A vanishing theorem for a class of systems with simple characteristics
Let \((X,O_ X)\) be a complex manifold of dimension n, and let \(E_ x\) be the sheaf of finite-order microdifferential operators. Let M be a coherent \(E_ X\)-module defined on an open subset U of \(T^*X\), and let \(V=char(M)\) be its characteristic variety. Let M be a real analytic manifold in X, such that X is a complexification of M, and let \(C_ M\) be the sheaf of Sato's microfunctions on the conormal bundle \(T^*_ MX\) to M in X. Assume that \(V\cap T^*_ MX\) is smooth, \(T(V\cap T^*_ MX)=TV\cap TT^*_ MX\), and the canonical 1-form \(\omega\) does not vanish on \(V\cap T^*_ MX\). Let \(L_{T^*_ MX}(V)\) be the Levi form of V with respect to \(T^*_ MX\). Then, the authors prove the following: Theorem. If the number of negative eigenvalues of \(L_{T^*_ MX}(V)\) is constant on \(V\cap T^*_ MX\), say r, then the groups \(Ext^ j_{E_ x}(M,C_ M)\) are zero for \(j\neq r\).
finite-order microdifferential operators
Sato's microfunctions on the conormal bundle
scientific article; zbMATH DE number 4017964
scientific article