Complex structures on product manifolds

Leonardo Biliotti; Alessandro Minuzzo

doi:10.1515/coma-2025-0015

Article Open Access

Complex structures on product manifolds

Leonardo Biliotti and Alessandro Minuzzo

Published/Copyright: July 17, 2025

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From the journal Complex Manifolds Volume 12 Issue 1

Abstract

Let M i , for i = 1 , 2, be a Kähler manifold, and let G be a compact Lie group acting on M i by Kähler isometries. Suppose that the action admits a momentum map μ i , and let N i ≔ μ i − 1 ( 0 ) be a regular-level set. When the action of G on N i is proper and free, the Meyer-Marsden-Weinstein quotient P i ≔ N i ∕ G is a Kähler manifold and π i : N i → P i is a principal fiber bundle with base P i and characteristic fiber G . In this article, we define an almost-complex structure on the manifold N 1 × N 2 and give necessary and sufficient conditions for its integrability. In the integrable case, we find explicit holomorphic charts for N 1 × N 2 . As applications, we consider a nonintegrable almost-complex structure on the product of two complex Stiefel manifolds and the infinite Calabi-Eckmann manifolds S 2 n + 1 × S ( ℋ ) , for n ≥ 1 , where S ( ℋ ) denotes the unit sphere of an infinite-dimensional complex Hilbert space ℋ .

Keywords: group action; momentum map; reduction; complex structures

MSC 2010: Primary 53C55; 57S20

1 Introduction

In a classical article by Calabi and Eckmann [2], it is shown that the compact manifold S 2 n + 1 × S 2 m + 1 admits a complex structure coming from the natural U ( 1 ) -action on C n × C m , but does not admit a Kähler structure for n ≥ 1 , as an immediate cohomological computation shows. For example, the second de Rham cohomology group H d R 2 ( S 2 n + 1 × S 2 m + 1 ) = { 0 } , precisely when n ≥ 1 or m ≥ 1 .

We generalize the idea of their construction by considering two Kähler manifolds ( M i , h i , J i , ω i ) , i = 1 , 2, where h i is the Riemannian metric, J i the complex structure, ω i the compatible symplectic structure, and letting G be a compact Lie group acting on each M i by Kähler isometries. This means that for all g ∈ G , we have

g * h i = h i , g * ∘ J i = J i ∘ g * , g * ω i = ω i ,

where g * and g * denote the pull-back and the push-forward associated with the diffeomorphism induced by g , respectively. We assume that each action admits a momentum map μ i : M i → g * , where g * denotes the dual of the Lie algebra of G , and we let N i ≔ μ i − 1 ( 0 ) be a regular level set of μ i . We further suppose that the action of G on N i is free and consider the orbit space P i ≔ N i ∕ G . We denote with π i : N i → P i the natural projection from N i onto its orbit space. Note that since G is compact and the action is free, π i : N i → P i is a principal fiber bundle with base P i and characteristic fiber G . The manifold P i is called the Meyer-Marsden-Weinstein quotient, and it is known to be a symplectic manifold (see, for an effective treatment, Part IX of the book [3], or the original articles [7,9]). Guillemin and Sternberg [5] proved that on P i , there exist a Riemannian metric h ˜ i , a complex structure J ˜ i , and a symplectic structure ω ˜ i , such that ( P i , h ˜ i , J ˜ i , ω ˜ i ) is a Kähler manifold. We call the process of passing from the manifold ( M i , h i , J i , ω i ) to the manifold ( P i , h ˜ i , J ˜ i , ω ˜ i ) , a process of Kähler reduction.

We now describe more precisely the structure of T N i and briefly review the reduction process. Since the action is free, the projection π i defines a distribution inside T N i , the so-called vertical distribution, V i ≔ ⨆ p i ∈ N i V p i , given point-wise by V p i ≔ ker ( ( d π i ) p i ) . This means that V i is integrable by definition, the integral leaves being the fibers π i − 1 ( π i ( p i ) ) , which correspond to the orbits of the action. In particular, it follows that at each point p i , V p i coincides with the tangent space to the G -orbit of p i . We further note that V i is a trivial sub-bundle of T N i , whose fiber is isomorphic to g . We then define a complementary subspace to V p i at p i ∈ N i inside T p i N i as

H p i ≔ { v ∈ T p i N i : ( h i ) p i ( v , w ) = 0 , ∀ w ∈ V p i } .

