• Shinder, Evgeny (School of Mathematics and Statistics University of Sheffield)
  • Received : 2016.08.17
  • Accepted : 2016.12.02
  • Published : 2017.09.30


For a complex smooth projective surface M with an action of a finite cyclic group G we give a uniform proof of the isomorphism between the invariant $H^1(G,\;H^2(M,\;{\mathbb{Z}}))$ and the first cohomology of the divisors fixed by the action, using G-equivariant cohomology. This generalizes the main result of Bogomolov and Prokhorov [4].


  1. D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), no. 3, 531-572.
  2. M. A. Armstrong, The fundamental group of the orbit space of a discontinuous group, Proc. Cambridge Philos. Soc. 64 (1968), 299-301.
  3. M. F. Atiyah, Character and cohomology of finite groups, Inst. Hautes tudes Sci. Publ. Math. No. 9 (1961), 23-64.
  4. F. Bogomolov and Yu. Prokhorov, On stable conjugacy of finite subgroups of the plane Cremona group. I, Cent. Eur. J. Math. 11 (2013), no. 12, 2099-2105.
  5. A. Borel, Seminar on transformation groups, with contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. Annals of Mathematics Studies, No. 46 Princeton University Press, Princeton, N.J. 1960.
  6. R. Bott and L. W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982.
  7. V. A. Iskovskikh, Minimal models of rational surfaces over arbitrary fields, Math. USSR-Izv. 14 (1980), no. 1, 17-39.
  8. B. Iversen, Cohomology of Sheaves, Universitext. Springer-Verlag, Berlin, 1986.
  9. Yu. I. Manin, Rational surfaces over perfect fields, Inst. Hautes Etudes Sci. Publ. Math. (30) (1966), 55-113.
  10. Yu. I. Manin, Cubic forms. Algebra, geometry, arithmetic, Translated from the Russian by M. Hazewinkel. North-Holland Mathematical Library, 4. North-Holland Publishing Co., Amsterdam, 1986.
  11. I. Moerdijk, Orbifolds as groupoids: an introduction, Orbifolds in mathematics and physics (Madison, WI, 2001), 205-222, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002
  12. C. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994.