1 ) and p r ( e ) = [ l ] , then there exists λ ∈ C such that e = λ l , where l is a representative for [ l ] , and h ( e ) = ∣ ∣ λ l ∣ ∣ C n − m − 1 ). We denote by h ′ = p r 2 * h the metric on E α .

Proposition 4.2

[26, Proposition 3.25] There exists E a vector bundle over B l Y ( X ) such that ( Ψ α ) * E = E α . Moreover:

E is trivial over B l Y ( X ) \ σ − 1 ( Y ) ;
If we make the identification σ − 1 ( Y ) ≃ P ( N Y ∣ X ) , we have E ∣ σ − 1 ( Y ) ≃ σ * N Y ∣ X .

Remark 4.3

Proposition 4.2, (1) implies that there exists η a nowhere vanishing global section. We can choose h ″ a metric on E ∣ B l Y ( X ) \ σ − 1 ( Y ) such that h ″ ( η ) = 1 . As σ − 1 ( Y ) is compact, then there exists W 1 , and W 2 relatively compact open sets, with W ¯ 1 ⊂ W 2 , such that σ − 1 ( Y ) ⊂ W 1 . We choose ρ 1 , ρ 2 C ∞ functions such that ρ 1 = 1 on W 1 and ρ 1 = 0 outside W 2 and ρ 2 = 0 on W 1 and ρ 2 = 1 outside W 2 .

We define H ≔ ρ 1 h ′ + ρ 2 h ″ metric on E .

Proposition 4.4

[8, p. 186] The curvature ω of the canonical connection associated with metric H on E has the following properties:

ω is zero outside W 2 ;
− ω is semi-negative definite on W 1 ;
the restriction of it at σ − 1 ( Y ) is negative definite on vectors that are tangent to the fibre of the bundle σ * N Y ∣ X ⟶ P ( N Y ∣ X ) .

4.2 Stability of EP-like metrics at blow-up

Here, we investigate the stability of an EP-like metric, generalizing the result for SKT metrics in [5].

Theorem 4.5

Let X be a complex manifold of dimension n endowed with a Hermitian metric g such that, for a fixed k > 0 , the fundamental two-form F of g satisfies ∂ ∂ ¯ F i = 0 , for any i = 1 , k ¯ .

Let Y ⊂ X be a compact complex submanifold.

Then the blow-up B l Y ( X ) of X along Y admits a Hermitian metric such that its fundamental two-form F ˜ satisfies ∂ ∂ ¯ F ˜ i = 0 , for any i = 1 , k ¯ .

Proof

Let σ : B l Y ( X ) ⟶ X . We know that σ : B l Y ( X ) \ σ − 1 ( Y ) ⟶ X \ Y is biholomorphism and σ − 1 ( Y ) ≃ P ( N Y ∣ X ) .

From Proposition 4.2, we know that there exists a holomorphic line bundle E over B l Y ( X ) which admits a Hermitian metric. This metric has the properties listed in Proposition 4.4.

We denote by ω the curvature of the canonical connection associated to this metric.

As Y is compact, there exists N ≫ 0 such that F ˜ = σ * F + N ω is positive definite (choose N strictly greater than the absolute value of all eigenvalues of σ * F on Y ).

Then,

F ˜ k = ( σ * F + N ω ) k = ∑ l = 0 k C k l ( σ * F ) l ∧ ( N ω ) k − l .

Hence,

∂ ∂ ¯ F ˜ k = ∑ l = 0 k C k l [ ∂ ∂ ¯ ( σ * F ) l ∧ ( N ω ) k − l − ∂ ¯ ( σ * F ) l ∧ ∂ ( N ω ) k − l + ∂ ( σ * F ) l ∧ ∂ ¯ ( N ω ) k − l + ( σ * F ) l ∧ ∂ ∂ ¯ ( N ω ) k − l ] .

Since ω is the curvature form of a connection, it is d -closed. Then ∂ ω = ∂ ¯ ω = 0 . Thus, the last three terms of the sum reduce to zero.

In conclusion,

∂ ∂ ¯ F ˜ k = ∑ l = 0 k C k l ∂ ∂ ¯ ( σ * F ) l ∧ ( N ω ) k − l = ∑ l = 0 k C k l N k − l ∂ ∂ ¯ ( σ * F ) l ∧ ω k − l = ∑ l = 0 k C k l N k − l ∂ ∂ ¯ σ * F l ∧ ω k − l = ∑ l = 0 k C k l N k − l σ * ∂ ∂ ¯ F l ∧ ω k − l .

From the hypothesis, we have ∂ ∂ ¯ F l = 0 . Hence, the conclusion.□

Corollary 4.6

Let X be a complex manifold endowed with an EP-like metric. Then the blow-up along a submanifold of X admits an EP-like metric.

Remark 4.7

The aforementioned computation for F ˜ k can be applied to any sum of two Hermitian metrics and shows that the product between a Kähler manifold and an EP-like manifold is EP-like. In particular, the problem being local, we derive that any principal torus bundle over an EP-like manifold is EP-like. This can be used to construct new examples, for instance, by taking the class of Endo–Pajitnov manifolds described in [2, Theorem 5.6] as a basis for a principal torus bundle.

Acknowledgment

My warm thanks to Alexandra Otiman for suggesting this research topic and to Liviu Ornea and Miron Stanciu for many useful discussions. I am grateful to the anonymous referees for their insightful and constructive suggestions, which have significantly improved the quality of this work.

Funding information: Author states that no funding information.
Author contribution: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.
Conflict of interest: The author states that no conflict of interest.

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Received: 2024-12-10

Revised: 2025-04-11

Accepted: 2025-05-06

Published Online: 2025-07-03

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/coma-2025-0014

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blow-up; deformation; Hermitian metric; pluriclosed; SKT

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