TOPICS
Search

Holomorphic Line Bundle


A complex line bundle is a vector bundle pi:E->M whose fibers pi^(-1)(m) are a copy of C. pi is a holomorphic line bundle if it is a holomorphic map between complex manifolds and its transition functions are holomorphic.

HolomorphicLineMap
HolomorphicLineBundle

On a compact Riemann surface, a variety divisor sumn_ip_i determines a line bundle. For example, consider 2p-q on X. Around p there is a coordinate chart U given by the holomorphic function z_p with z_p(p)=0. Similarly, z_q is a holomorphic function defining a disjoint chart V around q with z_q(q)=0. Then letting W=X-{p,q}, the Riemann surface is covered by X=U union V union W. The line bundle corresponding to 2p-q is then defined by the following transition functions,

g_(UW)(x)=z_p(x)^2 defined for x in U intersection W
(1)
g_(VW)(x)=z_q(x)^(-1) defined for x in V intersection W.
(2)

See also

Chern Class, Hermitian Metric, Holomorphic Function, Holomorphic Tangent Bundle, Holomorphic Vector Bundle, Line Bundle, Riemann Surface, Vector Bundle

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd. "Holomorphic Line Bundle." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HolomorphicLineBundle.html

Subject classifications