Therefore, we have a ( h i ) p i -orthogonal splitting T p i N i = H p i ⊕ V p i . The collection of the H p i defines a distribution H i ≔ ∪ p i ∈ N i H p i ⊂ T N i , which is called horizontal distribution. Since the action of G is isometric, we also have that for all g ∈ G ,

g * H i = H i .

This tells us that H i ⊂ T N i is a principal connection for the principal fiber bundle π i : N i → P i ; for a general treatment of principal bundles and principal connections, see Chapter II of the book [6]. We note that the distribution H i is J i -invariant, i.e.,

J i ( H i ) ⊆ H i .

To see this, we observe that for all p i ∈ N i , given v ∈ H p i , for all w ∈ V p i ,

( h i ) p i ( ( J i ) p i v , w ) = ( ω i ) p i ( v , w ) = 0 ,

because ( ω i ) p i is null along V p i (see point ii) of the lemma at page 123 in [7]). Therefore, ( J i ) p i v ∈ H p i .

Observe now that since ( d π i ) p i : H p i → T π i ( p i ) P i is an isomorphism, and since J i is G -equivariant, there exists a unique almost-complex structure J ˜ i on P i such that π i * ∘ ( J i ∣ H i ) = J ˜ i ∘ ( π i * ∣ H i ) . As it is proved in [5], we have that J ˜ i is actually a complex structure (see also the very clear treatment given in Section 2 of [4]). Similarly, from the G -invariance of ω i and h i , we find a unique Riemannian metric h ˜ i and a symplectic form ω ˜ i on P i such that h i ∣ N i = π i * h ˜ i , and ω i ∣ N i = π i * ω ˜ i . The compatibility between h i , J i , and ω i , easily descends to h ˜ i , J ˜ i , and ω ˜ i , making ( P i , h ˜ i , J ˜ i , ω ˜ i ) a Kähler manifold.

In the following, we will use the decomposition T N i ≃ H i ⊕ V i for both i = 1 , 2, and hence, the tangent bundle of N 1 × N 2 will be split as

(1) T ( N 1 × N 2 ) ≃ ( H 1 ⊕ V 1 ) ⊕ ( H 2 ⊕ V 2 ) .

One last fact we are going to use in the computations is that the action G × N i → N i defines a Lie algebra anti-homomorphism^[1] between the Lie algebra of G and the Lie algebra of smooth vector fields on N i . This is the homomorphism that associates with each element ξ ∈ g the infinitesimal generator of the action ξ N i , which is the smooth vector field defined at each p i ∈ N i as ξ N i ( p i ) ≔ d d t exp ( t ξ ) ⋅ p i ∣ t = 0 . Hence, we have for all ξ , η ∈ g that

[ ξ , η ] N i = − [ ξ N i , η N i ] .

In contrast, the bracket between sections of H i is not automatically a section of H i , i.e., in general, H i is not an integrable distribution.

We start with the following definition.

Definition 1.1

We define the following endomorphism J of T ( N 1 × N 2 ) as

J ≔ ( J 1 ⊕ T 1 ) ⊕ ( J 2 ⊕ T 2 ) ,

where we have used the splitting (1), and we have set for ξ ∈ g

T 1 ( ξ N 1 ) ≔ ξ N 2 T 2 ( ξ N 2 ) ≔ − ξ N 1 ,

where ξ N i denotes the infinitesimal generator of the action of G on N i associated with ξ .

Our first result is the following.

Theorem 1.2

The endomorphism J defines an almost-complex structure on N 1 × N 2 , i.e., J 2 = − id T ( N 1 × N 2 ) . Furthermore, J is integrable, i.e., complex, if and only if G is Abelian.

By the celebrated Newlander-Nirenberg theorem [10], the integrability of the almost-complex structure J on N 1 × N 2 is equivalent to the vanishing of the Nijenhuis tensor associated with J . This tensor is defined as

N J ( X , Y ) ≔ [ J X , J Y ] − [ X , Y ] − J [ J X , Y ] − J [ X , J Y ] ,

where X and Y are smooth vector fields on N 1 × N 2 .

