Algebra & Number Theory

Higher direct images of the structure sheaf in positive characteristic

Andre Chatzistamatiou and Kay Rülling

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We prove vanishing of the higher direct images of the structure (and the canonical) sheaf for a proper birational morphism with source a smooth variety and target the quotient of a smooth variety by a finite group of order prime to the characteristic of the ground field. We also show that for smooth projective varieties the cohomology of the structure sheaf is a birational invariant. These results are well known in characteristic zero.

Article information

Algebra Number Theory, Volume 5, Number 6 (2011), 693-775.

Received: 2 December 2009
Revised: 17 January 2011
Accepted: 1 March 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14E05: Rational and birational maps
Secondary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]

birational geometry rational singularities


Chatzistamatiou, Andre; Rülling, Kay. Higher direct images of the structure sheaf in positive characteristic. Algebra Number Theory 5 (2011), no. 6, 693--775. doi:10.2140/ant.2011.5.693.

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  • B. Conrad, Grothendieck duality and base change, Lecture Notes in Mathematics 1750, Springer, Berlin, 2000.
  • B. Conrad, “Clarifications and corrections for [Co?]”, 2001,
  • B. Conrad, “Deligne's notes on Nagata compactifications”, J. Ramanujan Math. Soc. 22:3 (2007), 205–257.
  • W. Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete $($3$)$ 2, Springer, Berlin, 1998.
  • A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA $2)$, Advanced Studies in Pure Mathematics 2, North-Holland Publishing Co., Amsterdam, 1968.
  • R. Hartshorne, Residues and duality, Lecture Notes in Mathematics 20, Springer, Berlin, 1966.
  • R. Hübl and G. Seibert, “The adjunction morphism for regular differential forms and relative duality”, Compositio Math. 106 (1997), 87–123.
  • G. Kempf, F. F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings, I, Lecture Notes in Mathematics 339, Springer, Berlin, 1973.
  • J. Kollár, “Higher direct images of dualizing sheaves, I”, Ann. of Math. $(2)$ 123:1 (1986), 11–42.
  • J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge University Press, 1998.
  • E. Kunz, Kähler differentials, Vieweg & Sohn, Braunschweig, 1986.
  • J. Lipman, Dualizing sheaves, differentials and residues on algebraic varieties, Astérisque 117, Société Mathématique de France, Paris, 1984.
  • J. Lipman and M. Hashimoto, Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Mathematics 1960, Springer, Berlin, 2009. 2010b:18001
  • S. Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics 5, Springer, New York, 1998.
  • E. Viehweg, “Rational singularities of higher dimensional schemes”, Proc. Amer. Math. Soc. 63:1 (1977), 6–8.