## A vanishing theorem for a class of systems with simple characteristics.(English)Zbl 0626.58028

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 $$j\neq r$$.
Reviewer: K.Taniguchi

### MSC:

 58J99 Partial differential equations on manifolds; differential operators
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### References:

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