Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\omega\partial_z\bar{\partial}_z\log|z|^2 $$
where $\partial_z,\bar{\partial}_z$ denote the differential operators on $\mathbb{P}^1$.
My question is how can we use Stokes's theorem (in the integration by parts) here given that the function $\log|z|^2$ has singularities at $0$ and $\infty$.
References: http://www.math.stonybrook.edu/~leontak/On%20Bott-Chern%20forms%20and%20their%20applications.pdf page 24, Soulé's "Lecture on Arakelov Geometry" page 81 (top).