Connections, The Kähler Condition, and Curvature – II. Connections for Holomorphic Vector Bundles

Before we discuss the case of a holomorphic vector bundle over a complex manifold, let us review the Cauchy-Riemann equations and fix the notations. Suppose we have a complex valued function {f(z)} of a complex variable {z = x + iy}. The complex derivative {f'(z)} of {f} is the limit of the difference quotient {{{f(z + \Delta z)-f(z)}\over{\Delta z}}} as {\Delta z} approaches 0 in {\mathbb{C}}. In particular, we get the same limit when {\Delta z} approaches {0} through {\Delta x} with {\Delta y = 0} and also the same limit when {\Delta z} approaches 0 through {i \Delta y} with {\Delta x = 0}. So we have {f'(z) = {{\partial f}\over{\partial x}} = {1\over i} {{\partial f}\over{\partial y}}}. The equation {{{\partial f}\over{\partial x}} = {1\over i} {{\partial f}\over{\partial y}}} is the Cauchy-Riemann equations. When we write {f = u + iv} and separate the real and imaginary parts, we get back the usual Cauchy-Riemann equations {{{\partial{u}\over{\partial{x}}}} = {{\partial{v}\over{\partial{y}}}}} and {{{\partial{u}\over{\partial{y}}}} = -{{\partial{v}\over{\partial{x}}}}}. We can rewrite {{{\partial{f}}\over{\partial{x}}} = {1\over{i}}{{\partial f}\over{\partial y}}} in the form {{\partial\over{\partial {\overline{z}}}}f = 0}, where {{\partial\over{\partial \overline{z}}} = {1\over2}\left({\partial\over{\partial x}} + i {\partial\over{\partial y}}\right)}. The reason for the factor {{1\over2}} is that we want the value of the 1-form {d{\overline{z}}} evaluated at {{\partial\over{\partial\overline{z}}}} to be 1. Note that {{\partial\over{\partial x}}}, {{\partial\over{\partial y}}} are tangent vectors and {{\partial\over{\partial\overline{z}}}} is a {\mathbb{C}}-linear combination of tangent vectors and therefore is an element of the complexification of the tangent space of {\mathbb{C}}. A holomorphic function can be defined either as a function expressible as a convergent power series of the complex variable or as a function satisfying the Cauchy-Riemann equations. In the higher dimensional case the situation is the same. A holomorphic function {f\left(z^1, \dots, z^n\right)} of complex variables {z^1, \dots, z^n} can be defined either as a function expressible as a convergent power series of the complex variables {z^1, \dots, z^n} or as a function satisfying the Cauchy-Riemann equations {{{\partial f}\over{\partial \overline{z^\nu}}} = 0} for {1 \le \nu \le n}, where {{{\partial f}\over{\partial \overline{z^\nu}}} = {1\over2}\left({\partial\over{\partial x^\nu}} + i {\partial\over{\partial y^nu}}\right)}. When we allow tangent vectors to have complex coefficients, {{\partial\over{\partial z^\nu}}}, {{\partial\over{\partial \overline{z^\nu}}}} {(1 \le \nu \le n)} form a basis of the complexified tangent space {T_{\mathbb{C}^n} \otimes \mathbb{C}} of {\mathbb{C}^n} over {\mathbb{C}} at every point of {\mathbb{C}^n}. Let {T^{1,\,0}} be the {\mathbb{C}}-vector subspace of {T_{\mathbb{C}^n} \otimes \mathbb{C}} spanned by {{\partial\over{\partial z^\nu}}} {(1 \le \nu \le n)} and let {T^{0,\,1}} be the {\mathbb{C}}-vector subspace of {T_{\mathbb{C}^n} \otimes \mathbb{C}} spanned by {{\partial\over{\partial \overline{z^\nu}}}} {(1 \le \nu \le n)}. Suppose we have another set of local holomorphic coordinates {w^1, \dots, w^n}. Because of the Cauchy-Riemann equations {{{\partial w^\mu}\over{\partial \overline{z^\nu}}}=0} and {{{\partial z^\mu}\over{\partial \overline{w^\nu}}}=0}, the subspaces {T^{1,\,0}} and {T^{1,\,0}} are independent of the choice of local holomorphic coordinates and one can do similar constructions on complex manifolds. Elements of {T^{1,\,0}} are called the {(1,0)}-directions and elements of {T^{0,\,1}} are called the {(0,1)}-directions. A {k}-form on {\mathbb{C}^n} with complex coefficients is an element of {\wedge^k\left(T_{\mathbb{C}^n} \otimes \mathbb{C}\right)^*}, where {\left(T_{\mathbb{C}^n} \otimes \mathbb{C}\right)^*} means the dual space of {T_{\mathbb{C}^n} \otimes \mathbb{C}} and {\wedge^k} means taking the {k}th exterior product. We have the direct sum decomposition

