# 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.