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manuscripta mathematica

, Volume 107, Issue 1, pp 43–58 | Cite as

Cohomology vanishing¶and a problem in approximation theory

  • Henry K. Schenck
  • Peter F. Stiller

Abstract:

For a simplicial subdivison Δ of a region in k n (k algebraically closed) and rN, there is a reflexive sheaf ? on P n , such that H 0(?(d)) is essentially the space of piecewise polynomial functions on Δ, of degree at most d, which meet with order of smoothness r along common faces. In [9], Elencwajg and Forster give bounds for the vanishing of the higher cohomology of a bundle ℰ on P n in terms of the top two Chern classes and the generic splitting type of ℰ. We use a spectral sequence argument similar to that of [16] to characterize those Δ for which ? is actually a bundle (which is always the case for n= 2). In this situation we can obtain a formula for H 0(?(d)) which involves only local data; the results of [9] cited earlier allow us to give a bound on the d where the formula applies. We also show that a major open problem in approximation theory may be formulated in terms of a cohomology vanishing on P 2 and we discuss a possible connection between semi-stability and the conjectured answer to this open problem.

Mathematics Subject Classification (2000): 14J60, 14Q10, 52B30 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Henry K. Schenck
    • 1
  • Peter F. Stiller
    • 1
  1. 1.Mathematics Department, Texas A&M University, College Station, TX 77843-3368, USA. e-mail: schenck@math.tamu.edu; stiller@math.tamu.eduUS

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