Cohomology vanishing and a problem in approximation theory

Henry K. Schenck, Peter F. Stiller

Research output: Contribution to journalArticle

Abstract

For a simplicial subdivison Δ of a region in kn (k algebraically closed) and r ∈ N, there is a reflexive sheaf script K sign on Pn such that H0 (script K sign (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 ε on Pn in terms of the top two Chern classes and the generic splitting type of ε. We use a spectral sequence argument similar to that of [16] to characterize those Δ for which script K sign is actually a bundle (which is always the case for n = 2). In this situation we can obtain a formula for H0(script K sign (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 P2 and we discuss a possible connection between semi-stability and the conjectured answer to this open problem.

Original languageEnglish (US)
Pages (from-to)43-58
Number of pages16
JournalManuscripta Mathematica
Volume107
Issue number1
DOIs
StatePublished - Dec 1 2002

ASJC Scopus subject areas

  • Mathematics(all)

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