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fixed typos for dimension
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DamienC
DamienC
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Let $M=\mathbb C^g/ \Gamma$ be a complex tori and $E$ a be a holomorphic vector bundle of rank $r$ over $M$. Then $E$ is characterised by factor of automorphy, i.e. a holomorphic map $J:\Gamma\times\mathbb C^r\to GL(n,\mathbb C)$$J:\Gamma\times\mathbb C^g\to GL(r,\mathbb C)$ such that $J(\gamma'\gamma,x)=J(\gamma',\gamma x)J(\gamma,x)$. If $f:M\to M$ is a holomorphic diffeomorphism of $M$, $f^*(E)$ is the pull-back bundle. Then can we deduce that $f^*(E)$ is given by a factor of automorphy $J_f(\gamma,x)=J(\gamma,f(x))$ ?

Let $M=\mathbb C^g/ \Gamma$ be a complex tori and $E$ a be a holomorphic vector bundle of rank $r$ over $M$. Then $E$ is characterised by factor of automorphy, i.e. a holomorphic map $J:\Gamma\times\mathbb C^r\to GL(n,\mathbb C)$ such that $J(\gamma'\gamma,x)=J(\gamma',\gamma x)J(\gamma,x)$. If $f:M\to M$ is a holomorphic diffeomorphism of $M$, $f^*(E)$ is the pull-back bundle. Then can we deduce that $f^*(E)$ is given by a factor of automorphy $J_f(\gamma,x)=J(\gamma,f(x))$ ?

Let $M=\mathbb C^g/ \Gamma$ be a complex tori and $E$ a be a holomorphic vector bundle of rank $r$ over $M$. Then $E$ is characterised by factor of automorphy, i.e. a holomorphic map $J:\Gamma\times\mathbb C^g\to GL(r,\mathbb C)$ such that $J(\gamma'\gamma,x)=J(\gamma',\gamma x)J(\gamma,x)$. If $f:M\to M$ is a holomorphic diffeomorphism of $M$, $f^*(E)$ is the pull-back bundle. Then can we deduce that $f^*(E)$ is given by a factor of automorphy $J_f(\gamma,x)=J(\gamma,f(x))$ ?

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Mjr
Mjr
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Pull-back of factor of automorphy

Let $M=\mathbb C^g/ \Gamma$ be a complex tori and $E$ a be a holomorphic vector bundle of rank $r$ over $M$. Then $E$ is characterised by factor of automorphy, i.e. a holomorphic map $J:\Gamma\times\mathbb C^r\to GL(n,\mathbb C)$ such that $J(\gamma'\gamma,x)=J(\gamma',\gamma x)J(\gamma,x)$. If $f:M\to M$ is a holomorphic diffeomorphism of $M$, $f^*(E)$ is the pull-back bundle. Then can we deduce that $f^*(E)$ is given by a factor of automorphy $J_f(\gamma,x)=J(\gamma,f(x))$ ?