Transition maps for holomorphic vector bundle
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In the definition of a holomorphic line bundle of rank $n$, it is required that the transition maps $g_ij:U_icap U_jtotextGL_n(Bbb C)$ are holomorphic.
What does this actually mean? Are we seeing $textGL_n(Bbb C)$ as a complex manifold? Otherwise I can't even make sense of this being continuous? Is it a problem that these may have highly different dimensions?
I am mainly thinking about Riemann surfaces.
vector-bundles complex-manifolds
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up vote
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In the definition of a holomorphic line bundle of rank $n$, it is required that the transition maps $g_ij:U_icap U_jtotextGL_n(Bbb C)$ are holomorphic.
What does this actually mean? Are we seeing $textGL_n(Bbb C)$ as a complex manifold? Otherwise I can't even make sense of this being continuous? Is it a problem that these may have highly different dimensions?
I am mainly thinking about Riemann surfaces.
vector-bundles complex-manifolds
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In the definition of a holomorphic line bundle of rank $n$, it is required that the transition maps $g_ij:U_icap U_jtotextGL_n(Bbb C)$ are holomorphic.
What does this actually mean? Are we seeing $textGL_n(Bbb C)$ as a complex manifold? Otherwise I can't even make sense of this being continuous? Is it a problem that these may have highly different dimensions?
I am mainly thinking about Riemann surfaces.
vector-bundles complex-manifolds
In the definition of a holomorphic line bundle of rank $n$, it is required that the transition maps $g_ij:U_icap U_jtotextGL_n(Bbb C)$ are holomorphic.
What does this actually mean? Are we seeing $textGL_n(Bbb C)$ as a complex manifold? Otherwise I can't even make sense of this being continuous? Is it a problem that these may have highly different dimensions?
I am mainly thinking about Riemann surfaces.
vector-bundles complex-manifolds
asked Aug 18 at 8:15
user488646
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