A holomorphic line bundle is a complex vector bundle 
whose fibers 
are copies of 
, whose projection is a holomorphic
map between complex manifolds and whose transition functions are holomorphic.
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On a Riemann surface that is a compact space, a variety divisor 
determines a line bundle.
For example, consider 
on 
. Around 
there is a coordinate chart 
given by the holomorphic
function 
with 
. Similarly, 
is a holomorphic function
defining a disjoint coordinate chart 
around 
with 
.
Then letting 
, the Riemann
surface is covered by 
.
The line bundle corresponding to 
is then defined by the following transition
functions,
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(1)
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(2)
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