Complex vector bundle

In mathematics, a complex vector bundle is a <a href="/facts/Vector_bundle/cXACAa1I">vector bundle</a> whose fibers are <a href="/facts/Complex_vector_space/M1pxMLx2">complex vector spaces</a>.
Any complex vector bundle can be viewed as a <a href="/facts/Real_vector_bundle/cXACAa1I">real vector bundle</a> through the <a href="/facts/Restriction_of_scalars/RYGsTsYk">restriction of scalars</a>. Conversely, any real vector bundle 
 
 
 
 E
 
 
 {\displaystyle E}
 
 can be promoted to a complex vector bundle, the <a href="/facts/Complexification/fDfFLccF">complexification</a>

E
        ⊗
        
          C
        
        ;
      
    
    {\displaystyle E\otimes \mathbb {C} ;}

whose fibers are 
 
 
 
 
 E
 
 x
 
 
 
 ⊗
 
 
 R
 
 
 
 
 C
 
 
 
 {\displaystyle E_{x}\otimes _{\mathbb {R} }\mathbb {C} }
 
.
Any complex vector bundle over a <a href="/facts/Paracompact_space/rjTrvF7j">paracompact space</a> admits a <a href="/facts/Hermitian_metric/69fYok0x">hermitian metric</a>.
The basic invariant of a complex vector bundle is a <a href="/facts/Chern_class/Tw2WD5hM">Chern class</a>. A complex vector bundle is canonically <a href="/facts/Oriented_vector_bundle/FXAzI2UD">oriented</a>; in particular, one can take its <a href="/facts/Euler_class/UnjlBrKe">Euler class</a>.
A complex vector bundle is a <a href="/facts/Holomorphic_vector_bundle/FeRrK9Bg">holomorphic vector bundle</a> if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a <a href="/facts/Complex_manifold/DlvPnpzm">complex manifold</a> and if the local trivializations are <a href="/facts/Biholomorphic/3g5kL1RD">biholomorphic</a>.

Complex vector bundle open-in-new

Complex vector bundle