# English reference for the Grauert–Riemenschneider vanishing theorem

What is the best reference in English for the following theorem of Grauert–Riemenschneider:

Theorem: Let $$\phi:X \to Y$$ be a proper bi-rational morphism of algebraic varieties over characteristic $$0$$ field. Assume that $$X$$ is smooth. Let $$\Omega_X$$ be the sheaf of top differential forms on $$X$$.Then $$R\phi_*(\Omega_X)=\phi_*(\Omega_X)$$, i.e. $$H^i(R\phi_*(\Omega_X))=0$$ for all $$i \neq 0$$.

I found the original Grauert–Riemenschneider: H. Grauert, O. Riemenschneider, "Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen" (MSN link in restricted access).

But I can't read German, and the review does not state this formulation of the theorem (it probably states a more general one, but I do not understand it and do not see how to deduce what I need).

• $\Omega_X$ usually denotes the sheaf of $1$-forms. The sheaf of top differentials is usually denoted by $\omega_X$ or possibly $\Omega_X^n$ where $n=\dim X$. Dec 8, 2012 at 19:53

Dear Rami,

You could see Kollar-Mori, Birational geometry of algebraic varieties. (Page 73)

or

Lazarsfeld, Positivity in Algebraic Geometry I and II. (Page 257)

• Lazarsfeld, Positivity in Algebraic Geometry I. (Page 257, Theorem 4.3.9.) is exactly what I need. I could not deduce from the statement in Kollar-Mori the formulation that I need, may be I just did not find the correct statement. Anyhow it dose not matter. Thank you again
– Rami
Dec 14, 2012 at 20:35

nowadays it also holds in positive characteristic:

Higher direct images of the structure sheaf in positive characteristic. Algebra & Number Theory, vol 5, No. 6 (2011), 693-775