We then study the action by the Abelian group ( C , + ) on N 1 × N 2 defined as

(2) Ψ : C × ( N 1 × N 2 ) → ( N 1 × N 2 ) , ( a + i b , ( p 1 , p 2 ) ) ↦ ( e 2 π i ( a + b ) ⋅ p 1 , e 2 π i ( a − b ) ⋅ p 2 ) ,

where the dot denotes a free U ( 1 ) -action on the two factors. We prove the following proposition, telling us that, up to a sign, the map Ψ is holomorphic with respect to the complex structure J of Definition 1.1 associated with the U ( 1 ) -action.

Proposition 1.3

The map Ψ ( p 1 , p 2 ) : C → ( N 1 × N 2 ) , Ψ ( p 1 , p 2 ) ( a + i b ) ≔ Ψ ( a + i b , ( p 1 , p 2 ) ) is holomorphic with respect to the complex structure of C and the complex structure − J .
The map Ψ a + i b : ( N 1 × N 2 ) → ( N 1 × N 2 ) , Ψ a + i b ( p 1 , p 2 ) ≔ Ψ ( a + i b , ( p 1 , p 2 ) ) is holomorphic with respect to the complex structure J.

Let now

(3) Λ ≔ n 1 + i 2 + m 1 − i 2 : n , m ∈ Z .

Then, C ∕ Λ ≃ T 2 = U ( 1 ) × U ( 1 ) is a one-dimensional complex torus. This is a compact Abelian group, and the quotient action by T 2 is free. Without using Theorem 1.2, we are able to prove the following theorem.

Theorem 1.4

Given the action (2), the natural projection π : ( N 1 × N 2 ) → ( P 1 × P 2 ) is a principal holomorphic bundle with characteristic fiber T 2 . Moreover, the construction can be easily generalized to the case of an action of C n ∕ Λ n = T 2 n .

2 Proofs of the results

Proof

(Theorem 1.2) We start by proving that J is an almost-complex structure. Consider for ( p 1 , p 2 ) ∈ ( N 1 × N 2 ) a basis element for T ( p 1 , p 2 ) ( N 1 × N 2 ) ≃ ( H p 1 ⊕ g ) ⊕ ( H p 2 ⊕ g ) , denoted as ( v 1 , ξ 1 , v 2 , ξ 2 ) , where v i ∈ H p i and ξ i ∈ g are nonzero elements.

Since J i is a complex structure on H p i , we have that

J 2 ( v 1 , ξ 1 , v 2 , ξ 2 ) = J ( J 1 v 1 , − ξ 2 , J 2 v 2 , ξ 1 ) = ( J 1 2 v 1 , − ξ 1 , J 2 2 v 2 , − ξ 2 ) = − ( v 1 , ξ 1 , v 2 , ξ 2 ) .

We conclude J 2 = − id T ( N 1 × N 2 ) .

To prove integrability, we show the vanishing of the Nijenhuis tensor of J , denoted as N J . We consider each case comprised in the splitting

T ( N 1 × N 2 ) ≃ ( H 1 ⊕ V 1 ) ⊕ ( H 2 ⊕ V 2 ) .

The integrability of J i implies N J ( X i , X ˜ i ) = 0 for all sections X i , X ˜ i of H i .

Let Y 1 ≔ ξ N 1 , Y 2 ≔ η N 2 be vertical vector fields, we have

N J ( Y 1 , Y 2 ) = [ J Y 1 , J Y 2 ] − [ Y 1 , Y 2 ] − J [ J Y 1 , Y 2 ] − J [ Y 1 , J Y 2 ] = − J [ J Y 1 , Y 2 ] − J [ Y 1 , J Y 2 ] = − J ( [ ξ N 2 , η N 2 ] − [ ξ N 1 , η N 1 ] ) = − J ( [ ξ , η ] N 1 − [ ξ , η ] N 2 ) .

Therefore, N J ( Y 1 , Y 2 ) = 0 for all ξ , η ∈ g if and only if G is Abelian.

Now, we check that N J ( X 1 , Y i ) = 0 . For i = 1 ,

N J ( X 1 , Y 1 ) = − [ X 1 , Y 1 ] − J [ J X 1 , Y 1 ] = − [ X 1 , Y 1 ] − J 2 [ X 1 , Y 1 ] = [ X 1 , Y 1 ] − [ X 1 , Y 1 ] = 0 .

For i = 2 ,

N J ( X 1 , Y 2 ) = [ J X 1 , J Y 2 ] − J [ X 1 , J Y 2 ] = J [ X 1 , J Y 2 ] − J [ X 1 , J Y 2 ] = 0 .

