Non smooth Lagrangian sets and estimations of micro-support (Q1817366)

The authors give the characterization of the sheaves whose micro-support is contained in a smooth Lagrangian manifold. Let \(X\) be a real \(C^1\)-manifold and \(Y\subset X\) be a closed submanifold. Denote by \(T^*_YX\) the conormal bundle to \(Y\) in \(X\) and denote by \(D^b(Y)\) the derived category of the category of bounded complexes of sheaves of \(C\)-vector spaces on \(X\). For \(F\) an object of \(D^b(X)\) denote by \(SS(F)\) its micro-support of \(T^*T\). Then one proves the following theorem. Theorem. Let \(A\subset X\) be a closed \(C\)-convex subset at \(x_0\) and take \(p\in (T^*_AX)_{x_0}\). Let \(F\), \(G\) be objects of \(D^b(X)\) such that \(SS(F)\), \(SS(G)\subset T^*_AX\). Then (i) \(\mu\hom(F,G)\cong N_{T^*_AX}\) for a complex \(N\) of \(C\)-vector spaces; (ii) \(F\cong M\) in \(D^b(X,p)\) for a complex of \(C\)-vector spaces; (iii) for \(M\) as in (ii), one has \(M\cong\mu\hom(C_A,F)_p\).