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edited Dec 16, 2011 at 6:09

$\Gamma (X\times Y,p_1^{}E\otimes p_2^{}F)=\Gamma (X,E)\otimes\Gamma (Y,F)?$ Global sections of tensor product of pull-back of two vector bundles

Suppose X,Y are two complex manifolds and E,F are vector bundles over them respective, what can I say about their global sections? Does this formula $\Gamma (X\times Y,p_1^{*}E\otimes p_2^{*}F)=\Gamma (X,E)\otimes\Gamma (Y,F)$ hold？

If it holds，then we will get every holomorphic function of two complex variables will be the form $f_1(z_1)g_1(z_2)+\cdots+f_n(z_1)g_n(z_2)$. It seems plausibile.

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asked Dec 15, 2011 at 6:00

MZWang

$\Gamma (X\times Y,p_1^{}E\otimes p_2^{}F)=\Gamma (X,E)\otimes\Gamma (Y,F)?$

Suppose X,Y are two complex manifolds and E,F are vector bundles over them respective, what can I say about their global sections?

complex-geometry

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$\Gamma (X\times Y,p_1^{*}E\otimes p_2^{*}F)=\Gamma (X,E)\otimes\Gamma (Y,F)?$ Global sections of tensor product of pull-back of two vector bundles

$\Gamma (X\times Y,p_1^{*}E\otimes p_2^{*}F)=\Gamma (X,E)\otimes\Gamma (Y,F)?$

Global sections of tensor product of pull-back of two vector bundles

$\Gamma (X\times Y,p_1^{*}E\otimes p_2^{*}F)=\Gamma (X,E)\otimes\Gamma (Y,F)?$

$\Gamma (X\times Y,p_1^{}E\otimes p_2^{}F)=\Gamma (X,E)\otimes\Gamma (Y,F)?$ Global sections of tensor product of pull-back of two vector bundles

$\Gamma (X\times Y,p_1^{}E\otimes p_2^{}F)=\Gamma (X,E)\otimes\Gamma (Y,F)?$

$\Gamma (X\times Y,p_1^{}E\otimes p_2^{}F)=\Gamma (X,E)\otimes\Gamma (Y,F)?$