Yang–Mills bar connection and holomorphic structure

1.   Introduction

Let $E$ be a $C^{\infty}$ complex vector bundle of rank $r$ over a compact complex manifold $X$ and $H$ be some reference Hermitian inner product in the fibres of $E$, i.e., $(E,H)$ defines an Hermitian vector bundle. We shall sometimes consider $E$ issued from its associated $GL_{n}(\mathbb{C})$ principal bundle or from its associated unitary principal bundle. The classical Newlander– Nirenberg theorem^[9] states that given an almost complex structure $J$ over an even dimensional smooth manifold $X$ then the torsion of $J$ (also called the Nijenhuis tensor) vanishes if and only if $J$ defines a complex structure. We denote by $F_{A}$ the curvature $2$-form of a smooth connection $A$ of $(E,H)$ over a complex manifold $X$. We will be interested in the bundle version of the Newlander-Nirenberg theorem as first proven in Ref. [8] (also shown in Ref. [3, Theorem 2.1.53]). It states that unitary connections satisfying $F_{A}^{0,2} = 0$ are in one to one correspondence with holomorphic structures:

Koszul–Malgrange criterion. Let $A$ be a smooth unitary connection of a $C^{\infty}$ Hermitian bundle $(E,H)$ over a complex manifold $X$. Then $E$ has a holomorphic structure if and only if $F_{A}^{0,2} = 0$.

The calculus of variations of Yang–Mills in four-dimensions has naturally led to the definition of Sobolev connections. One of the goals of Ref. [10] is to extend this identification to Sobolev connections. More precisely, the authors analyzed the weak holomorphic structures, that is Sobolev connections (see Ref. [10, Definition 1.1]) satisfying the integrability condition $F_{A}^{0,2} = 0$.

We note that in the decomposition for the curvature of unitary connection $A$:

The Kozsul–Malgrange criterion suggests that we consider the Yang–Mills bar functional

which is the square of the $L^{2}$-norm of the $(0,2)$-component $F_{A}^{0,2}$ of the curvature on $(E,H)$. Ref. [6] introduced the Yang–Mills bar equation as the Euler-Lagrange equation for the Yang–Mills bar functional. The solutions of the Yang–Mills bar equation are called Yang–Mills bar connections.

Definition 1.1.^{[5, 6]} A connection $A$ on a compact complex manifold is said to be a Yang–Mills bar connection if the $(0,2)$-part of its curvature is harmonic, i.e.,

Since a holomorphic connection on a complex bundle of rank $r\geq2$ over a compact complex manifold $X$, $\dim_{\mathbb{C}}X\geq2$, is overdetermined, and the Yang–Mills bar connection is moduli invariant under the complex gauge group of the complex vector bundle $E$. The Yang–Mills bar connection has an advantage over the holomorphic connection. Thus, Ref. [6] suggested that we can use the Yang–Mills bar equation to find useful sufficient conditions under which a complex vector bundle carries a holomorphic structure. The existence of a holomorphic structure on complex vector bundles over projective algebraic manifolds could be a key step in solving the Hodge conjecture. A particular result (see Ref. [6, Theorem 4.25]) which states that any Yang–Mills bar connection on a compact Kähler surface with positive Ricci curvature is holomorphic. In higher-dimensional cases, Stern proved that on a compact Calabi–Yau $3$-fold $X$ with ${\rm{Hol}}(X) = SO(3)$, if the connection $A$ on a principal $G$-bundle $P$ over $X$ is a stable critical point of $E'(A)$, then $F_{A}^{0,2} = 0$, i.e., $(P,\bar{\partial}_{A})$ is holomorphic (see Ref. [11, Theorem 6.21]).

