Generalized Leray formula on positive complex Lagrange-Grassmann manifolds (Q1072178)

The authors introduce the notions concerning canonical argument and signature of a triple of Lagrangian planes in a complex phase space. They give the full proof of a matrix lemma which generalizes the lemma used by \textit{J. J. Duistermaat} [Fourier integral operators (1973; Zbl 0272.47028)]. With the aid of this lemma a formula is established, this is a natural extension of Leray's famous formula in real phase space [\textit{J. Leray}, Lagrangian analysis and quantum mechanics (1981; Zbl 0483.35002)]. The generalized Hörmander cross index is introduced. It is still the key point in establishing the notion of almost analytic Maslov line bundles [\textit{A. Melin} and \textit{J. Sjöstrand}, Lect. Notes Math. 459, 120-223 (1975; Zbl 0306.42007)], just as in the real phase case. Finally, an invariant formulation for a one dimensional Čech co-cycle is given, which was used without rigour [\textit{A. S. Mishchenko}, \textit{B. Yu. Sternin} and \textit{V. E. Shatalov}, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 8, 5-39 (1977; Zbl 0425.58010)].