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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.

HolomorphicLineMapHolomorphicLineBundle

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-Roch Theorem, Riemann Surface, Vector Bundle

This entry contributed by Todd Rowland

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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

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