In this note we consider the Yang–Mills bar connection $A$ on a $SU(2)$ or $SO(3)$-bundle $P$ over a compact complex surface. A self-dual two-form $B\in\varOmega^{+}(X,\mathfrak{g}_{P})$ which takes value in $\mathfrak{g}_{P}$ is said to be of rank $r$ if, when considered as a section of ${\rm{Hom}}(\varLambda^{+,\ast})$, $B(x)$ has rank less than or equal to $r$ at every point $x\in X$ (see Ref. [12, Defintion 1.5]). If the structure group of the principal bundle $P$ is either $SU(2)$ or $SO(3)$, then the rank of any $B$ must be less than or equal to $3$. The key point in the proof of the following result is that $F_{A}^{0,2}+F_{A}^{2,0}$ has at most rank one (see Proposition 3.1).

Theorem 1.1. Let $(X,g)$ be a compact complex surface with $H^{1}(X,\mathbb{Z}_{2}) = 0$, $P$ be a $SU(2)$ or $SO(3)$-bundle over $X$ and $A$ be a connection on $P$. Suppose that $A$ is an irreducible Yang–Mills bar connection. Then $F_{A}^{0,2} = 0$, i.e., $(P,\bar{\partial}_{A})$ is holomorphic

Remark 1.1. The following example shows that the condition for irreducible connection in Theorem 1.1 is necessary. Let $T^{4}$ be a 2-dimensional complex torus with coordinates $z^{1} = x^{1}+\sqrt{-1}y^{1}$, $z^{2} = x^{2}+\sqrt{-1}y^{2}$. It is easy to see that $\pi_{1}(T^{4})\cong\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}$. Therefore, $H^{1}(T^{4},\mathbb{Z}_{2}) = 0$. Let $L\rightarrow T^{4}$ be a complex line bundle whose Chern class is represented by the cohomology class $c_{1}(L)$ of $dz^{1}\wedge dz^{2}+d\bar{z}^{1}\wedge d\bar{z}^{2}$. Let $A$ be a unitary connection of $L$. Then the curvature

where $a\in\varOmega^{1}(T^{4})$. The new connection $A' = A-a$ has the curvature $\sqrt{-1}(dz^{1}\wedge dz^{2}+d\bar{z}^{1}\wedge d\bar{z}^{2})$. However, according to the Hodge theorem, we observe that $L$ has no holomorphic structure (see Ref. [6, 3.21]). One can see that the bundle $L\oplus L^{-1}$ also carries no holomorphic structure. In general, the Hodge theory implies that on any Hermitian complex line bundle over a compact complex manifold there is a Yang–Mills bar connection which realizes the infimum of the energy $E'(A)$.

2.   Preliminaries

2.1   Yang–Mills bar connection

Let $(X,g)$ be a smooth complex surface with a $(1,1)$-form $\omega$ and $P$ be a principal $G$-bundle over $X$ with $G$ being a compact Lie group. We denote by $\mathcal{A}_{P}$ the set of all connections. For any connection $A$ on $P$. We have the covariant exterior derivatives $d_{A}:\varOmega^{k}(X,\mathfrak{g}_{P})\rightarrow\varOmega^{k+1}(X,\mathfrak{g}_{P})$. Like the canonical splitting the exterior derivatives $d = \partial+\bar{\partial}$, decomposes over $X$ into $d_{A} = \partial_{A}+\bar{\partial}_{A}$. We also denote by $\varOmega^{p,q}(X,\mathfrak{g}_{P}^{\mathbb{C}})$ the space of $C^{\infty}$-$(p,q)$ forms on $\mathfrak{g}_{P}^{\mathbb{C}}: = \mathfrak{g}_{P}\otimes\mathbb{C}$.

