Equivariant Holomorphic Hermitian Vector Bundles over a Projective Space

Abstract

The aim here is to describe all isomorphism classes of $\mathrm{SU}(n+1)$-equivariant Hermitian holomorphic vector bundles on the complex projective space $\mathbb{C}{\mathbb{P}}^n$. If $G\subset \mathrm{SU}(n+1)$ is the isotropy subgroup of a chosen point $x_0\in \mathbb{C}{\mathbb{P}}^n$, and $\rho :G⟶\mathrm{GL}\left(V\right)$ is a unitary representation, we obtain $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundles on $\mathbb{C}{\mathbb{P}}^n$. Next, given any $v\in \mathrm{End}\left(V_{\rho }\right)\otimes {\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$ satisfying certain conditions, a new structure of an $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle on this underlying $C^{\infty }$ holomorphic Hermitian bundle is obtained. It is shown that all $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundles on $\mathbb{C}{\mathbb{P}}^n$ arise this way.

1. Introduction

Take any holomorphic vector bundle E of rank r defined over $\mathbb{C}{\mathbb{P}}^1$. Then, E holomorphically decomposes as

$$E=\underset{i=1}{\overset{r}{⨁}}{L}_i,$$

where each $L_i$ is a holomorphic line bundle on the complex projective line $\mathbb{C}{\mathbb{P}}^1$ [1]. We know that each $L_i$ is isomorphic to ${\mathcal{O}}_{\mathbb{C}{\mathbb{P}}^1}\left(d_i\right)$ for some $d_i\in \mathbb{Z}$. Therefore, the above theorem of Grothendieck describes all holomorphic vector bundles on $\mathbb{C}{\mathbb{P}}^1$.

Here, we consider a special class of holomorphic vector bundles on the complex projective space $\mathbb{C}{\mathbb{P}}^n$ of dimension n. To explain this class, first note that $\mathrm{SU}(n+1)$ acts transitively on $\mathbb{C}{\mathbb{P}}^n$ via holomorphic automorphisms. We consider holomorphic Hermitian vector bundles on $\mathbb{C}{\mathbb{P}}^n$ that are equivariant for the action of $\mathrm{SU}(n+1)$. More precisely, we take a holomorphic vector bundle E on the complex projective space $\mathbb{C}{\mathbb{P}}^n$ and assume that E is equipped with a Hermitian structure h; furthermore, let

$$\rho :\mathrm{SU}(n+1)\times E⟶E$$

be a $C^{\infty }$ action of $\mathrm{SU}(n+1)$ on E such that the following conditions hold:

• for any $g\in \mathrm{SU}(n+1)$, the map $v⟼\rho (g,v)$ is a holomorphic automorphism of E over the automorphism of $\mathbb{C}{\mathbb{P}}^n$ given by the action of g;
• the above map $v⟼\rho (g,v)$ preserves the Hermitian structure h.

Holomorphic Hermitian vector bundles E on $\mathbb{C}{\mathbb{P}}^n$ equipped with such an action of $\mathrm{SU}(n+1)$ are called $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundles.

Our aim here is to describe all the isomorphism classes of $\mathrm{SU}(n+1)$-equivariant Hermitian holomorphic vector bundles defined over $\mathbb{C}{\mathbb{P}}^n$.

Let us take two $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundles on $\mathbb{C}{\mathbb{P}}^n$, say $(E,h,\rho )$ and $(E^{\prime },h^{\prime },{\rho }^{\prime })$. An isomorphism between $(E,h,\rho )$ and $(E^{\prime },h^{\prime },{\rho }^{\prime })$ is a holomorphic isomorphism of vector bundles

$$\mathsf{\Phi }:E⟶E^{\prime }$$

such that

• $h(v,w)=h^{\prime }(\mathsf{\Phi }\left(v\right),\mathsf{\Phi }\left(w\right))$ for all $v,w\in E_x$ and all $x\in \mathbb{C}{\mathbb{P}}^n$,
• $\mathsf{\Phi }\left(\rho (g,v)\right)={\rho }^{\prime }(g,\mathsf{\Phi }\left(v\right))$ for all $g\in \mathrm{SU}(n+1)$ and $v\in E$.

We describe the space of all isomorphism classes of $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundles on $\mathbb{C}{\mathbb{P}}^n$.

To explain the isomorphism classes of $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundles on $\mathbb{C}{\mathbb{P}}^n$, we fix a point $z_0\in \mathbb{C}{\mathbb{P}}^n$. Let $G\subset \mathrm{SU}(n+1)$ be the isotropy subgroup of the point $z_0$ for the action of $\mathrm{SU}(n+1)$ on $\mathbb{C}{\mathbb{P}}^n$. We note that G is isomorphic to the intersection of the subgroup $\mathrm{U}\left(1\right)\times \mathrm{U}\left(n\right)\subset \mathrm{U}(n+1)$ with the subgroup $\mathrm{SU}(n+1)\subset \mathrm{U}(n+1)$. Let

$$Z_G\subset G$$

be the center of G. We note that $Z_G$ is identified with $\mathrm{U}\left(1\right)$.

Consider all pairs of the form $(\rho ,v)$, where

$$\rho :G⟶\mathrm{GL}\left(V_{\rho }\right)$$

is a finite dimensional complex representation of the group $G\subset \mathrm{SU}(n+1)$ and

$$v\in {(\mathrm{End}\left(V_{\rho }\right)\otimes {\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast })}^{Z_G}\subset \mathrm{End}\left(V_{\rho }\right)\otimes {\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$$

is an invariant for the action of $Z_G$ on $\mathrm{End}\left(V_{\rho }\right)\otimes {\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$ given by the actions of $Z_G$ on $V_{\rho }$ and $T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n$. For any such pair $(\rho ,v)$,

$$[v,v]\in \mathrm{End}\left(V_{\rho }\right)\otimes {\left(T_{z_0}^{0,2}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }.$$

Let $\mathbf{S}$ denote the space of all pairs $(\rho ,v)$ of the above type such that $[v,v]=0$.

Two elements $(\rho ,v)$ and $({\rho }^{\prime },v^{\prime })$ of $\mathbf{S}$ will be called equivalent if there is a $\mathbb{C}$–linear isomorphism

$$\psi :V_{\rho }⟶V_{{\rho }^{\prime }}$$

such that the following two conditions hold:

• $\psi \circ \rho \left(g\right)={\rho }^{\prime }\left(g\right)\circ \psi$ for all $g\in G$ (note that both $\psi \circ \rho \left(g\right)$ and ${\rho }^{\prime }\left(g\right)\circ \psi$ are homomorphisms from $V_{\rho }$ to $V_{{\rho }^{\prime }}$),
• $v^{\prime }\circ \psi =(\psi \otimes {\mathrm{Id}}_{{\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }})\circ v$ (note that both sides are homomorphisms from $V_{\rho }$ to $V_{{\rho }^{\prime }}\otimes {\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$).

Let

$$\mathcal{S}$$

denote the equivalence classes of elements of $\mathbf{S}$.

The following theorem is proved here (see Theorem 2):

Theorem 1.

There is a canonical bijection between $\mathcal{S}$ (see (38)) and the space of all isomorphisms classes of $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle on $\mathbb{C}{\mathbb{P}}^n$.

