Show that a space of holomorphic sections of a line bundle is isomorphic to the space of meromorphic...
$begingroup$
Let $Sigma$ be a Riemann surface, $x in Sigma$ and let $z: U rightarrow D subset mathbb{C}$ be a local coordinate system centered at $x$. For every $k in mathbb{Z}$, a holomorphic line bundle $L_{kx}$ on $Sigma$ is defined by gluing the trivial vector bundles $(Sigma backslash {x}) times mathbb{C} rightarrow Sigma$ and $U times mathbb{C} rightarrow U$ via the holomorphic transition function
$$(U backslash {x}) times mathbb{C} rightarrow (U backslash {x}) times mathbb{C}, (p, v) mapsto (p, z(p)^k v)$$
Can someone give me some hints how to prove the following claims ?
The space of global holomorphic sections of $L_{kx}$ is isomorphic to the space of meromorphic functions on $Sigma$, holomorphic outside of $x$ and has a pole of order at most $k$ if $k geq 0$ and a zero of order $k$ if $k < 0$.
Thanks for your help.
complex-geometry vector-bundles riemann-surfaces holomorphic-bundles line-bundles
asked Jan 12 at 10:52
CrystalCrystal
33319
$endgroup$
add a comment |
$begingroup$
Let $Sigma$ be a Riemann surface, $x in Sigma$ and let $z: U rightarrow D subset mathbb{C}$ be a local coordinate system centered at $x$. For every $k in mathbb{Z}$, a holomorphic line bundle $L_{kx}$ on $Sigma$ is defined by gluing the trivial vector bundles $(Sigma backslash {x}) times mathbb{C} rightarrow Sigma$ and $U times mathbb{C} rightarrow U$ via the holomorphic transition function
$$(U backslash {x}) times mathbb{C} rightarrow (U backslash {x}) times mathbb{C}, (p, v) mapsto (p, z(p)^k v)$$
Can someone give me some hints how to prove the following claims ?
The space of global holomorphic sections of $L_{kx}$ is isomorphic to the space of meromorphic functions on $Sigma$, holomorphic outside of $x$ and has a pole of order at most $k$ if $k geq 0$ and a zero of order $k$ if $k < 0$.
Thanks for your help.
complex-geometry vector-bundles riemann-surfaces holomorphic-bundles line-bundles
asked Jan 12 at 10:52
CrystalCrystal
33319
$endgroup$
add a comment |
$begingroup$
Let $Sigma$ be a Riemann surface, $x in Sigma$ and let $z: U rightarrow D subset mathbb{C}$ be a local coordinate system centered at $x$. For every $k in mathbb{Z}$, a holomorphic line bundle $L_{kx}$ on $Sigma$ is defined by gluing the trivial vector bundles $(Sigma backslash {x}) times mathbb{C} rightarrow Sigma$ and $U times mathbb{C} rightarrow U$ via the holomorphic transition function
$$(U backslash {x}) times mathbb{C} rightarrow (U backslash {x}) times mathbb{C}, (p, v) mapsto (p, z(p)^k v)$$
Can someone give me some hints how to prove the following claims ?
The space of global holomorphic sections of $L_{kx}$ is isomorphic to the space of meromorphic functions on $Sigma$, holomorphic outside of $x$ and has a pole of order at most $k$ if $k geq 0$ and a zero of order $k$ if $k < 0$.
Thanks for your help.
complex-geometry vector-bundles riemann-surfaces holomorphic-bundles line-bundles
asked Jan 12 at 10:52
CrystalCrystal
33319
$endgroup$
Let $Sigma$ be a Riemann surface, $x in Sigma$ and let $z: U rightarrow D subset mathbb{C}$ be a local coordinate system centered at $x$. For every $k in mathbb{Z}$, a holomorphic line bundle $L_{kx}$ on $Sigma$ is defined by gluing the trivial vector bundles $(Sigma backslash {x}) times mathbb{C} rightarrow Sigma$ and $U times mathbb{C} rightarrow U$ via the holomorphic transition function
$$(U backslash {x}) times mathbb{C} rightarrow (U backslash {x}) times mathbb{C}, (p, v) mapsto (p, z(p)^k v)$$
Can someone give me some hints how to prove the following claims ?
The space of global holomorphic sections of $L_{kx}$ is isomorphic to the space of meromorphic functions on $Sigma$, holomorphic outside of $x$ and has a pole of order at most $k$ if $k geq 0$ and a zero of order $k$ if $k < 0$.
Thanks for your help.
complex-geometry vector-bundles riemann-surfaces holomorphic-bundles line-bundles
complex-geometry vector-bundles riemann-surfaces holomorphic-bundles line-bundles
asked Jan 12 at 10:52
CrystalCrystal
33319
asked Jan 12 at 10:52
CrystalCrystal
33319
asked Jan 12 at 10:52
CrystalCrystal
33319
asked Jan 12 at 10:52
CrystalCrystal
33319
asked Jan 12 at 10:52
CrystalCrystal
33319
33319
add a comment |
add a comment |
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