We define a Hermitian inner product $\langle\cdot,\cdot\rangle$ on $\varOmega^{p,q}(X,\mathfrak{g}_{P}^{\mathbb{C}})$ by

where $\ast$ is the $\mathbb{C}$-linear extension of the Hodge operator over complex forms and $\bar{\bullet}$ is the conjugation on the bundle $\mathfrak{g}_{P}\otimes\mathbb{C}$-forms which is defined naturally. One can also see Ref. [5, Page 99] or Ref. [4]. Denote by $L_{\omega}$ the operator of exterior multiplication by the Kähler form $\omega$:

and, as usual, let $\varLambda_{\omega}$ denote its pointwise adjoint, i.e.,

It is well known that $\varLambda_{\omega} = \ast^{-1}\circ L_{\omega}\circ\ast$. We can decompose the curvature, $F_{A}$, as

where $F_{A0}^{1,1} = F_{A}^{1,1}-\frac{1}{2}\varLambda_{\omega}F_{A}\otimes\omega$. Ref. [11] defined two new energies:

We can write the Yang–Mills functional as

The energy functional $\|\varLambda_{\omega}F_{A}\|^{2}$ plays an important role in the study of Hermitian-Einstein connections, see Refs. [2, 3, 13]. If the connection $A$ is a Hermitian–Yang–Mills connection, i.e.,

where $\varLambda$ is a constant, then the Yang–Mills functional is minimum. Suppose that an integrable connection $A\in\mathcal{A}_{P}^{1,1}$ on a holomorphic bundle over a Kähler surface is Yang–Mills, then $\varLambda_{\omega}F_{A}$ is parallel, i.e., $\nabla_{A}\varLambda_{\omega}F_{A} = 0$.

Using the formula

we get the first variation of energy $E'(A)$ is given by

One can see that the Yang–Mills bar connection

is a critical point of $E'(A)$. Using the Bianchi identity $\bar{\partial}_{A}F_{A}^{0,2} = 0$, a Yang–Mills bar connection $A$ is equivalent to the $(0,2)$-part, $F_{A}^{0,2}$, of the curvature of the connection $A$ is harmonic with respect to the Laplacian operator $\Delta_{\bar{\partial}_{A}}$, i.e.,

2.2   Irreducible connection

In this section, we first recall a definition of irreducible connection on a principal $G$-bundle $P$, where $G$ being a compact, semisimple Lie group. Given a connection $A$ on a principal $G$-bundle $P$ over $X$. We can define the stabilizer $\varGamma_{A}$ of $A$ in the gauge group $\mathcal{G}_{P}$ by

One can also see Ref. [3, Section 4.2.2]. A connection $A$ is called reducible if the connection $A$ whose stabilizer $\varGamma_{A}$ is larger than the center $C(G)$ of $G$. Otherwise, the connections are irreducible, they satisfy $\varGamma_{A}\cong C(G)$. It is easy to see that a connection $A$ is irreducible when it admits no nontrivial covariantly constant Lie algebra-value $0$-form, i.e.,

The most useful definition of reducibility in our note is the following.

Definition 2.1.^{[12, Definition B.1]} A connection $A$ on a principal $SU(2)$ or $SO(3)$ bundle $P\rightarrow X$ is reducible if one of the following equivalent conditions is satisfied:

① The stabilizer of $A$ under the group of gauge transformations has a positive dimension.

② There exists a nonzero $\varGamma\in\varOmega^{0}(X,\mathfrak{g}_{P})$ such that $d_{A}\varGamma = 0$.

③ The holonomy of $A$ is contained in some $SO(2)$ subgroup.

We recall the definition of locally reducible connection on a principal $SU(2)$ or $SO(3)$ bundle $P$.

Definition 2.2.^{[12, Definition 2.1]} A connection $A$ on a principal $SU(2)$ or $SO(3)$ bundle $P$ over a smooth closed Riemanian manifold $X$ is locally reducible if there is an open cover of $X$ such that on each of the open subsets, there is a nonzero, covariantly constant section of $\mathfrak{g}_{P}$.

With regard to the local reducibility as in Definition 2.2, Tanaka observed that

Proposition 2.1.^{[12, Proposition B.3]} A connection $A$ on a principal $SU(2)$ or $SO(3)$-bundle $P\rightarrow X$ is locally reducible if and only if the holonomy of $A$ is contained in some $O(2)$ subgroup.

