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I read the fact somewhere that the dimension of cohomology group $H^i(X,\mathcal{O}_X)$ is a birational invariant.

For surfaces, the reference for this fact is Hartshorne Proposition 3.4 saying $H^i(X, \mathcal{O}_X) \cong H^i(\tilde{X}, \mathcal{O}_{\tilde{X}})$ for all $i \geq 0$ where $\tilde{X}$ is a blow-up of $X$ at $1$ point, combined with the fact that any birational morphism between surfaces factors as a finite number of blow-ups (of a point) and an isomorphism (Beauville-Complex Algebraic Surfaces, Theorem II.11).

What is the source for varieties of higher dimensions?

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Unfortunately, the reference you provide is only valid for smooth surfaces. I don't know if this should hold for blow-ups at singular points. At the start of Chapter 5 on Surfaces, Hartshorne assumes that all surfaces in the Chapter are smooth. (This should be a comment, although I don't have the reputation to post this as one.)

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