A Hermitian metric on a complex vector bundle assigns a Hermitian inner product to every
fiber. The basic example is the trivial
bundle 
, where 
is an open set in 
. Then a positive
definite Hermitian matrix 
defines a Hermitian metric by
 |
where 
is the complex conjugate of 
. By a partition of
unity, any complex vector bundle has
a Hermitian metric.
In the special case of a complex manifold, the holomorphic tangent bundle 
may have a Hermitian metric, in which case its real
part is a Riemannian metric and the negative
of its imaginary part is a nondegenerate alternating multilinear form 
. When 
is closed, i.e., in this
case a symplectic form, then 
is a Kähler form.
On a holomorphic vector bundle with a Hermitian metric 
, there is a unique vector bundle connection compatible with
and the complex
structure. Its 
component is 
. In a local holomorphic trivialization,
write a bundle section 
as a column of coefficient functions and let 
be the matrix representing 
in the convention 
. Then
 |
where 
is the exterior derivative and 
is its 
component.
