Birational invariance of the geometric genus of a surface
= Birational invariance of the geometric genus of a surface
If $\pi:\widetilde X\to X$ is the blowup of a smooth surface at a point, then $K_{\widetilde X}=\pi^*K_X+E$ and $\pi_*K_{\widetilde X}=K_X$. Pullback therefore identifies the global holomorphic two-forms and gives $p_g(\widetilde X)=p_g(X)$.