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If $ M $ is a differentiable manifold equipped with an Atlas $ \mathcal{A} = ( U_i , \varphi_i )_{ i \in I} $, we can then calculate the integral of a differential form $ \omega $ over $ M $ with the partition of unity $ \{\rho_1 , \dots , \rho_n \} $ by the formula : $ \int_M \omega = \displaystyle \sum_{ i = 1}^{ n} \int_{M } \rho_i \omega $ with : $ \rho_i \omega \in \Omega_c ( U_i ) $.

My question is whether there is a way to calculate the integral of a differential form $ \omega $ over a projective algebraic variety $ X = V ( f_1 , \dots , f_n ) $ defined by: $ \int_X \omega $ ? and what type are $ \omega $ which admit integration $ \int_X \omega $, when $ X $ is a complex projective algebraic variety ?

Thank you in advance for your help.

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