The very special case of $n=0$ was addressed earlier in [2]. Note that that condition in (37) is automatically satisfied when $n=1$, because $T^{0,2}\mathbb{C}{\mathbb{P}}^1=0$.

See [3,4,5,6,7,8,9,10,11] for results on related topics.

2. Properties of Equivariant Vector Bundles on Projective Space

Consider the complex projective space $\mathbb{C}{\mathbb{P}}^n$ of dimension n that parametrizes all one-dimensional complex linear subspaces of ${\mathbb{C}}^{n+1}$. There is a natural projection ${\mathbb{C}}^{n+1}\setminus \left\{0\right\}⟶\mathbb{C}{\mathbb{P}}^n$ that sends any nonzero vector of $0\ne v\in {\mathbb{C}}^{n+1}$ to the line $\mathbb{C}·v$ in ${\mathbb{C}}^{n+1}$ generated by it. Consider the standard action of the special unitary group $\mathrm{SU}(n+1)$ on the complex vector space ${\mathbb{C}}^{n+1}$. This action evidently induces an action of $\mathrm{SU}(n+1)$ on $\mathbb{C}{\mathbb{P}}^n$. Let

$$f:\mathrm{SU}(n+1)⟶\mathrm{Aut}\left(\mathbb{C}{\mathbb{P}}^n\right)$$

(1)

be the morphism that defines the above action of $\mathrm{SU}(n+1)$ on $\mathbb{C}{\mathbb{P}}^n$. Clearly, for every $U\in \mathrm{SU}(n+1)$, the image $f\left(U\right)$ is a holomorphic automorphism of the projective space $\mathbb{C}{\mathbb{P}}^n$.

Take a holomorphic vector bundle E on the projective space $\mathbb{C}{\mathbb{P}}^n$; assume that E is equipped with a Hermitian structure h.

In the special case where $n=1$, Definitions 1 and 2 coincide with [2] (p. 248, Definition 2.1) and [2] (p. 248, Definition 2.2), respectively.

Definition 1.

The holomorphic Hermitian vector bundle $(E,h)$ on $\mathbb{C}{\mathbb{P}}^n$ is called $\mathrm{SU}(n+1)$-homogeneous if for each $U\in \mathrm{SU}(n+1)$, the pulled-back holomorphic Hermitian vector bundle $(f{\left(U\right)}^{\ast }E,f{\left(U\right)}^{\ast }h)$ is holomorphically isometric to $(E,h)$, where f is the homomorphism in (1).

Two $\mathrm{SU}(n+1)$-homogeneous vector bundles $(E,h)$ and $(E^{\prime },h^{\prime })$ are called isomorphic if there is a holomorphic isomorphism $E⟶E^{\prime }$ that takes h to $h^{\prime }$.

Definition 2.

An $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle on $\mathbb{C}{\mathbb{P}}^n$ is a triple $(E,h,\rho )$, where

• $E\stackrel{p}{⟶}\mathbb{C}{\mathbb{P}}^n$ is a holomorphic vector bundle on $\mathbb{C}{\mathbb{P}}^n$,
• h is a Hermitian structure on E,
• ρ is an action of $\mathrm{SU}(n+1)$ on the total space of E

  $$\rho :\mathrm{SU}(n+1)\times E⟶E$$

  (2)

  satisfying the following two conditions:

  1. 

  $p\circ \rho \left(\right(U,v\left)\right)=f\left(U\right)\left(p\right(v\left)\right)$ for all $(U,v)\in \mathrm{SU}(n+1)\times E$, where f is the homomorphism in (1) and p is the above projection of E to $\mathbb{C}{\mathbb{P}}^n$,

  2. 

  for each $U\in \mathrm{SU}(n+1)$, the action of U on E is a holomorphic isometry of the pulled-back holomorphic Hermitian vector bundle $(f{\left(U^{-1}\right)}^{\ast }E,f{\left(U^{-1}\right)}^{\ast }h)$ with $(E,h)$.

Two $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundles $(E,h,\rho )$ and $(E^{\prime },h^{\prime },{\rho }^{\prime })$ are called isomorphic if there is a holomorphic isometry $E⟶E^{\prime }$ that intertwines the two actions ρ and ${\rho }^{\prime }$ of $\mathrm{SU}(n+1)$.

An $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle on $\mathbb{C}{\mathbb{P}}^n$ is evidently $\mathrm{SU}(n+1)$-homogeneous. The following lemma shows that the converse holds.

In the special case where $n=1$, Lemma 1 reduces to [2] (p. 248, Lemma 2.3).

Lemma 1.

Let $(E,h)$ be a $\mathrm{SU}(n+1)$-homogeneous holomorphic Hermitian vector bundle on $\mathbb{C}{\mathbb{P}}^n$. Then, E admits an action ρ of $\mathrm{SU}(n+1)$ such that the triple $(E,h,\rho )$ is an $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle.

Proof. 

For any $U\in \mathrm{SU}(n+1)$, let $T\left(U\right)$ denote the space of all holomorphic isometries of the holomorphic Hermitian vector bundle $(f{\left(U^{-1}\right)}^{\ast }E,f{\left(U^{-1}\right)}^{\ast }h)$ with $(E,h)$, where f is the homomorphism in (1). The union

$$U_E:=\bigcup _{U\in \mathrm{SU}(n+1)}T\left(U\right)$$

(3)

has a natural structure of a finite dimensional Lie group. The group operation is defined as follows: for $A\in T\left(U\right)$ and $A^{\prime }\in T\left(U^{\prime }\right)$,

$$A{A}^{\prime }=A\circ A^{\prime }\in T\left(U{U}^{\prime }\right)$$

is simply the composition of maps. Since $(E,h)$ is $\mathrm{SU}(n+1)$-homogeneous, the homomorphism from the group $U_E$ in (3)

$$\gamma :U_E⟶\mathrm{SU}(n+1),$$

(4)

that sends any $A\in T\left(U\right)$ to U, is surjective. Therefore, we have a short exact sequence of groups

$$e⟶\mathrm{Aut}\left((E,h)\right)⟶U_E\stackrel{\gamma }{⟶}\mathrm{SU}(n+1)⟶e,$$

(5)

where $\gamma$ is constructed in (4), and $\mathrm{Aut}\left(\right(E,h\left)\right)$ is the group of all holomorphic isometries of the holomorphic Hermitian vector bundle $(E,h)$.

The Lie algebra of $U_E$ (respectively, $\mathrm{Aut}\left(\right(E,h\left)\right)$) will be denoted by $\mathfrak{g}$ (respectively, ${\mathfrak{g}}_0$). Let

$$0⟶{\mathfrak{g}}_0⟶\mathfrak{g}\stackrel{{\gamma }^{\prime }}{⟶}su(n+1)⟶e$$

(6)

be the short exact sequence of Lie algebras associated to (5), where ${\gamma }^{\prime }$ is the homomorphism of Lie algebras associated to the homomorphism $\gamma$ in (4).

The Lie algebra $su(n+1)$ of $\mathrm{SU}(n+1)$ is simple. Hence, the homomorphism ${\gamma }^{\prime }$ in (6) splits (see [12] (p. 91, Corollaire 3)). In other words, there is Lie algebra homomorphism

$${\gamma }^{″}:su(n+1)⟶\mathfrak{g}$$

(7)

such that ${\gamma }^{\prime }\circ {\gamma }^{″}={\mathrm{Id}}_{su(n+1)}$. Fix a splitting ${\gamma }^{″}$ as in (7).

