Negativity and vanishing of microfunction solution sheaves at the boundary
Nobuyuki Tose, Motoo Uchida · Proceedings of the Japan Academy Series A Mathematical Sciences · 1989
Introduction.Let M be a real analytic manifold with a complexifica- tion X.Let V be C-conic involutive submanifold of *X(--T*X\X), and let be a coherent 'x-module with constant multiplicity along V.Moreover let 9 be an open subset of M with real analytic boundary N=32.The aim of this note is to give vanishing theorems for the cohomology groups of the complex R Homx(Y2, C,x) where C,.r is the complex o microunctions at the boundary introduced by P. Schapira [8] (see 1.1 for the definition).The vanishing of the complex R Homcx (3, C) has been studied by M. Sato et al. [6], M. Kshiwara [3] and Kashiwara-Schapira [5], and we study in this note an analogous problem at the boundary.1. Preliminary and a lemma.1.1.Let M be a real analytic mani- fold o dimension n with a complexification X, and let 2 be an open subset of M with real analytic boundary N---q.The cotangent bundle T*X of X is endowed with the sheaf 'x of microdifferential operators of finite order.Refer to M. Sato et al.[6] and P. Schapira [7] for detailed account of .Let T*X denote the micro- support of Z, due to [4], and let L,t be the complex of microfunctions along T*X introduced by P. Schapira [8].With the bifunctor ffhom(., .)con- structed by Kashiwara-Schapira [4], the complex C,Ix is explicitly given by Cl-ffhom (Z, 0)(R) or [n] where or, denotes the orientation sheaf on M. 1.2.We follow the notation in 1.1.Let V be a (C)-conic involutive submanifold of *X.We recall the Levi form 3(V)(p) of V along A= TX at p e A ( V. Take a system of functions (f, ..., f) so that V={q e *X; f(q) f(q)=0} locally in a neighborhood p. Then .(V)(p)denotes the Hermitian form given by the matrix ({f, f})_<,,_<.Here f is the complex conjugate of f and {., } is the Poisson bracket.We remark that the signature of ATA(V)(p) is independent of the choice of (f,, ..., f).Refer to M. Sato et al.[6] and Kashiwara-Schapira [5].1.3.Let X be a C manifold.Then D(X) denotes the derived caregory o the category of bounded complexes of sheaves on X.For F e Ob(D(X)), SS(F) is its micro-support.Let Z and Z be two subsets in X.Then C(Z, Z) is the tangent cone for the pair (Z, Z).Refer to Kashiwara- Schpira [4] for all in this 1.3.