holomorphic vector bundle= vector bundle over a complex manifold with a connection that has holomorphic transition functions up to equivalence under tex2html_wrap130 transition functions.


tex2html_wrap132 : holomorphic principle bundle representation

Fiber is tex2html_wrap133

consider sections of tex2html_wrap132 , tex2html_wrap135

f is holomorphic iff tex2html_wrap136 .

necessary condition for existence of f is tex2html_wrap137 This means tex2html_wrap138 in other words, a portion of the curvature must be 0.

curvature = 0 for the antiholomorphic part means tex2html_wrap139 with tex2html_wrap140 , not necessarily in G.

If you generalize gauge to tex2html_wrap130 then you can choose: tex2html_wrap142 and tex2html_wrap143 .

Then the covariant derivatives: tex2html_wrap144 , satisfy:

Then tex2html_wrap147 simplifies too tex2html_wrap148 .

Example of holomorphic vector bundle: tangent bundles

Theorem: holomorphic principle bundle, P(M,G), then for every representation p of G, tex2html_wrap149 an associated holomorphic vector bundle tex2html_wrap150 .

To prove this, relate tex2html_wrap151 to tex2html_wrap152 via transition functions. Then derive that tex2html_wrap153 .

To holomorphic transition functions tex2html_wrap154 and tex2html_wrap155 are equivalent iff tex2html_wrap156 where tex2html_wrap157 are holomorphic.

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