Since the Lie group $\mathrm{SU}(n+1)$ is simply connected, there is a unique homomorphism of Lie groups

$${\rho }^{\prime }:\mathrm{SU}(n+1)⟶U_E$$

whose differential, at the identity element, is the homomorphism ${\gamma }^{″}$ in (7). This homomorphism ${\rho }^{\prime }$ defines an action of $\mathrm{SU}(n+1)$

$$\rho :\mathrm{SU}(n+1)\times E⟶E$$

(8)

on the total space of E. More precisely, the action of any $U\in \mathrm{SU}(n+1)$ on E defined by $\rho$ coincides with ${\rho }^{\prime }\left(U\right)$.

Now it is straightforward to verify that $(E,h,\rho )$ is an $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle, where $\rho$ is constructed in (8) (see the conditions on $\rho$ in (2)). This completes the proof of the lemma. □

Take any point $z\in \mathbb{C}{\mathbb{P}}^n$. Let

$$z^{\perp }=:S_z\subset {\mathbb{C}}^{n+1}$$

be the orthogonal complement of the line z in ${\mathbb{C}}^{n+1}$. The group of linear automorphisms of $S_z$ preserving the Hermitian structure on it induced by the Hermitian structure on ${\mathbb{C}}^{n+1}$, will be denoted by $\mathrm{U}\left(S_z\right)$. The group of linear automorphisms of the line z in ${\mathbb{C}}^{n+1}$ preserving the Hermitian structure on it induced by the Hermitian structure on ${\mathbb{C}}^{n+1}$, will be denoted by $\mathrm{U}\left(z\right)$. We note that $\mathrm{U}\left(z\right)$ is canonically identified with

$$\mathrm{U}\left(1\right)=\{\lambda \in \mathbb{C}|\left|\lambda \right|=1\}$$

The element in $\mathrm{U}\left(z\right)$ corresponding to any $\lambda \in \mathrm{U}\left(1\right)$ is uniquely determined by the condition that it acts as multiplication by $\lambda$ on the line in ${\mathbb{C}}^{n+1}$ represented by z. Note that $\mathrm{U}\left(z\right)\times \mathrm{U}\left(S_z\right)$ is a subgroup of $\mathrm{U}(n+1)$. Let

$$H_z\subset \mathrm{SU}(n+1)$$

(9)

be the isotropy subgroup of z for the action of $\mathrm{SU}(n+1)$ on $\mathbb{C}{\mathbb{P}}^n$. We note that $H_z$ is canonically identified with the subgroup

$$\mathrm{S}(\mathrm{U}\left(z\right)\times \mathrm{U}\left(S_z\right))=\mathrm{S}(\mathrm{U}\left(1\right)\times \mathrm{U}\left(S_z\right))\subset \mathrm{SU}(n+1),$$

where $\mathrm{S}(\mathrm{U}\left(z\right)\times \mathrm{U}\left(S_z\right)):=(\mathrm{U}\left(z\right)\times \mathrm{U}\left(S_z\right))\bigcap \mathrm{SU}(n+1)\subset \mathrm{U}(n+1)$; note that both $\mathrm{U}\left(z\right)\times \mathrm{U}\left(S_z\right)$ and $\mathrm{SU}(n+1)$ are subgroups of $\mathrm{U}(n+1)$.

Take any $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle $(E,h,\rho )$ on $\mathbb{C}{\mathbb{P}}^n$. For any two points $z,z^{\prime }\in \mathbb{C}{\mathbb{P}}^n$, the action of $H_z=\mathrm{S}(\mathrm{U}\left(z\right)\times \mathrm{U}\left(S_z\right))$ on the fiber $E_z$ given by $\rho$ can be shown to be a conjugate of the action of $H_{z^{\prime }}=\mathrm{S}(\mathrm{U}\left(z^{\prime }\right)\times \mathrm{U}\left(S_{z^{\prime }}\right))$ on $E_{z^{\prime }}$. To see this, take any $A\in \mathrm{SU}(n+1)$ such that $f\left(A\right)\left(z\right)=z^{\prime }$, where f is the homomorphism in (1); there is such an element A because the action of $\mathrm{SU}(n+1)$ on $\mathbb{C}{\mathbb{P}}^n$ is transitive. Then, we have $H_{z^{\prime }}=A{H}_z{A}^{-1}$. Let

$$\widehat{A}:H_z⟶H_{z^{\prime }},g⟼Ag{A}^{-1}$$

be the isomorphism. Now, the isomorphism

$$A_z:E_z\stackrel{\sim }{⟶}{E}_{z^{\prime }}$$

given by the action of A on E has the following property:

$$A_z\left(\rho (g,v)\right)=\rho (\widehat{A}\left(g\right),A_z\left(v\right))$$

(10)

for all $v\in E_z$ and $g\in H_z$.

Let

$$Z\left(H_z\right)\subset H_z$$

(11)

be the center of the group $H_z$. We note that $Z\left(H_z\right)$ is identified with $\mathrm{U}\left(1\right)=\{\lambda \in \mathbb{C}|\left|\lambda \right|=1\}$. To see this, for any $\lambda \in \mathbb{C}$ with $\left|\lambda \right|=1$, consider the automorphism of ${\mathbb{C}}^{n+1}$

$$g_{\lambda }:={\lambda }^n·{\mathrm{Id}}_z\oplus {\lambda }^{-1}·{\mathrm{Id}}_{z^{\perp }}\in \mathrm{GL}(n+1,\mathbb{C});$$

so $g_{\lambda }$ acts on the line in ${\mathbb{C}}^{n+1}$ defined by z as multiplication by ${\lambda }^n$, and $g_{\lambda }$ acts on the orthogonal complement of this line as multiplication by ${\lambda }^{-1}$. It is easy to see that $g_{\lambda }\in Z\left(H_z\right)$, where $Z\left(H_z\right)$ is defined in (11). Now, we have an isomorphism

$$\mathrm{U}\left(1\right)\stackrel{\sim }{⟶}Z\left(H_z\right)$$

(12)

that sends any $\lambda \in \mathrm{U}\left(1\right)$ to $g_{\lambda }\in Z\left(H_z\right)$ constructed above from $\lambda$.

In the special case where $n=1$, Proposition 1 reduces to [2] (p. 250, Proposition 2.4).

Proposition 1.

Let $(E,h,\rho )$ and $(E^{\prime },h^{\prime },{\rho }^{\prime })$ be two $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundles on $\mathbb{C}{\mathbb{P}}^n$. Fix a point $z_0\in \mathbb{C}{\mathbb{P}}^n$. Assume that for the action of the center $Z\left(H_{z_0}\right)=\mathrm{U}\left(1\right)$ (see (11) and (12)) on $E_{z_0}$, every $\lambda \in \mathrm{U}\left(1\right)$ acts as multiplication by ${\lambda }^d$ for some fixed d. Let

$$B:E_{z_0}⟶E_{z_0}^{\prime }$$

(13)

be an isomorphism such that the following two conditions hold:

1. 

$h(v,w)=h^{\prime }(B\left(v\right),B\left(w\right))$ for all $v,w\in E_{z_0}$ (in other words, B takes the Hermitian structure $h_{z_0}$ to $h_{z_0}^{\prime }$),

2. 