In fact, for X i horizontal and Y i = ξ N i , vertical, we have

[ J X i , Y i ] = [ J i X i , Y i ] = J i [ X i , Y i ] = J [ X i , Y i ] .

This is because J i ∘ g * = g * ∘ J i implies ℒ ξ N i J i = 0 for all ξ ∈ g , and hence,

[ Y i , J i X i ] = ℒ Y i ( J i X i ) = ( ℒ Y i J i ) ( X i ) + J i ( ℒ Y i X i ) = J i ( ℒ Y i X i ) = J i [ Y i , X i ] .

Let now X i be a section of H i . Then, since [ H 1 , H 2 ] = 0 , we have N J ( X 1 , X 2 ) = 0 .

Finally, we consider Y i = ξ N i , Y ˜ i = η N i .

N J ( Y 1 , Y ˜ 1 ) = [ J ξ N 1 , J η N 1 ] − [ ξ N 1 , η N 1 ] − J [ J ξ N 1 , η N 1 ] − J [ ξ N 1 , J η N 1 ] = [ ξ N 2 , η N 2 ] − [ ξ N 1 , η N 1 ] − J [ ξ N 2 , η N 1 ] − J [ ξ N 1 , η N 2 ] = [ ξ , η ] N 1 − [ ξ , η ] N 2 .

We obtain N J ( Y 1 , Y ˜ 1 ) = 0 for all ξ , η ∈ g if and only if G is Abelian. Similarly, N J ( Y 2 , Y ˜ 2 ) = 0 if and only if G is Abelian. The remaining cases are comprised in the previous ones because of the skew-symmetry of N J .

Thus, we have proved that if G is Abelian, N J = 0 , and thanks to the Newlander-Nirenberg theorem, N 1 × N 2 is a complex manifold with the holomorphic structure induced by J .□

Proof

(Proposition 1.3)

(1) Consider the curve t ↦ ( a + t + i b ) ∈ C , so that d d t ( a + t + i b ) ∣ t = 0 = 1 ∈ T ( a + i b ) C . Therefore, setting ( p 1 ′ , p 2 ′ ) ≔ Ψ a + i b ( p 1 , p 2 ) , we have
( d Ψ ( p 1 , p 2 ) ) a + i b ( 1 ) = d d t Ψ ( p 1 , p 2 ) ( a + t + i b ) ∣ t = 0 = d d t ( e 2 π i ( a + t + b ) ⋅ p 1 , e 2 π i ( a + t − b ) ⋅ p 2 ) t = 0 = d d t ( e 2 π i t ⋅ p 1 ′ , e 2 π i t ⋅ p 2 ′ ) t = 0 = ( ξ N 1 ( p 1 ′ ) , ξ N 2 ( p 2 ′ ) ) ,
where ξ = 2 π i ∈ u ( 1 ) , and ξ N i is the infinitesimal generator of the U ( 1 ) -action on N i . For the tangent vector i , consider the curve t ↦ ( a + i ( t + b ) ) ∈ C , so that d d t ( a + i ( t + b ) ) ∣ t = 0 = i ∈ T ( a + i b ) C .
Now,
( d Ψ ( p 1 , p 2 ) ) a + i b ( 1 ) = d d t Ψ ( p 1 , p 2 ) ( a + i ( b + t ) ) t = 0 = d d t ( e 2 π i ( a + t + b ) ⋅ p 1 , e 2 π i ( a − t − b ) ⋅ p 2 ) t = 0 = d d t ( e 2 π i t ⋅ p 1 ′ , e − 2 π i t ⋅ p 2 ′ ) t = 0 = ( ξ N 1 ( p 1 ′ ) , − ξ N 2 ( p 2 ′ ) ) , = − J ( p 1 ′ , p 2 ′ ) ( ξ N 1 ( p 1 ′ ) , ξ N 2 ( p 2 ′ ) ) .
The result follows by observing that i = i ⋅ 1 is the action of the complex structure of C on the tangent vector 1.
(2) We use the decomposition of T N i ≃ H i ⊕ V i , for i = 1 , 2. Let us write Ψ a + i b = ( ψ a + i b 1 , ψ a + i b 2 ) . It follows that for i = 1 , 2, one has
( ψ a + i b i ) * ∘ J i = J i ∘ ( ψ a + i b i ) * ∣ H i .
For the vertical vector fields ξ N 1 , η N 2 , since they are invariant under ( ψ a + i b 1 ) * and ( ψ a + i b 2 ) * , respectively, one obtains
J ( ( Ψ a + i b ) * ) ( ξ N 1 , η N 2 ) = J ( ( ψ a + i b 1 ) * ξ N 1 , ( ψ a + i b 2 ) * η N 2 ) = J ( ξ N 1 , η N 2 ) = ( − η N 1 , ξ N 2 ) .
We also have
( Ψ a + i b ) * ( J ( ξ N 1 , η N 2 ) ) = ( Ψ a + i b ) * ( − η N 1 , ξ N 2 ) = ( − ( ψ a + i b 1 ) * η N 1 , ( ψ a + i b 2 ) * ξ N 2 ) = ( − η N 1 , ξ N 2 ) .
In conclusion,
□ ( Ψ a + i b ) * ∘ J = J ∘ ( Ψ a + i b ) * .

