Cohomology vanishing¶and a problem in approximation theory

Abstract

For a simplicial subdivisonof a region in k n (k algebraically closed) and r ∈ N, there is a reflexive sheaf K on P n , such that H 0 (K(d)) is essentially the space of piecewise polynomial functions on � , 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 thosefor which K is actually a bundle (which is always the case for n = 2). In this situation we can obtain a formula for H 0 (K(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.