$B\left(\rho (g,v)\right)={\rho }^{\prime }(g,B\left(v\right))$ for all $g\in H_{z_0}$ and $v\in E_{z_0}$.

Then, the $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle $(E,h,\rho )$ is isomorphic to $(E^{\prime },h^{\prime },{\rho }^{\prime })$.

Proof. 

For every $U\in \mathrm{SU}(n+1)$, let

$$B^U:E_{f\left(U\right)\left(z_0\right)}⟶E_{f\left(U\right)\left(z_0\right)}^{\prime }$$

be the linear isometry obtained from the isometry B in (13) using the actions of U on E and $E^{\prime }$, where f is the homomorphism in (1). More precisely,

$$B^U\left(\rho (U,v)\right)={\rho }^{\prime }(U,B\left(v\right))$$

for all $v\in E_{z_0}$. From the second condition in the statement of the proposition, it follows immediately that we have $B^U=B^{U^{\prime }}$ if $f\left(U\right)=f\left(U^{\prime }\right)$.

Therefore, all the isometries ${\left\{B^U\right\}}_{U\in \mathrm{SU}(n+1)}$ patch together to give a $C^{\infty }$ isomorphism

$$\mathcal{B}\in C^{\infty }(\mathbb{C}{\mathbb{P}}^n,E^{\ast }\otimes E^{\prime })$$

(14)

between the $C^{\infty }$ vector bundles which intertwines the actions of $\mathrm{SU}(n+1)$ on E and $E^{\prime }$. In particular, the section $\mathcal{B}$ is left invariant by the action of $\mathrm{SU}(n+1)$ on the vector bundle $E^{\ast }\otimes E^{\prime }$ induced by the actions of $\mathrm{SU}(n+1)$ on E and $E^{\prime }$.

From the construction of $\mathcal{B}$, it follows immediately that $\mathcal{B}$ takes the Hermitian structure h on E to $h^{\prime }$ on $E^{\prime }$.

The holomorphic structures on E and $E^{\prime }$ together define a holomorphic structure on the $C^{\infty }$ vector bundle $E^{\ast }\otimes E^{\prime }$. Let

$${\overline{\partial }}_{E,E^{\prime }}:C^{\infty }(\mathbb{C}{\mathbb{P}}^n,E^{\ast }\otimes E^{\prime })⟶C^{\infty }(\mathbb{C}{\mathbb{P}}^n,{\mathsf{\Omega }}_{\mathbb{C}{\mathbb{P}}^n}^{0,1}\otimes E^{\ast }\otimes E^{\prime })=C^{\infty }(\mathbb{C}{\mathbb{P}}^n,{\mathsf{\Omega }}_{\mathbb{C}{\mathbb{P}}^n}^{0,1}(E^{\ast }\otimes E^{\prime }))$$

be the Dolbeault operator for the holomorphic structure on the vector bundle $E^{\ast }\otimes E^{\prime }$. So we have

$${\overline{\partial }}_{E,E^{\prime }}\left(\mathcal{B}\right)\in C^{\infty }(\mathbb{C}{\mathbb{P}}^n,{\mathsf{\Omega }}_{\mathbb{C}{\mathbb{P}}^n}^{0,1}(E^{\ast }\otimes E^{\prime })),$$

(15)

where $\mathcal{B}$ is the section in (14).

Consider the action of $\mathrm{SU}(n+1)$ on the vector bundle $E^{\ast }\otimes E^{\prime }$ induced by the actions of $\mathrm{SU}(n+1)$ on E and $E^{\prime }$. Since $\mathcal{B}$ is left invariant by the action of $\mathrm{SU}(n+1)$ on $E^{\ast }\otimes E^{\prime }$, we conclude that the $(0,1)$-form ${\overline{\partial }}_{E,E^{\prime }}\left(\mathcal{B}\right)$ with values in $E^{\ast }⨂E^{\prime }$ (see (15)) is also left invariant by the action of $\mathrm{SU}(n+1)$ on the $C^{\infty }$ vector bundle

$${\left(T^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }\otimes E^{\ast }\otimes E^{\prime }$$

on $\mathbb{C}{\mathbb{P}}^n$; note that the action of $\mathrm{SU}(n+1)$ on $\mathbb{C}{\mathbb{P}}^n$ produces an action of $\mathrm{SU}(n+1)$ on ${\left(T^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$, and this action of $\mathrm{SU}(n+1)$ on ${\left(T^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$ combines together with the action of $\mathrm{SU}(n+1)$ on $E^{\ast }\otimes E^{\prime }$ to produce an action of $\mathrm{SU}(n+1)$ on ${\left(T^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }\otimes E^{\ast }\otimes E^{\prime }$.

Recall that for the action of the center $Z\left(H_{z_0}\right)=\mathrm{U}\left(1\right)$ (see (11) and (12)) on $E_{z_0}$, every $\lambda \in \mathrm{U}\left(1\right)$ acts as multiplication by ${\lambda }^d$ for some fixed d. Therefore, from the second condition in the proposition, it follows immediately that for the action of the center $Z\left(H_{z_0}\right)=\mathrm{U}\left(1\right)$ on $E_{z_0}^{\prime }$, every $\lambda \in \mathrm{U}\left(1\right)$ acts as multiplication by ${\lambda }^d$. Consequently, the group $Z\left(H_{z_0}\right)$ acts trivially on the fiber ${(E^{\ast }\otimes E^{\prime })}_{z_0}$.

On the other hand, the group $Z\left(H_{z_0}\right)=\mathrm{U}\left(1\right)$ acts on the fiber ${\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$ as follows: any $\lambda \in \mathrm{U}\left(1\right)$ acts as multiplication with $1/{\lambda }^{n+1}$. Since the section ${\overline{\partial }}_{E,E^{\prime }}\left(\mathcal{B}\right)$ in (15) is left invariant by the action of $\mathrm{SU}(n+1)$, in particular, it is left invariant by the action of $Z\left(H_{z_0}\right)$, we now conclude that the section ${\overline{\partial }}_{E,E^{\prime }}\left(\mathcal{B}\right)$ vanishes at $z_0$. The section ${\overline{\partial }}_{E,E^{\prime }}\left(\mathcal{B}\right)$ being $\mathrm{SU}(n+1)$-invariant, it follows that ${\overline{\partial }}_{E,E^{\prime }}\left(\mathcal{B}\right)$ vanishes identically; recall that the action of $\mathrm{SU}(n+1)$ on $\mathbb{C}{\mathbb{P}}^1$ is transitive. Consequently, the $C^{\infty }$ isomorphism $\mathcal{B}$ is actually holomorphic.

