• Received : 2009.01.18
  • Published : 2010.11.01


In this paper we compare a torsion free sheaf F on $P^N$ and the free vector bundle $\oplus^n_{i=1}O_{P^N}(b_i)$ having same rank and splitting type. We show that the first one has always "less" global sections, while it has a higher second Chern class. In both cases bounds for the difference are found in terms of the maximal free subsheaves of F. As a consequence we obtain a direct, easy and more general proof of the "Horrocks' splitting criterion", also holding for torsion free sheaves, and lower bounds for the Chern classes $c_i$(F(t)) of twists of F, only depending on some numerical invariants of F. Especially, we prove for rank n torsion free sheaves on $P^N$, whose splitting type has no gap (i.e., $b_i{\geq}b_{i+1}{\geq}b_i-1$ 1 for every i = 1,$\ldots$,n-1), the following formula for the discriminant: $$\Delta(F):=2_{nc_2}-(n-1)c^2_1\geq-\frac{1}{12}n^2(n^2-1)$$. Finally in the case of rank n reflexive sheaves we obtain polynomial upper bounds for the absolute value of the higher Chern classes $c_3$(F(t)),$\ldots$,$c_n$(F(t)) for the dimension of the cohomology modules $H^iF(t)$ and for the Castelnuovo-Mumford regularity of F; these polynomial bounds only depend only on $c_1(F)$, $c_2(F)$, the splitting type of F and t.


  1. T. Abe and M. Yoshinaga, Splitting criterion for reflexive sheaves, Proc. Amer. Math. Soc. 136 (2008), no. 6, 1887-1891.
  2. C. Bertone and M. Roggero, Positivity of Chern classes for reflexive sheaves on $P^N$, Geom. Dedicata 142 (2009), 121-138.
  3. F. A. Bogomolov, Stability of vector bundles on surfaces and curves, Einstein metrics and Yang-Mills connections (Sanda, 1990), 35-49.
  4. G. Elencwajg and O. Forster, Bounding cohomology groups of vector bundles on $P_n$. Math. Ann. 246 (1979/80), no. 3, 251-270.
  5. D. Gieseker, On a theorem of Bogomolov on Chern classes of stable bundles, Amer. J. Math. 101 (1979), no. 1, 77-85.
  6. R. Hartshorne, Algebraic Geometry, Grad. Texts in Math., No. 52. Springer-Verlag, New York-Heidelberg, 1977.
  7. R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121-176.
  8. G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. Lond. Math. Soc. (3) 14 (1964), 689-713.
  9. D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Aspects Math., E31. Friedr. Vieweg & Sohn, Braunschweig, 1997.
  10. D. Mumford, Lectures on Curves on an Algebraic Surface, Princeton University Press, Princeton, N.J. 1966.
  11. C. Okonek, Reflexive Garben auf P4, Math. Ann. 260 (1982), no. 2, 211-237.
  12. C. Okonek, M. Schneider, and H. Splinder, Vector Bundles on Complex Projective Spaces, Progress in Mathematics, 3. Birkhauser, Boston, Mass., 1980.
  13. T. Sauer, Nonstable reflexive sheaves on P3, Trans. Amer. Math. Soc. 281 (1984), no. 2, 633-655.
  14. R. L. E. Schwarzenberger, Vector bundles on the projective plane, Proc. Lond. Math. Soc. (3) 11 (1961), 623-640.
  15. F. A. Bogomolov, Stability of vector bundles on surfaces and curves, Einstein metrics and Yang-Mills connections, Lect. Notes Pure Appl. Math., 145, Dekker, New York, 1993.

Cited by

  1. Representing stable complexes on projective spaces vol.400, 2014,