Remark 2.1. Let $A$ be a connection on a principal $SU(2)$ or $SO(3)$ bundle $P$. If $\pi_{1}(X)$ has no subgroup of index two, it follows that $H^{1}(X,\mathbb{Z}_{2}) = 0$, then every locally reducible connection A is reducible (see Ref. [12, Remark B.5]). In particular, a locally reducible connection on a closed simply connected manifold is reducible.

3.   Proof of main theorem

Let $(X,\omega)$ be a compact Kähler surface with a smooth Kähler $(1,1)$-form $\omega$. Given an orthonormal coframe $\{e_{0},e_{1},e_{2},e_{3}\}$ on $X$ for which $\omega = e^{01}+e^{23}$, where $e^{ij} = e^{i}\wedge e^{j}$. We define

so that

For any $B\in\varOmega^{+}(X,\mathfrak{g}_{P})$, we can write

where $B_{i}$, $i = 1,2,3$, takes value in $\mathfrak{g}_{P}$. Then we can define $\beta$ as follows:

Hence

We can rewrite $B$ as

We define a bilinear map

by $\dfrac{1}{2}[\cdot,\cdot]_{\varOmega^{2,+}}\otimes[\cdot,\cdot]_{\mathfrak{g}_{P}}$ (see Ref. [7, Appendxi A]). In a direct calculation (see Ref. [7, Section 7.1]),

Let $P$ be a principal $G$-bundle over a closed, smooth Riemannian four-dimensional manifold $(X,g)$ with Riemannian metric $g$. We recall a notion of rank of a section $B\in\varOmega^{+}(X,\mathfrak{g}_{P})$ (see Refs. [1] or [12, Definiton 1.5]. We denote $d = \dim G$. Choose local frames for $\mathfrak{g}_{P}$ and $\varLambda^{+}(T^{\ast}X)$, ($\dim\varLambda^{+}(T^{\ast}X) = 3)$, then the section $B$ is represented by a $d\times 3$ matrix-valued function with respect to the local frames. The rank of $B$ at a point $x\in X$ is the rank of the matrix at $x$. We denote by ${\rm{rank}}(B)$ the maximum of the pointwise rank over $X$. The pointwise rank of $B$ also provides a stratification of the manifold $X$, namely,

The top rank stratum is a nonempty open subset of $X$. If the structure group of the principal bundle $P$ is either $SU(2)$ or $SO(3)$, then the possibilities for the rank of $B$ are less than or equal to $3$. We next recall the following from Ref. [7, Section 4.11]

Lemma 3.1.^{[12, Lemma 1.6]} Let $P\rightarrow X$ be a principal $SU(2)$ or $SO(3)$ bundle over a closed four-dimensional Riemannian manifold $X$. If $B\in\varOmega^{+}(X,\mathfrak{g}_{P})$ satisfies $[B.B] = 0$, then the rank of $B$ is at most one. Furthermore,

Proof. Since the rank of $B$ is at most one, it is easy to see $X = X_{1}(B)\cup X_{0}(B)$. Noting that $X_{0}(B)$ is the zero set of B. Therefore, $X^{1}(B) = \{x\in X:B(x)\neq0 \}$.

We then obtain that

Proposition 3.1. Let $A$ be a connection on a principal $SU(2)$ or $SO(3)$-bundle over a compact Kähler surface. If the connection $A$ is a Yang–Mills bar connection, then $F^{0,2}_{A}+F_{A}^{2,0}$ has at most rank one.

Proof. Since $\bar{\partial}_{A}^{\ast}F_{A}^{0,2} = 0$, we have

In an orthonormal coframe, we can write $F_{A}^{0,2}$ as

where $B_{1}$ and $B_{2}$ take value in Lie algebra $\mathfrak{su}(2)$ or $\mathfrak{so}(3)$. Thus

Following Eq. (2), we obtain that

Hence

Note that $F_{A}^{0,2}+F_{A}^{2,0}$ is a self-dual two form that takes value in $\mathfrak{g}_{P}$. Following Eq. (3), we obtain

Therefore, following Lemma 3.1, $[F_{A}^{0,2}+F_{A}^{2,0}]$ has at most rank one. □

We now prove a useful lemma that will be crucial in the proof of Theorem 1.1. The idea follows from Ref. [3, Lemma 4.3.25].