We already noted that $\mathcal{B}$ takes the Hermitian metric h to $h^{\prime }$, and it also intertwines the actions of $\mathrm{SU}(n+1)$ on E and $E^{\prime }$. Therefore, $\mathcal{B}$ is an isomorphism between the two $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundles $(E,h,\rho )$ and $(E^{\prime },h^{\prime },{\rho }^{\prime })$. This completes the proof of the proposition. □

3. An Equivariant Principal Bundle

Fix a point $z_0\in \mathbb{C}{\mathbb{P}}^n$. Let

$$G:=H_{z_0}\subset \mathrm{SU}(n+1)$$

(16)

be the corresponding isotropy subgroup (see (9)). Since the action of $\mathrm{SU}(n+1)$ on $\mathbb{C}{\mathbb{P}}^n$ is transitive, the quotient $\mathrm{SU}(n+1)/G$, where G is the subgroup in (16), is identified with $\mathbb{C}{\mathbb{P}}^n$. This identification is constructed by sending any $g\in \mathrm{SU}(n+1)$ to $f\left(g\right)\left(z_0\right)\in \mathbb{C}{\mathbb{P}}^n$, where f is the homomorphism in (1). Consider the quotient map

$$q:\mathrm{SU}(n+1)⟶\mathrm{SU}(n+1)/G=\mathbb{C}{\mathbb{P}}^n.$$

(17)

It gives a $C^{\infty }$ principal G-bundle on $\mathbb{C}{\mathbb{P}}^n$. We will show that this principal G-bundle

$$\mathcal{P}:=\left(\mathrm{SU}\right(n+1),q)$$

(18)

on $\mathbb{C}{\mathbb{P}}^n$ has a tautological connection.

To construct the tautological connection on the principal G-bundle $\mathcal{P}$ in (18), consider the Lie algebra $su(n+1)$ of $\mathrm{SU}(n+1)$; the Killing form on it will be denoted by $\kappa$. We note that $-\kappa$ is a positive definite inner product on $su(n+1)$. Let

$$\mathfrak{g}=\mathrm{Lie}\left(G\right)\subset su(n+1)$$

be the Lie subalgebra of the Lie subgroup G in (16). We have the orthogonal complement

$$W:={\mathfrak{g}}^{\perp }\subset su(n+1)$$

(19)

of $\mathfrak{g}$ for the inner product $-\kappa$ on $su(n+1)$. Let

$$\delta :su(n+1)⟶\mathfrak{g}$$

(20)

be the orthogonal projection for the inner product $-\kappa$ on $su(n+1)$. So $\mathrm{kernel}\left(\delta \right)=W$ (see (19)). Using the left-translation action of $\mathrm{SU}(n+1)$ on itself, the real tangent bundle $T^{\mathbb{R}}\mathrm{SU}(n+1)⟶\mathrm{SU}(n+1)$ is identified with the trivial vector bundle $\mathrm{SU}(n+1)\times su(n+1)⟶\mathrm{SU}(n+1)$ with fiber $su(n+1)$. Using this identification, the homomorphism $\delta$ produces a $C^{\infty }$$\mathfrak{g}$-valued 1-form on $\mathrm{SU}(n+1)$; this 1-form is evidently preserved by the left-translation action of $\mathrm{SU}(n+1)$ on itself. Let

$$\omega \in C^{\infty }(\mathrm{SU}(n+1),{\left(T^{\mathbb{R}}\mathrm{SU}(n+1)\right)}^{\ast }\otimes \mathfrak{g})$$

be this $\mathrm{SU}(n+1)$-invariant $\mathfrak{g}$-valued 1-form on $\mathrm{SU}(n+1)$ given by $\delta$. Next, note that the restriction of $\omega$ to any fiber of the projection q in (17) coincides with the Maurer–Cartan form on the fiber of the principal G-bundle $\mathcal{P}$ on $\mathbb{C}{\mathbb{P}}^n$ in (18). Moreover, $\omega$ is G-equivariant for the right action of G on $\mathrm{SU}(n+1)$ when we equip $\mathfrak{g}$ with the adjoint action of G. Consequently, $\omega$ defines a connection on the principal G-bundle $\mathcal{P}$ on $\mathbb{C}{\mathbb{P}}^n$; see [13]. Let

$${\nabla }^0$$

(21)

be this connection on principal G-bundle $\mathcal{P}$ given by $\omega$. We note that the left-translation action of $\mathrm{SU}(n+1)$ on itself preserves the connection ${\nabla }^0$ in (21) because the 1-form $\omega$ is $\mathrm{SU}(n+1)$-invariant.

The adjoint vector bundle for the principal G-bundle $\mathcal{P}$ in (18) will be denoted by $\mathrm{ad}\left(\mathcal{P}\right)$. Let

$$\mathcal{K}\left({\nabla }^0\right)\in C^{\infty }(\mathbb{C}{\mathbb{P}}^n,\mathrm{ad}\left(\mathcal{P}\right)\otimes {\bigwedge }^2{\left(T^{\mathbb{R}}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast })$$

(22)

be the curvature of the connection ${\nabla }^0$ in (21). Since the left-translation action of $\mathrm{SU}(n+1)$ on itself preserves the connection ${\nabla }^0$, we conclude that the section $\mathcal{K}\left({\nabla }^0\right)$ in (22) is $\mathrm{SU}(n+1)$-invariant. Let

$$\mathcal{K}\left({\nabla }^0\right)=\mathcal{K}{\left({\nabla }^0\right)}^{2,0}\oplus \mathcal{K}{\left({\nabla }^0\right)}^{1,1}\oplus \mathcal{K}{\left({\nabla }^0\right)}^{0,2}$$

(23)

be the Hodge type decomposition of the $\mathrm{ad}\left(\mathcal{P}\right)$-valued 2-form $\mathcal{K}\left({\nabla }^0\right)$ in (22).

Proposition 2.

In (23),

$$\mathcal{K}{\left({\nabla }^0\right)}^{2,0}=0=\mathcal{K}{\left({\nabla }^0\right)}^{0,2};$$

in other words, the equality

$$\mathcal{K}\left({\nabla }^0\right)=\mathcal{K}{\left({\nabla }^0\right)}^{1,1}$$

holds.

Proof. 

Since $\mathcal{K}\left({\nabla }^0\right)$ in (22) is $\mathrm{SU}(n+1)$-invariant, and the action of each element of $\mathrm{SU}(n+1)$ on $\mathbb{C}{\mathbb{P}}^n$ preserves the complex structure of $\mathbb{C}{\mathbb{P}}^n$, we conclude that $\mathcal{K}{\left({\nabla }^0\right)}^{2,0}$, $\mathcal{K}{\left({\nabla }^0\right)}^{1,1}$ and $\mathcal{K}{\left({\nabla }^0\right)}^{0,2}$ in (23) are all $\mathrm{SU}(n+1)$-invariant.

The action of $\mathrm{SU}(n+1)$ on $\mathcal{P}$ induces an action of $\mathrm{SU}(n+1)$ on the adjoint vector bundle $\mathrm{ad}\left(\mathcal{P}\right)$.