\displaystyle \bigwedge^k\left(T_{\mathbb{C}^n} \otimes \mathbb{C}\right)^* = \bigoplus_{\nu=0}^k \left(\bigwedge^nu\left(T^{1,\,0}\right)^*\right) \otimes \left(\bigwedge^{k-\nu}\left(T^{0,\,1}\right)^*\right).

An element of {\left(\wedge^p\left(T^{1,\,0}\right)^*\right) \otimes \left(\wedge^q\left(T^{0,\,1}\right)^*\right)} is called a {(p,q)}-form. The notion of {(p,q)}-forms is independent of the choice of local holomorphic coordinates and can be carried over to complex manifolds. A {k}-form is uniquely decomposable into a sum of {(\nu, k-\nu)}-forms with {0 \le \nu \le k}.

Let {M} be a complex manifold of dimension {n} and {V} be a holomorphic vector bundle of {\mathbb{C}}-rank {r} over {M}. Let {H} be a Hermitian metric along the fibers of {V}. With respect to a local trivialization of {V} the Hermitian metric {H} is a positive Hermitian matrix {\left(H_{\alpha\overline{\beta}}\right)_{1\le \alpha,\,\beta \le r}}. We are going to use the first index {\alpha} as the row index and the second index {\overline{\beta}} as the column index for the matrix {\left(H_{\alpha\overline{\beta}}\right)_{1\le \alpha,\,\beta \le r}}.

We introduce the concept of a complex metric connection for the Hermitian vector bundle {V} with metric {H}. Suppose {e_\alpha}, {1 \le \alpha \le r}, is a smooth local basis of {V}. Suppose we have a connection and use {D} to denote the operator of differentiating sections. The result {De_{\alpha}} of applying {D} to {e_\alpha} is a local {E}-valued 1-form on {M}. We express {De_\alpha} in terms of the basis {e_\beta}, {1\le \beta \le r} and get {De_\alpha = \omega_\alpha^\beta e_\beta}, where {\omega_\alpha^\beta} is a local 1-form on {M} and the summation convention of summing over repeated indices is used. We use {\omega} to denote the matrix {\left(\omega_\alpha^\beta\right)_{1 \le \alpha,\,\beta \le r}} and regard the subscript {\alpha} as the row index and the superscript {\beta} as the column index of the matrix {\left(\omega_{\alpha}^\beta\right)_{1\le \alpha,\,\beta \le r}}.

Since the bundle {V} is holomorphic, it is possible to define partial differentiation in the {(0,1)} direction in a natural way, namely the {(0,1)} derivative of a local holomorphic section of {V} is defined to be zero and the {(0, 1)} derivative of any smooth section is defined by expressing it in terms of a local holomorphic basis and using the Leibniz rule of differentiating products. A connection is said to be complex if its partial differentiation in the {(0,1)} direction is the natural one just described. When the local basis {e_\alpha} {(1 \le \alpha \le r)} is holomorphic, a connection {\left(\omega_{\alpha}^\beta\right)_{1\le \alpha,\,\beta \le r}} is complex if and only if the local 1-forms {\omega_\alpha^\beta} are all of type {(1,0)}.

