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Microlocal Lefschetz classes of graph trace kernels. (English) Zbl 1343.32006

Let \(X\) be a \(C^\infty\)-manifold and let \(\phi: X \rightarrow X\) be a morphism. The author defines a \(\phi\)-graph trace kernel slightly generalizing the corresponding construction on manifolds introduced by M. Kashiwara and P. Schapira [J. Inst. Math. Jussieu 13, No. 3, 487–516 (2014; Zbl 1327.14083)]. He then discusses some basic properties of the microlocal Lefschetz class in this setting. Indeed, his main result is the functoriality of microlocal Lefschetz classes with respect to the composition of graph trace kernels. As an application, the microlocal Lefschetz fixed point formula for constructible sheaves on a real analytic manifolds is obtained [Y. Matsui and K. Takeuchi, Int. Math. Res. Not. 2010, No. 5, 882–913 (2010; Zbl 1198.32003)].

MSC:

32C38 Sheaves of differential operators and their modules, \(D\)-modules
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
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