Now, consider the center $Z_G:=Z\left(H_{z_0}\right)\subset H_{z_0}=G$ (see (11) and (16)). The action of $\mathrm{SU}(n+1)$ on $\mathcal{P}$ produces an action of the group $Z_G$ on the fiber $\mathrm{ad}{\left(\mathcal{P}\right)}_{z_0}$ of $\mathrm{ad}\left(\mathcal{P}\right)$ over the point $z_0$. It is easy to see that this action of $Z_G$ on $\mathrm{ad}{\left(\mathcal{P}\right)}_{z_0}$ is the trivial one. On the other hand, as noted in the proof of Proposition 1, for the action of the group $Z_G=\mathrm{U}\left(1\right)$ (see (12)) on the fiber ${\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$, any $\lambda \in Z_G=\mathrm{U}\left(1\right)$ acts as multiplication with $1/{\lambda }^{n+1}$. Therefore, any $\lambda \in Z_G=\mathrm{U}\left(1\right)$ acts on ${\bigwedge }^2{\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$ as multiplication with $1/{\lambda }^{2(n+1)}$. Hence, any $\lambda \in Z_G$ acts on $\mathrm{ad}{\left(\mathcal{P}\right)}_{z_0}\otimes {\bigwedge }^2{\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$ as multiplication with $1/{\lambda }^{2(n+1)}$. This implies that no nonzero element of $\mathrm{ad}{\left(\mathcal{P}\right)}_{z_0}\otimes {\bigwedge }^2{\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$ is fixed by the action of $Z_G$. Since $\mathcal{K}{\left({\nabla }^0\right)}^{0,2}$ in (23) is $\mathrm{SU}(n+1)$-invariant, and no nonzero element of $\mathrm{ad}{\left(\mathcal{P}\right)}_{z_0}\otimes {\bigwedge }^2{\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$ is fixed by the action of $Z_G$, we conclude that

$$\mathcal{K}{\left({\nabla }^0\right)}^{0,2}=0.$$

For the action of the group $Z_G=\mathrm{U}\left(1\right)$ on the fiber ${\left(T_{z_0}^{1,0}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$, any $\lambda \in Z_G=\mathrm{U}\left(1\right)$ acts as multiplication with ${\lambda }^{n+1}$. Therefore, repeating the above argument, we conclude that no nonzero element of $\mathrm{ad}{\left(\mathcal{P}\right)}_{z_0}\otimes {\bigwedge }^2{\left(T_{z_0}^{1,0}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$ is fixed by the action of $Z_G$. Hence, we have

$$\mathcal{K}{\left({\nabla }^0\right)}^{2,0}=0$$

as well. This completes the proof. □

4. Equivariant Vector Bundle for a Representation

Consider the subgroup $G\subset \mathrm{SU}(n+1)$ in (16). Let V be a finite dimensional complex vector space and

$$\rho :G⟶\mathrm{GL}\left(V\right)$$

(24)

a homomorphism. Let

$$E_V=\mathcal{P}{\times }^{G}V⟶\mathbb{C}{\mathbb{P}}^n$$

(25)

be the vector bundle associated to the principal G-bundle $\mathcal{P}$ (see (18)) for the G-module V in (24). We recall that $E_V$ is a quotient $\mathcal{P}\times V$ where two elements $(y_1,v_1)$ and $(y_2,v_2)$ of $\mathcal{P}\times V$ are identified if there is an element $g\in G$ such that $y_2=y_{1}g$ and $v_2=\rho \left(g^{-1}\right)\left(v_1\right)$. The action of $\mathrm{SU}(n+1)$ on $\mathcal{P}$ produces an action of $\mathrm{SU}(n+1)$ on $E_V$. More precisely, consider the following action of $\mathrm{SU}(n+1)$ on $\mathcal{P}\times V=\mathrm{SU}(n+1)\times V$: the action of any $g\in \mathrm{SU}(n+1)$ sends any $(y,v)\in \mathcal{P}\times V$ to $(gy,v)$. This action of $\mathrm{SU}(n+1)$ on $\mathcal{P}\times V=\mathrm{SU}(n+1)\times V$ produces an action of $\mathrm{SU}(n+1)$ on the quotient space $E_V$.

Since the group G is compact, there is a positive definite Hermitian structure $h_0$ on V such that the image of the homomorphism $\rho$ in (24) is contained in the subgroup of $\mathrm{GL}\left(V\right)$ consisting of all automorphisms of V that preserve the Hermitian structure $h_0$. Now, it can be seen that $h_0$ produces a Hermitian structure on the vector bundle $E_V$ in (25). Indeed, the Hermitian structure $h_0$ on V produces a Hermitian structure on the trivial vector bundle $\mathcal{P}\times V⟶\mathcal{P}$ with fiber V. Since the action of G on V given by $\rho$ preserves $h_0$, this Hermitian structure on the trivial vector bundle $\mathcal{P}\times V⟶\mathcal{P}$ descends to a Hermitian structure on the vector bundle $E_V$ over $\mathbb{C}{\mathbb{P}}^n$. The Hermitian structure on $E_V$ obtained this way will be denoted by h.

The above action of $\mathrm{SU}(n+1)$ on $E_V$ clearly preserves the Hermitian structure h on $E_V$.

Next, consider the connection ${\nabla }^0$ on the principal G-bundle $\mathcal{P}$ constructed in (21). It induces a connection on any vector bundle associated to $\mathcal{P}$. In particular, ${\nabla }^0$ induces a connection on the vector bundle $E_V$ in (25). This induced connection on $E_V$ will be denoted by ${\nabla }^V$. We note that the action of $\mathrm{SU}(n+1)$ on $E_V$ preserves the connection ${\nabla }^V$ because the action of $\mathrm{SU}(n+1)$ on $\mathcal{P}$ preserves the connection ${\nabla }^0$.

The connection ${\nabla }^V$ on $E_V$ clearly preserves the Hermitian structure h on $E_V$.

Let $\mathcal{K}\left({\nabla }^V\right)\in C^{\infty }(\mathbb{C}{\mathbb{P}}^n,\mathrm{End}\left(E_V\right)\otimes {\bigwedge }^2{\left(T^{\mathbb{R}}\mathrm{SU}(n+1)\right)}^{\ast })$ be the curvature of the connection ${\nabla }^V$. Since the curvature $\mathcal{K}\left({\nabla }^V\right)$ is given by the curvature of ${\nabla }^0$, from Proposition 2, we conclude that $\mathcal{K}\left({\nabla }^V\right)$ is of Hodge type $(1,1)$. From this, it follows that the $(0,1)$-component of the connection ${\nabla }^V$ produces a holomorphic structure on the vector bundle $E_V$ (see [14] (p. 9, Prop. 3.5) and [14] (p. 9, Prop. 3.7)).

The action of elements on $\mathrm{SU}(n+1)$ on $E_V$ preserves the above holomorphic structure on $E_V$ because the action of $\mathrm{SU}(n+1)$ on $E_V$ preserves the connection ${\nabla }^V$.

Consequently, $(E_V,h)$ is an $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle on $\mathbb{C}{\mathbb{P}}^n$.

The following lemma summarizes the above construction.

Lemma 2.

Given a finite dimensional complex vector space V and a homomorphism

$$\rho :G⟶\mathrm{GL}\left(V\right),$$

there is a naturally associated $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle $(E_V,h)$ on $\mathbb{C}{\mathbb{P}}^n$.

5. Representation for an Equivariant Vector Bundle

Let $(\mathcal{E},\widehat{h},\widehat{\rho })$ be an $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle on $\mathbb{C}{\mathbb{P}}^n$. As in Section 3, fix a point $z_0\in \mathbb{C}{\mathbb{P}}^n$. V denotes the fiber ${\mathcal{E}}_{z_0}$ of $\mathcal{E}$ over the point $z_0$. Consider the action of the subgroup $G\subset \mathrm{SU}(n+1)$ (see (16)) on ${\mathcal{E}}_{z_0}=V$. Let

$$\rho :G⟶\mathrm{GL}\left(V\right)$$

(26)

be the homomorphism given by the action of G on V.

Lemma 2 associates an $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle $(E_V,h)$ to $\rho$ in (26).

Lemma 3.

