# 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:

http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=Verschwindungss%C3%A4tze+f%C3%BCr+analytische+Kohomologiegruppen+auf+komplexen+R%C3%A4umen&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&s7=&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&review_format=html

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)

Thank you

• $\Omega_X$ usually denotes the sheaf of $1$-forms. The sheaf of top differentials is usually denoted by $\omega_X$ or possibly $\Omega_X^n$ where $n=\dim X$. – Sándor Kovács Dec 8 '12 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 '12 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