holomorphic vector bundle= vector bundle over a complex manifold with a connection that has holomorphic transition functions up to equivalence under transition functions.

: holomorphic principle bundle representation

Fiber is

consider sections of ,

f is holomorphic iff .

necessary condition for existence of f is This means in other words, a portion of the curvature must be 0.

curvature = 0 for the antiholomorphic part means with , not necessarily in G.

If you generalize gauge to then you can choose: and .

Then the covariant derivatives: , satisfy:

• 
• 

Then simplifies too .

Example of holomorphic vector bundle: tangent bundles

Theorem: holomorphic principle bundle, P(M,G), then for every representation p of G, an associated holomorphic vector bundle .

To prove this, relate to via transition functions. Then derive that .

To holomorphic transition functions and are equivalent iff where are holomorphic.

source psfile jl@crush.caltech.edu index holomorphic_section quotient_bundle