There is a canonical $C^{\infty }$ isomorphism

$$\varphi :E_V\stackrel{\sim }{⟶}\mathcal{E}$$

such that

1. 

ϕ is $\mathrm{SU}(n+1)$-equivariant,

2. 

ϕ takes the Hermitian structure h on $E_V$ to the Hermitian structure $\widehat{h}$ on $\mathcal{E}$.

Proof. 

Consider the quotient map q in (17). The pulled-back vector bundle $q^{\ast }{E}_V⟶\mathrm{SU}(n+1)$ is canonically identified with the trivial vector bundle

$$\mathrm{SU}(n+1)\times V=\mathcal{P}\times V⟶\mathrm{SU}(n+1);$$

this follows immediately from the construction of $E_V$ from $\rho$. We will construct an isomorphism

$$\tilde{\varphi }:\mathrm{SU}(n+1)\times V⟶q^{\ast }\mathcal{E}$$

(27)

of vector bundles over $\mathrm{SU}(n+1)=\mathcal{P}$. Take any

$$(g,v)\in \mathrm{SU}(n+1)\times V=\mathrm{SU}(n+1)\times {\mathcal{E}}_{z_0}.$$

The map $\tilde{\varphi }$ sends $(g,v)$ to $(g,\widehat{\rho }(g,v))\in q^{\ast }\mathcal{E}$, where $\widehat{\rho }$ is the map giving the action of $\mathrm{SU}(n+1)$ on $\mathcal{E}$. Recall from Section 4 that $E_V$ is a quotient of $\mathrm{SU}(n+1)\times V$. It is easy to see that the isomorphism $\tilde{\varphi }$ in (27) descends to an isomorphism

$$\varphi :E_V\stackrel{\sim }{⟶}\mathcal{E}$$

of vector bundles over $\mathbb{C}{\mathbb{P}}^n$.

From the construction of $\varphi$, it follows immediately that $\varphi$ is $\mathrm{SU}(n+1)$-equivariant.

Since the action of $\mathrm{SU}(n+1)$ on $\mathcal{E}$ preserves the Hermitian structure $\widehat{h}$ on $\mathcal{E}$, it follows that $\varphi$ takes the Hermitian structure h on $E_V$ to the Hermitian structure $\widehat{h}$ on $\mathcal{E}$. □

From Lemma 2, we know that $(E_V,h)$ is an $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle. It should be clarified that the isomorphism $\varphi$ in Lemma 3 need not be holomorphic.

The Dolbeault operators on the holomorphic vector bundles $\mathcal{E}$ and $E_V$ will be denoted by ${\overline{\partial }}_{\mathcal{E}}$ and ${\overline{\partial }}_{E_V}$, respectively (see [14] (p. 10, Prop. 3.9)). We have

$$\theta :={\overline{\partial }}_{\mathcal{E}}-{\overline{\partial }}_{E_V}\in C^{\infty }(\mathbb{C}{\mathbb{P}}^n,\mathrm{End}\left(E_V\right)\otimes {\left(T^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }).$$

(28)

Since ${\overline{\partial }}_{\mathcal{E}}$ and ${\overline{\partial }}_{E_V}$ define holomorphic structures, it follows that

$${\overline{\partial }}_{E_V}\theta +[\theta ,\theta ]=0$$

(29)

[14] (p. 9, Prop. 3.5).

Proposition 3.

Let

$$\beta \in C^{\infty }{(\mathbb{C}{\mathbb{P}}^n,\mathrm{End}\left(E_V\right)\otimes {\left(T^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast })}^{\mathrm{SU}(n+1)}\subset C^{\infty }(\mathbb{C}{\mathbb{P}}^n,\mathrm{End}\left(E_V\right)\otimes {\left(T^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast })$$

be an $\mathrm{SU}(n+1)$-invariant $C^{\infty }$$(0,1)$-form on $\mathbb{C}{\mathbb{P}}^n$ with values in $\mathrm{End}\left(E_V\right)$. Then,

$${\overline{\partial }}_{E_V}\beta =0,$$

where ${\overline{\partial }}_{E_V}$ is the Dolbeault operator in (29).

Proof. 

Consider the center

$$\mathrm{U}\left(1\right)=Z_G=Z\left(H_{z_0}\right)=G\subset \mathrm{SU}(n+1)$$

of G (see (16), (11) and (12)). The action of $\mathrm{SU}(n+1)$ on $E_V$ produces an action of $\mathrm{U}\left(1\right)=Z_G$ on the fiber ${\mathcal{E}}_{z_0}=:V$. Let

$$V=\underset{i=1}{\overset{\ell }{⨁}}{V}_i$$

(30)

be the isotypical decomposition of the $\mathrm{U}\left(1\right)$-module V. For each $1\le i\le \ell$, let

$${\nu }_i\in \mathbb{Z}$$

(31)

be the unique integer satisfying the following condition: The action of any $\lambda \in \mathrm{U}\left(1\right)=Z_G$ on $V_i$ sends any $v\in V_i$ to ${\lambda }^{{\nu }_i}·v$.

Since $\mathrm{U}\left(1\right)=Z_G$ is the center of G, the action of G on V, given by the action of $\mathrm{SU}(n+1)$ on $E_V$, preserves the decomposition in (30). In other words, the action of G on V preserves each $V_i$ in (30).

Note that Lemma 2 associates to each G-module $V_i$ an $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle on $\mathbb{C}{\mathbb{P}}^n$. For each $1\le i\le \ell$, let $(E_{V_i},h_i)$ be the $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle on $\mathbb{C}{\mathbb{P}}^n$ for $V_i$. From (30), we have

$$(E_V,h)=\underset{i=1}{\overset{\ell }{⨁}}(E_{V_i},h_i).$$

(32)

We have the maps

$$E_{V_i}\stackrel{t_i}{⟶}{E}_V\stackrel{q_i}{⟶}{E}_{V_i}$$

(33)

where $t_i$ is the inclusion map and $q_i$ is the projection map constructed using the decomposition in (32).

Consider the section $\beta$ in the proposition. For every $1\le i,j\le \ell$, let

$${\beta }_{i,j}:E_{V_i}⟶E_{V_j}\otimes {\left(T^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$$

be the homomorphism given by the following composition of homomorphisms

$$E_{V_i}\stackrel{t_i}{⟶}{E}_V\stackrel{\beta }{⟶}{E}_V\otimes {\left(T^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }\stackrel{q_j\otimes \mathrm{id}}{⟶}{E}_{V_j}\otimes {\left(T^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast },$$

where $t_i$ and $q_j$ are the homomorphisms in (33). Then, we have

$$\beta =\underset{i,j=1}{\overset{\ell }{⨁}}{\beta }_{i,j}.$$

(34)

Note that each section ${\beta }_{i,j}$ is preserved by the action of $\mathrm{SU}(n+1)$ because

• $\beta$ is preserved by the action of $\mathrm{SU}(n+1)$,
• the homomorphisms $t_i$ and $q_i$ in (33) are also preserved by the action of $\mathrm{SU}(n+1)$.

Recall that for the action of the group $Z_G=\mathrm{U}\left(1\right)$ on the fiber ${\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$, any $\lambda \in Z_G=\mathrm{U}\left(1\right)$ acts as multiplication with $1/{\lambda }^{n+1}$. Therefore, if ${\beta }_{i,j}\ne 0$, then we have

$${\nu }_i={\nu }_j-n-1$$

(35)

(see (31)).