Remark 2.1

We observe that, in general, given A ∈ G L ( 2 , R ) , the action Ψ : C × ( N 1 × N 2 ) → ( N 1 × N 2 ) defined as Ψ ( a + i b , ( x , y ) ) ≔ ( e 2 π i ( A 11 a + A 21 b ) ⋅ x , e 2 π i ( A 12 a + A 22 b ) ⋅ y ) is holomorphic with respect to J , and induces a free holomorphic action of C ∕ Λ A , where

Λ A ≔ n A 22 − i A 12 det ( A ) + m − A 21 + i A 11 det ( A ) : n , m ∈ Z .

We retrieve the lattice (3) by letting A = 1 1 1 − 1 .

Proof

(Theorem 1.4) We find holomorphic trivializations of π : ( N 1 × N 2 ) → ( P 1 × P 2 ) for the case of U ( 1 ) -actions and the corresponding ( C , + ) -action defined in equation (2) of Section 1.

Since the quotient action by C ∕ Λ is proper and free, we know that the natural projection π : N 1 × N 2 → P 1 × P 2 is a principal T 2 -bundle. In particular, given U ⊆ P 1 × P 2 open subset, there is a local trivialization

ϕ : π − 1 ( U ) → U × T 2 .

Let us define Σ ≔ ϕ − 1 ( U × { e } ) , where e is the identity of T 2 , and let s ≔ ϕ − 1 ( x , e ) ∈ Σ , where x ∈ U . Since ϕ is a local trivialization, it easily follows that the map

Φ : Σ × T 2 → N 1 × N 2 , ( s , g ) ↦ g . s ,

where g . s now denotes the C ∕ Λ ≃ T 2 -action on N 1 × N 2 , is a smooth open embedding. The map Φ is also equivariant, where we consider the T 2 -action on the second factor of Σ × T 2 , given by a left translation. By Proposition 1.3, since Σ is complex (it is biholomorphic to U ), this map is also holomorphic, hence a biholomorphism. Together with the holomorphic charts of P 1 × P 2 and of T 2 , the map Φ allows us to construct holomorphic charts for N 1 × N 2 .

We now look for the transition maps. Let U ˜ ⊆ P 1 × P 2 be another open subset, with U ˜ ∩ U ≠ ∅ , and let

ϕ ˜ : π − 1 ( U ˜ ) → U ˜ × T 2

be the corresponding trivialization. We define Σ ˜ ≔ ϕ ˜ − 1 ( U ˜ × { e } ) . Again, we have an analogous map Φ ˜ : Σ ˜ × T 2 → N 1 × N 2 , which is a biholomorphism with the image.

Define maps Θ , Θ ˜ as follows:

Φ − 1 : Φ ( Σ × T 2 ) → Σ × T 2 , p ↦ ( Π − 1 ( π ( p ) ) , Θ ( p ) ) , Φ ˜ − 1 : Φ ˜ ( Σ ˜ × T 2 ) → Σ ˜ × T 2 , p ↦ ( Π ˜ − 1 ( π ( p ) ) , Θ ˜ ( p ) ) ,

where Π ≔ π ∣ Σ and Π ˜ ≔ π ∣ Σ ˜ are smooth, invertible, biholomorphic maps. Then, we can write (inverse) trivializations as

H ( x , g ) ≔ Φ ( Π − 1 ( x ) , g ) .

