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Reflexive sheaf
About: Reflexive sheaf is a research topic. Over the lifetime, 70 publications have been published within this topic receiving 1092 citations.
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717 citations
TL;DR: In this article, the flatness of leaves for sufficiently stable foliations with numerically trivial canonical bundles was proved under certain stability conditions, which implies the algebraicity of leaves in the case of minimal models with trivial canonical class.
Abstract: Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent works of Druel and Greb–Guenancia–Kebekus this establishes the Beauville–Bogomolov decomposition for minimal models with trivial canonical class.
89 citations
TL;DR: For a simplicial subdivison of 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.
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.
29 citations
TL;DR: For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne as mentioned in this paper showed that the equivariant Chow cohomology of X_p is the Sym(N)-algebra C^0(P) of integral piecewise polynomial functions on P.
Abstract: For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf on Proj(N), showing that the Chern classes depend on subtle geometry of P and giving criteria for the splitting of the sheaf as a sum of line bundles. For certain fans associated to the reflection arrangement A_n, we describe a connection between C^0(P) and logarithmic vector fields tangent to A_n.
28 citations
11 Feb 2008
TL;DR: In this article, the authors studied the structure of reflexive sheaves over projective spaces through hyperplane sections and gave a criterion for a sheaf to split into a direct sum of line bundles.
Abstract: The purpose of this paper is to study the structure of reflexive sheaves over projective spaces through hyperplane sections. We give a criterion for a reflexive sheaf to split into a direct sum of line bundles. An application to the theory of free hyperplane arrangements is also given.
24 citations