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 of a complex variable . The complex derivative of is the limit of the difference quotient as approaches 0 in . In particular, we get the same limit when approaches through with and also the same limit when approaches 0 through with . So we have . The equation is the Cauchy-Riemann equations. When we write and separate the real and imaginary parts, we get back the usual Cauchy-Riemann equations and . We can rewrite in the form , where . The reason for the factor is that we want the value of the 1-form evaluated at to be 1. Note that , are tangent vectors and is a -linear combination of tangent vectors and therefore is an element of the complexification of the tangent space of . 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 of complex variables can be defined either as a function expressible as a convergent power series of the complex variables or as a function satisfying the Cauchy-Riemann equations for , where . When we allow tangent vectors to have complex coefficients, , form a basis of the complexified tangent space of over at every point of . Let be the -vector subspace of spanned by and let be the -vector subspace of spanned by . Suppose we have another set of local holomorphic coordinates . Because of the Cauchy-Riemann equations and , the subspaces and are independent of the choice of local holomorphic coordinates and one can do similar constructions on complex manifolds. Elements of are called the -directions and elements of are called the -directions. A -form on with complex coefficients is an element of , where means the dual space of and means taking the th exterior product. We have the direct sum decomposition
An element of is called a -form. The notion of -forms is independent of the choice of local holomorphic coordinates and can be carried over to complex manifolds. A -form is uniquely decomposable into a sum of -forms with .
Let be a complex manifold of dimension and be a holomorphic vector bundle of -rank over . Let be a Hermitian metric along the fibers of . With respect to a local trivialization of the Hermitian metric is a positive Hermitian matrix . We are going to use the first index as the row index and the second index as the column index for the matrix .
We introduce the concept of a complex metric connection for the Hermitian vector bundle with metric . Suppose , , is a smooth local basis of . Suppose we have a connection and use to denote the operator of differentiating sections. The result of applying to is a local -valued 1-form on . We express in terms of the basis , and get , where is a local 1-form on and the summation convention of summing over repeated indices is used. We use to denote the matrix and regard the subscript as the row index and the superscript as the column index of the matrix .
Since the bundle is holomorphic, it is possible to define partial differentiation in the direction in a natural way, namely the derivative of a local holomorphic section of is defined to be zero and the 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 direction is the natural one just described. When the local basis is holomorphic, a connection is complex if and only if the local 1-forms are all of type .
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
for any local smooth section and of , where denotes the pointwise inner product defined by the metric and the equation means that both sides give the same value when evaluated at any tangent vector of . For and the above equation reads
as one can easily verify by evaluating both sides at a tangent vector of . In matrix notations this means that
where is the complex conjugate of and the superscript of means the transpose of . By breaking down the equation into the and components, we get two equations and , because , being a complex connection, is a matrix of forms. The two equations and are equivalent, because is a Hermitian matrix, as one can easily see by evaluating at a tangent vector of and taking complex conjugates and transposes of the matrices.
Given any Hermitian metric along the fibers of a holomorphic vector bundle there exists one and only one complex metric connection . We verify the statement by taking a local holomorphic basis . The two conditions are: (i) is a matrix of -forms; and (ii) . Thus is the unique complex metric connection.
There is a another interpretation of this connection. A Hermitian metric defines an isomorphism from to the complex conjugate of its dual . The transition function are the complex conjugate of those of . Since the transition functions on are anti-holomorphic, there is a natural connection of for the directions. Take a local smooth section of . We can define as . We claim that this is a complex metric connection. Take a local holomorphic basis of and write . Then , where is the local basis of corresponding to the basis . It follows from that
and the connection is the complex metric connection.