Lemma 3.2. Let $X$ be a smooth closed Riemannian four-manifold with $H^{1}(X,\mathbb{Z}_{2}) = 0$, and let $P\rightarrow X$ be a principal $G$-bundle with structure group $G$ being either $SU(2)$ or $SO(3)$. Let $B\in\ker d^{+,\ast}_{A}\cap\varOmega^{+}(X,\mathfrak{g}_{P})$ be a nonzero self-dual $2$-form that takes value in $\mathfrak{g}_{P}$. If $B$ has at most rank $1$, the connection $A$ is reducible.

Proof. Let $Z^{c}$ denote the complement of the zero set of $B$. By unique continuation of the elliptic equation $d^{+,\ast}_{A}B = 0$, $Z^{c}$ is either empty or dense. If $Z^{c}$ is not empty, then $\phi$ has rank one and is nowhere vanishing on $Z^{c}$ (see Lemma 3.1). We denote by $\{U_{\alpha }\}$ a finite open cover of $X$. Locally, in each open set $U_{\alpha }$, we can write

where $s_{\alpha }$ is a section $\varGamma(\mathfrak{g}_{P},U_{\alpha })$ with $|s_{\alpha }| = 1$ and $\omega_{\alpha }\in\varOmega^{+}(U_{\alpha })$. It is easy to see $\omega_{\alpha }(x)\neq 0$, $x\in Z^{c}\cap U_{\alpha }$. Now the condition $|s_{\alpha }| = 1$ implies that $\langle d_{A}s_{\alpha },s_{\alpha }\rangle = 0$ on $U_{\alpha }$. The equation $d^{+,\ast}_{A}B = 0$ implies that

Therefore, we obtain

Since a nonvanishing, pure self-dual $2$-form $\omega_{\alpha }$ gives an isomorphism from $\varOmega^{1}(Z^{c}\cap U_{\alpha })$ to $\varOmega^{3}(Z^{c}\cap U_{\alpha })$, $s_{\alpha }$ is covariant, i.e., $\nabla_{A}s_{\alpha } = 0$ on $Z^{c}\cap U_{\alpha }$. Since $Z^{c}$ is dense on $X$, $\nabla_{A}s_{\alpha } = 0$ all over $U_{\alpha }$. Hence, the connection $A$ is locally reducible. Since $H^{1}(X,\mathbb{Z}_{2}) = 0$, $A$ is reducible (see Remark 2.1). □

Following Lemma 3.2, we then have

Corollary 3.1. Let $X$ be a smooth closed Riemannian four-manifold with $H^{1}(X,\mathbb{Z}_{2}) = 0$, and let $P\rightarrow X$ be a principal $G$-bundle with structure group $G$ being either $SU(2)$ or $SO(3)$. Let $B\in\ker d^{+,\ast}_{A}\cap\varOmega^{+}(X,\mathfrak{g}_{P})$. If $d_{A}^{+,\ast}B = 0$ and $[B.B] = 0$, the either $B$ vanishes or the connection $A$ is reducible.

Proof of Theorem 1.1. We now begin to prove Theorem 1.1. Following Proposition 3.1, $(F_{A}^{0,2}+F_{A}^{2,0})\in\varOmega^{+}(X,\mathfrak{g}_{P})$ has at most rank one. Noting that

According to Corollary 3.1, $F_{A}^{0,2}$ vanishes over all of $X$ since connection $A$ is irreducible. □

Acknowledgements

This work was supported by the National Natural Science Foundation of China (12271496), the Youth Innovation Promotion Association CAS, the Fundamental Research Funds of the Central Universities, and the USTC Research Funds of the Double First-Class Initiative.

Conflict of interest

The author declares that he has no conflict of interest.