Similarly,

H ˜ ( x , g ) ≔ Φ ˜ ( Π ˜ − 1 ( x ) , g ) .

Therefore, we have that H − 1 ( p ) = ( π ( p ) , Θ ( p ) ) . Now,

( H − 1 ∘ H ˜ ) ( x , g ) = H − 1 ( Φ ˜ ( Π ˜ − 1 ( x ) , g ) ) = ( π ( Φ ˜ ( Π ˜ − 1 ( x ) , g ) ) , Θ ( Φ ˜ ( Π ˜ − 1 ( x ) , g ) ) ) = ( x , Θ ( Φ ˜ ( Π ˜ − 1 ( x ) , g ) ) ) ,

which shows that H − 1 ∘ H ˜ is a biholomorphism. We can also see that the transition maps are provided by a right translation in the group. Indeed, let h ( x ) be such that Φ ˜ ( Π ˜ − 1 ( x ) , e ) = h ( x ) . Φ ( Π − 1 ( x ) , e ) . Then, by equivariance,

( H − 1 ∘ H ˜ ) ( x , g ) = ( x , Θ ( Φ ˜ ( Π ˜ − 1 ( x ) , g ) ) ) = ( x , Θ ( g . Φ ˜ ( Π − 1 ( x ) , e ) ) ) = ( x , Θ ( g h ( x ) . Φ ( Π − 1 ( x ) , e ) ) ) = ( x , Θ ( Φ ( Π − 1 ( x ) , g h ( x ) ) ) ) = ( x , g h ( x ) ) .

In summary, we have proved that the transition functions of the principal fiber bundle π : ( N 1 × N 2 ) → ( P 1 × P 2 ) are holomorphic. Hence, N 1 × N 2 is a holomorphic principal bundle. It also follows that the natural complex structure induced by the complex atlas is, in fact, J . Moreover, if we realize P 1 × P 2 as Meyer-Marsden-Weinstein reduction by an action of S 1 × ⋅ ⋅ ⋅ k − times × S 1 , then the same argument proves that π : ( N 1 × N 2 ) → ( P 1 × P 2 ) is a ( T 2 × ⋅ ⋅ ⋅ k − times × T 2 ) -principal holomorphic bundle. As before, N 1 × N 2 admits a complex atlas and J is precisely the complex structure induced by the atlas.□

3 Applications

3.1 Complex Stiefel manifolds

Consider the action of the unitary group in complex dimension k , U ( k ) , on the space of linear maps Hom ( C k , C n ) , for n ≥ k ,

Ψ : U ( k ) × Hom ( C k , C n ) → Hom ( C k , C n ) , Ψ ( u , A ) ≔ A ∘ u † ,

where we denoted with the superscript † the complex conjugate transposed. This is a proper left action^[2] We identify the tangent space at any point of Hom ( C k , C n ) with Hom ( C k , C n ) itself, and let J : T ( Hom ( C k , C n ) ) → T ( Hom ( C k , C n ) ) correspond to the scalar multiplication by i ∈ C defined on Hom ( C k , C n ) as a complex vector space. The action Ψ is isometric with respect to the Hermitian metric on Hom ( C k , C n ) given by

Tr ( A ∘ B † ) .

In fact, for all u ∈ U ( k ) , we have

( Ψ u * ) ( Tr ( A ∘ B † ) ) = Tr ( A ∘ u † ∘ u ∘ B † ) = Tr ( A ∘ B ) .

The associated J -invariant Riemannian metric is Re ( Tr ( A ∘ B † ) ) . The associated J -invariant symplectic form is ω ( A , B ) ≔ − Im ( Tr ( A ∘ B † ) ) . ω is exact and equal to the differential of the 1-form θ ∣ A ( B ) ≔ − 1 2 Im ( Tr ( A ∘ B † ) ) . To make this clear, we change the notation. If z denotes the natural holomorphic coordinates on Hom ( C k , C n ) , in matrix notation, we verify that we can write ω = i 2 Tr ( ( d z ) ∧ ( d z † ) ) . In fact,

ω = i 2 Tr ( ( d z ) ∧ ( d z † ) ) = i 2 Tr ( ( d x + i d y ) ∧ ( d x T − i d y T ) ) = i 2 Tr ( ( d x ∧ d x T − d y ∧ d y T + i ( d y ∧ d x T − d x ∧ d y T ) ) ) = i 2 Tr ( − 2 i d x ∧ d y T ) = Tr ( d x ∧ d y T ) ,

which is the standard symplectic structure of C k n ≃ R 2 k n . In particular, if A = A x + A y and B = B x + B y are tangent vectors, we have

ω ( A , B ) = Tr ( A x B y T − B x A y T ) = − Im ( Tr ( A ∘ B † ) ) .

