×

On the microlocal cut-off of sheaves. (English) Zbl 0901.58070

Let \(E\) be a finite-dimensional real vector-space, and let \(\pi\colon T^*E\simeq E\times E^*\rightarrow E\) be the canonical projection. Take \(p\in T^*E\setminus\{0\}\), set \(x_0=\pi(p)\in E\), and consider \(\gamma\subset T^*_{x_0}E\), a proper convex conic open neighborhood of \(p\). The “microlocal cut-off lemma” gives a functorial way of associating with a sheaf \(F\) on \(E\) another sheaf \(F'\) which is isomorphic to \(F\) in \(\gamma\), and whose micro-support over \(x_0\) is contained in \(\overline\gamma\) and close to that of \(F\).
The author gives a refined version of the aforementioned microlocal cut-off lemma, in which a class of cones which are not necessarily convex nor proper is considered. This class is wide enough to allow dealing with the case of parameters and the case of complex manifolds.
He says that \((U,\gamma)\) is a refined cutting pair (at \(0\in E)\), where \(U\subset E\) is a relatively open neighborhood of \(0\) and \(\gamma\subset E^*\) is an open cone, if for any \(x\in\partial U\cap\partial^\circ\gamma\) there exists \(\xi\in E^*\setminus\{0\}\) such that \(N_x^*(U)={\mathbb{R}}_{\geq 0}\xi\) and \(\chi(\text{SS}({\mathbb{C}}_\gamma))\cap\pi^{-1}(x)={\mathbb{R}}_{\leq 0}\xi\).
Here \(N_x(U)=N^*(U)\cap\pi^{-1}(x)\), \(N^*(U)\) being the conormal cone to \(U\), \(\partial^\circ\gamma=\partial\gamma^{\circ a}\setminus\{0\}\), \(\gamma^{\circ}\) being the polar cone to \(\gamma\) and \(\gamma^{\circ a}\) being the antipodal cone to \(\gamma^\circ\), and \(\chi\colon T^*E^*\rightarrow T^*E\) is the map \((\xi,x)\mapsto (x,-\xi)\).
Denote now by \(D^b(E)\) the derived category of the category of bounded complexes of sheaves of \({\mathbb{C}}\)-vector spaces on \(E\), and let \(F\in\text{Ob}(D^b(E))\). Denote by \(\Phi_{U,\gamma}(F)=({\mathbb{C}}_{\gamma^a}^\wedge\otimes\omega_E)*F_U\), where \(\omega_E\) is the dualizing complex of \(E\) and \(^\wedge\) is the Fourier-Sato transform of \({\mathbb{C}}_{\gamma^a}\). Then the refined version of the microlocal cut-off lemma the author proves is the following
Theorem. Let \((U,\gamma)\) be a refined cutting pair. For \(F\in\text{Ob}(D^b(E))\) one has \[ \bigl(\text{SS}(\Phi_{U,\gamma})\cap\pi^{-1}(0)\bigr)\setminus\{0\}\subset\{\xi\in\gamma|\;(0,\xi) \in\text{SS}(F)\}\cup\{\xi\in\partial\gamma|\;\exists x\in\overline{U},\;(x,\xi)\in\text{SS}(F)\}. \]

MSC:

58J99 Partial differential equations on manifolds; differential operators
18F99 Categories in geometry and topology
PDFBibTeX XMLCite
Full Text: DOI