Holomorphic Line Bundle

A complex line bundle is a vector bundle  whose fibers  are a copy of .  is a holomorphic line bundle if it is a holomorphic map between complex manifolds and its transition functions are holomorphic.

On a compact Riemann surface, 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 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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