As with the real case we have the concept of compatibility with the metric. A connection is compatible with the metric if and only if

\displaystyle d\langle u,v\rangle = \langle Du, v \rangle + \langle u, Dv \rangle

for any local smooth section {u} and {v} of {V}, where {\langle\cdot,\cdot\rangle} denotes the pointwise inner product defined by the metric {H} and the equation means that both sides give the same value when evaluated at any tangent vector of {M}. For {u = e_\alpha} and {v = e_\beta} the above equation reads

\displaystyle dH_{\alpha\overline{\beta}} = \omega_\alpha^\gamma H_{\gamma\overline{\beta}} + H_{\alpha\overline{\gamma}}\overline{\omega_\beta^\gamma}

as one can easily verify by evaluating both sides at a tangent vector of {M}. In matrix notations this means that

\displaystyle dH = \omega H + H\overline{\omega}^t,

where {\overline{\omega}} is the complex conjugate of {\omega} and the superscript {t} of {\overline{\omega}} means the transpose of {\overline{\omega}}. By breaking down the equation into the {(1, 0)} and {(0,1)} components, we get two equations {\partial H = \omega H} and {\overline{\partial}H = H\overline{\omega}^t}, because {\omega}, being a complex connection, is a matrix of {(1, 0)} forms. The two equations {\partial H = \omega H} and {\overline{\partial} H = H \overline{\omega}^t} are equivalent, because {H} is a Hermitian matrix, as one can easily see by evaluating at a tangent vector of {M} and taking complex conjugates and transposes of the matrices.

Given any Hermitian metric {H} along the fibers of a holomorphic vector bundle {V} there exists one and only one complex metric connection {\omega}. We verify the statement by taking a local holomorphic basis {e_\alpha} {(1 \le \omega \le r)}. The two conditions are: (i) {A} is a matrix of {(1, 0)}-forms; and (ii) {\partial H = \omega}. Thus {\omega = (\partial H)H^{-1}} is the unique complex metric connection.

There is a another interpretation of this connection. A Hermitian metric defines an isomorphism {\Psi} from {V} to the complex conjugate of its dual {\overline{V^*}}. The transition function {\overline{V^*}} are the complex conjugate of those of {V^*}. Since the transition functions on {\overline{V^*}} are anti-holomorphic, there is a natural connection of for the {(1, 0)} directions. Take a local smooth section {s} of {V}. We can define {D^{1,0}s} as {\Psi^{-1}\left(D^{0,1}\Psi s\right)}. We claim that this is a complex metric connection. Take a local holomorphic basis {e_\alpha} {(1 \le \alpha \le r)} of {V} and write {s = s^\alpha e_\alpha}. Then {\Psi s = \left(H_{\alpha\overline{\beta}} s^\alpha\right)\overline{e_*^\beta}}, where {\overline{e_*^\beta}} {(1 \le \beta \le r)} is the local basis of {\overline{V^*}} corresponding to the basis {e_\alpha} {(1 \le \alpha \le r)}. It follows from {D^{1, 0}\Psi s = \left(\partial\left(H_{\alpha\overline{\beta}}s^\alpha\right)\right)\overline{e_*^\beta}} that

\displaystyle \Psi^{-1}\left(D^{1,0}\Psi s\right) = H^{\overline{\beta}\gamma}\left(\partial\left(H_{\alpha\overline{\beta}}s^\alpha\right)\right)e_\gamma = \left(\partial s^\gamma + H^{\overline{\beta}\gamma}\left(\partial H_{\alpha\overline{\beta}}\right)s^\alpha\right)e_\gamma

and the connection is the complex metric connection.

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