m at h . A G ] 2 9 Ju n 19 99 Ind - Sheaves , distributions , and microlocalization
نویسنده
چکیده
If C is an abelian category, the category Ind(C) of ind-objects of C has many remarkable properties: it is much bigger than C, it contains C, and furthermore it is dual (in a certain sense) to C. We introduce here the category of ind-sheaves on a locally compact space X as the category of ind-objects of the category of sheaves with compact supports. This construction has some analogy with that of distributions: the space of distributions is bigger than that of functions, and is dual to that of functions with compact support. This last condition implies the local nature of distributions, and similarly, one proves that the category of ind-sheaves defines a stack. In our opinion, what makes the theory of ind-sheaves on manifolds really interesting is twofold. (i) Ind-sheaves allows us to treat in the formalism of sheaves (the “six operations”) functions with growth conditions. For example, on a complex manifold X, one can define the ind-sheaf of “tempered holomorphic functions” O X or the indsheaf of “Whitney holomorphic functions” O X , and obtain for example the sheaf of Schwartz’s distributions using Sato’s construction of hyperfunctions, simply replacing OX with O t X . (ii) On a real manifold, one can construct a microlocalization functor μX which sends sheaves (i.e. objects of the derived category of sheaves) on X to ind-sheaves on T X, and the Sato functor of microlocalization along a submanifold M ⊂ X (see [9]) becomes the usual functor Hom (π(kM), ·) (where π : T X −→ X is the projection) composed with μX(·). When combining (i) and (ii), one can treat in a unified way various objects of classical analysis. The results presented here are extracted from [7]. We refer to [5] for an exposition on derived categories and sheaves, and to [3] for the theory of ind-objects.
منابع مشابه
ar X iv : m at h / 99 07 19 2 v 1 [ m at h . G N ] 3 0 Ju l 1 99 9 ASYMPTOTIC TOPOLOGY
We establish some basic theorems in dimension theory and absolute extensor theory in the coarse category of metric spaces. Some of the statements in this category can be translated in general topology language by applying the Higson corona functor. The relation of problems and results of this ‘Asymptotic Topology’ to Novikov and similar conjectures is discussed.
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ar X iv : m at h / 06 06 28 9 v 1 [ m at h . A G ] 1 2 Ju n 20 06 On correspondences of a K 3 surface with itself . IV
Let X be a K3 surface with a polarization H of the degree H 2 = 2rs, r, s ≥ 1, and the isotropic Mukai vector v = (r, H, s) is primitive. The moduli space of sheaves over X with the isotropic Mukai vector (r, H, s) is again a K3 surface, Y. In [11] the second author gave necessary and sufficient conditions in terms of Picard lattice N (X) of X when Y is isomorphic to X (some important particula...
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