We now verify that θ = i 4 Tr ( z ( d z † ) − ( d z ) z † ) = 1 2 Tr ( x d y T − y d x T ) , which is clearly a primitive for ω . In fact, for any point A = A x + A y and tangent vector B = B x + B y

θ ∣ A ( B ) = 1 2 Tr ( A x B y T − A y B x T ) = − 1 2 Im ( Tr ( A ∘ B † ) ) .

The infinitesimal generator of the action at A ∈ Hom ( C k , C n ) for ξ ∈ u ( k ) is given by

d d t ( A ∘ exp t ξ ) ∣ t = 0 = − A ∘ ξ .

Note that Ψ u * θ = θ , then we can compute the momentum map as follows:^[3]

2 ⟨ μ ( A ) ∣ ξ ⟩ = − 2 θ A ( − A ∘ ξ ) = − Im ( Tr ( A ∘ ξ † ∘ A † ) ) = Im ( Tr ( A † ∘ A ∘ ξ ) ) = − i Tr ( A † ∘ A ∘ ξ ) .

Hence, ⟨ μ ( A ) ∣ ξ ⟩ = − i 2 Tr ( A † ∘ A ∘ ξ ) . This means that

μ ( A ) = − i 2 ( A † ∘ A ) ∈ u ( k ) * .

The momentum map is Ad U ( k ) * -equivariant; indeed,

μ ( A ∘ u † ) = − i 2 ( u ∘ A † ∘ A ∘ u † ) = u ∘ μ ( A ) ∘ u † .

Note that

μ − 1 − i 2 I = { A ∈ Hom ( C k , C n ) : A † ∘ A = I }

is equal to the set of unitary k -frames in C n , the so-called Stiefel manifold V k ( C n ) , and that U ( k ) acts on V k ( C n ) freely. Then, since − i 2 I is central, we can perform a Kähler reduction. We also note that

V k ( C n ) ∕ U ( k ) = Gr k ( C n ) .

This shows in particular that the complex Grassmannians are Kähler manifolds.

Thanks to our construction, the product V k ( C n ) × V k ( C n ) carries a nonintegrable almost-complex structure.

We can also consider a p -dimensional torus in U ( k ) , p ≤ k , and obtain a free action by C p ∕ Λ on V k ( C n ) × V k ( C n ) , which is then foliated by tori. In particular, if p = 1 , the orbits are J -holomorphic curves [8].

3.2 Infinite Calabi-Eckmann manifolds

Consider a complex Hilbert space ℋ and its unit sphere S ( ℋ ) . Theorem 1.4 tells us that the product S 2 n + 1 × S ( ℋ ) is holomorphic with complex structure J , and then, we have the following.

Theorem 3.1

There cannot be a J-invariant symplectic structure ω on S 2 n + 1 × S ( ℋ ) such that ω ( ⋅ , J ⋅ ) is Riemannian. In other words, S 2 n + 1 × S ( ℋ ) cannot be Kähler with respect to the complex structure J.

Proof

If such a symplectic form ω does exist, the complex submanifolds j : S 2 n + 1 × S 2 m + 1 ↪ S 2 n + 1 × S ( ℋ ) would be Kähler with respect to j * ω . This is a contradiction since S 2 n + 1 × S 2 m + 1 cannot be Kähler, e.g., as we have already recalled, for cohomological reasons, when n ≥ 1 or m ≥ 1 .□

Acknowledgments

We would like to thank the anonymous referees for carefully reading our manuscript and for giving such constructive comments that substantially helped improve the quality of the article.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and gave consent to its submission to the journal, reviewed all results, and approved the final version of the manuscript. Both authors contributed in finding the results and carrying out the proofs.
Conflict of interest: The authors declare no conflict of interest.

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Received: 2025-01-08

Revised: 2025-06-11

Accepted: 2025-06-15

Published Online: 2025-07-17

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

Articles in the same Issue

https://doi.org/10.1515/coma-2025-0015

Keywords for this article

group action; momentum map; reduction; complex structures

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