Microlocal study of sheaves
Microlocal study of sheaves
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滑轮的微局部研究
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发表时间：
1985
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通讯作者：
P. Schapira
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作者：
柏原 正樹;P. Schapira
0.导论.在[2]中，我们定义了真实的流形X上层复形F的微支撑，并研究了它的函子性质。有了这个工具，我们现在可以对任何层进行接触变换。我们证明了这样的q.c.t.与Sato微局部化可换，并且当流形是复解析流形时，我们证明了结构层不受q.c.t. 1.设X是C”（2<：_a__ oo，or --o）类的真实的流形，T*X是它的余切丛，T*X到X的投影.设A是交换环.我们用D/（X）（resp. D（X））X上的A-模层复形的导出范畴的全子范畴，其上同调是下有界的（分别是：bounded）。设F ∈ Ob（D/（X））. F的微支撑SS（F）是T*X的闭圆锥子集，定义在[2]中。设f 2是T*X的子集。设S（/2）是D/（X）中的态射集，u”F-G，使得u的映射锥属于（/2）.则S（9）满足[1]中的公理，这使我们能够用S（/2）局部化D/（X）.我们用D+（X，12）表示三角范畴so。构造（对于p ∈ T*X，我们写D+（X，p）而不是D/（X，{p}））。设q是X的第j个投影，（j-1，2），设z是XX的对角线。对于F和GeOb（D/（X）），我们定义”/ hom（F，G）=/（Rq（o（q;1F，qF））.回想一下，对于子流形YcX，/r（.）是Sato微定位沿着Y的函子（[4]）。因此/hom（F，G）是T*X T*（X X）上的层复形，命题1。令p ∈ T*X。那么就存在一个自然的同构“Hom.+”（，，）（F，G）q（（Z horn（F，G）），. 2.设（E，a）是一个真实的辛向量空间，2，2是E中的三个拉格朗日平面.设q是上的二次形式，由”
0. Introduction. In [2] we defined the micro-support of a complex of sheaves F on a real manifold X and studied its functorial properties. With this tool, we are now able to quantize contact transformations for any sheaves. We prove that such q.c.t, commute with the Sato microlocalization, and when the manifolds are complex analytic we prove that the structure sheaf ) is invariant by q.c.t. 1. Let X be a real manifold of class C" (2<:_a___ oo, or --o), T*X its cotangent bundle, and the projection from T*X to X. Let A be a commutative ring. We denote by D/(X) (resp. D(X)) the full subcategory of the derived category of complexes of sheaves of A-modules on X whose cohomology is bounded from below (resp. bounded). Let F e Ob(D/(X)). The micro-support of F, SS(F), is a closed conic subset of T*X defined in [2]. Let/2 be a subset of T*X. We set" ([2)-(F e Ob(D+(X)) SS(F) Let S(/2) be the set of morphisms in D/(X), u" F--.G, such that the mapping cone of u belongs to (/2). Then S(9) satisfies the axioms of [1], which enable us to localize D/(X) by S(/2). We denote by D+(X, 12) the triangulated category so. constructed (for p e T*X we write D+(X, p) instead of D/(X, {p})). Let q be the ]-th projection from X X, (]--1, 2) and let z/be the diagonal of XX. For F and G e Ob(D/(X)), we define" / hom (F, G)=/ (R q(o (q;1F, qF)). Recall that for a submanifold YcX,/r(.) is the functor of the Sato microlocalization along Y([4]). Thus /hom (F, G)is a complex of sheaves on T*X T* (X X), Proposition 1. Let p e T*X. Then there exists a natural isomorphism" Hom.+(,,) (F, G)q((Z horn (F, G)),. 2. Let (E, a) be a real symplectic vector space, and , 2, 2 three Lagrangian planes in E. Let q be the quadratic form on. given by"