From (34), it follows that

$${\overline{\partial }}_{E_V}\beta =\underset{i,j=1}{\overset{\ell }{⨁}}{\overline{\partial }}_{E_V}{\beta }_{i,j}.$$

(36)

Note that

$${\overline{\partial }}_{E_V}{\beta }_{i,j}\in C^{\infty }(\mathbb{C}{\mathbb{P}}^n,\mathrm{Hom}(E_{V_i},E_{V_j})\otimes {\left(T^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }),$$

and ${\overline{\partial }}_{E_V}{\beta }_{i,j}$ is preserved by the action of $\mathrm{SU}(n+1)$ because ${\beta }_{i,j}$ is preserved by the action of $\mathrm{SU}(n+1)$. Since for the action of the group $Z_G=\mathrm{U}\left(1\right)$ on the fiber ${\left(T_{z_0}^{0,2}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$, any $\lambda \in Z_G=\mathrm{U}\left(1\right)$ acts as multiplication with $1/{\lambda }^{2n+2}$, it follows that if ${\overline{\partial }}_{E_V}{\beta }_{i,j}\ne 0$; then, we have

$${\nu }_i={\nu }_j-2n-2.$$

Comparing this with (35) we conclude that ${\overline{\partial }}_{E_V}{\beta }_{i,j}=0$ if ${\beta }_{i,j}\ne 0$. Now, the proposition follows from (36). □

Corollary 1.

The term ${\overline{\partial }}_{E_V}\theta$ in (29) vanishes identically.

Proof. 

The section $\theta$ in (28) is preserved by the action of $\mathrm{SU}(n+1)$ because the two Dolbeault operators ${\overline{\partial }}_{\mathcal{E}}$ and ${\overline{\partial }}_{E_V}$ are both preserved by the action of $\mathrm{SU}(n+1)$. Hence, from Proposition 3, it follows that ${\overline{\partial }}_{E_V}\theta =0$. □

6. Equivariant Holomorphic Hermitian Vector Bundles

As before, $G:=H_{z_0}\subset \mathrm{SU}(n+1)$.

Consider all pairs of the form $(\rho ,v)$, where

$$\rho :G⟶\mathrm{GL}\left(V_{\rho }\right)$$

is a finite dimensional complex representation of the group G and

$$v\in {(\mathrm{End}\left(V_{\rho }\right)\otimes {\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast })}^{Z_G}\subset \mathrm{End}\left(V_{\rho }\right)\otimes {\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast },$$

where $Z_G=\mathrm{U}\left(1\right)\subset G$ is the center of G. For any such pair $(\rho ,v)$, note that we have

$$[v,v]\in \mathrm{End}\left(V_{\rho }\right)\otimes {\left(T_{z_0}^{0,2}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }.$$

Let $\mathbf{S}$ denote the space of all pairs $(\rho ,v)$ of the above type such that

$$[v,v]=0.$$

(37)

Two elements $(\rho ,v)$ and $({\rho }^{\prime },v^{\prime })$ of $\mathbf{S}$ will be called equivalent if there is an isomorphism

$$\psi :V_{\rho }⟶V_{{\rho }^{\prime }}$$

such that the following two conditions hold:

• $\psi \circ \rho \left(g\right)={\rho }^{\prime }\left(g\right)\circ \psi$ for all $g\in G$ (note that both $\psi \circ \rho \left(g\right)$ and ${\rho }^{\prime }\left(g\right)\circ \psi$ are homomorphisms from $V_{\rho }$ to $V_{{\rho }^{\prime }}$),
• $v^{\prime }\circ \psi =(\psi \otimes {\mathrm{Id}}_{{\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }})\circ v$ (note that both sides are homomorphisms from $V_{\rho }$ to $V_{{\rho }^{\prime }}\otimes {\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$).

Let

$$\mathcal{S}$$

(38)

denote the equivalence classes of elements of $\mathbf{S}$.

Theorem 2.

There is a canonical bijection between $\mathcal{S}$ (see (38)) and the space of all isomorphisms classes of $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle on $\mathbb{C}{\mathbb{P}}^n$.

Proof. 

As in Section 5, take any $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle $(\mathcal{E},\widehat{h})$ on $\mathbb{C}{\mathbb{P}}^n$.

Consider $\rho$ in (26) and

$$\theta \left(z_0\right)\in \mathrm{End}\left(E_V\right)\otimes {\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast })$$

constructed in (28). From (29) and Corollary 1, it follows immediately that

$$[\theta \left(z_0\right),\theta \left(z_0\right)]=0.$$

Hence, we have $(\rho ,\theta \left(z_0\right))\in \mathbf{S}$.

We send $(\mathcal{E},\widehat{h})$ to the element of $\mathcal{S}$ given by the pair $(\rho ,\theta \left(z_0\right))$. This produces a map to $\mathcal{S}$ from the space of all isomorphism classes of $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundles on $\mathbb{C}{\mathbb{P}}^n$.

For the inverse map, take any $(\rho ,v)\in \mathcal{S}$. Using Lemma 2, the homomorphism $\rho$ produces an $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle $(E_{V_{\rho }},h)$ on $\mathbb{C}{\mathbb{P}}^n$. Since v is fixed by the action of $Z_G$ on $\mathrm{End}\left(V_{\rho }\right)\otimes {\left(T_{z_0}^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast }$, using the actions of $\mathrm{SU}(n+1)$ on $E_{V_{\rho }}$ and $\mathbb{C}{\mathbb{P}}^n$, the element v produces a $\mathrm{SU}(n+1)$-invariant section

$$\theta \in C^{\infty }{(\mathbb{C}{\mathbb{P}}^n,\mathrm{End}\left(E_{V_{\rho }}\right)\otimes {\left(T^{0,1}\mathbb{C}{\mathbb{P}}^n\right)}^{\ast })}^{\mathrm{SU}(n+1)}.$$

The given condition in (37) that $[v,v]=0$ implies that

$$[\theta ,\theta ]=0$$

(39)

because $\theta$ is preserved by the action of $\mathrm{SU}(n+1)$ and the action of $\mathrm{SU}(n+1)$ is transitive.

Let ${\overline{\partial }}_{E_{V_{\rho }}}$ be the Dolbeault operator for the holomorphic vector bundle $E_{V_{\rho }}$. Consider the Dolbeault operator ${\overline{\partial }}_{E_{V_{\rho }}}+\theta$ on the $C^{\infty }$ vector bundle $E_{V_{\rho }}$. From (39) and Proposition 3, we conclude that

$${\overline{\partial }}_{E_V}\theta +[\theta ,\theta ]=0.$$

Therefore, ${\overline{\partial }}_{E_{V_{\rho }}}+\theta$ defines a holomorphic structure on the $C^{\infty }$ vector bundle $E_{V_{\rho }}$.

The Hermitian structure h on $E_{V_{\rho }}$ remains unchanged. This new holomorphic Hermitian vector bundle is evidently $\mathrm{SU}(n+1)$-equivariant. This produces the inverse map from $\mathcal{S}$ to the space of all isomorphisms classes of the $\mathrm{SU}(n+1)$-equivariant holomorphic Hermitian vector bundle on $\mathbb{C}{\mathbb{P}}^n$. □