# Integration over a variety

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 ?