Cohomology vanishing¶and a problem in approximation theory |
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Authors: | Henry K Schenck Peter F Stiller |
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Institution: | (1) Mathematics Department, Texas A&M University, College Station, TX 77843-3368, USA. e-mail: schenck@math.tamu.edu; stiller@math.tamu.edu, US |
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Abstract: | For a simplicial subdivison Δ of a region in k
n
(k algebraically closed) and r∈N, 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.
Received: 9 April 2001 |
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Keywords: | Mathematics Subject Classification (2000): 14J60 14Q10 52B30 |
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