The singular support of sheaves is \(\gamma\)-coisotropic (Q6581837)

This paper studies the microsupport of sheaves and coisotropicness of the microsupport. The paper considers two different versions of coisotropicness: \(\gamma\)-coisotropicness [\textit{C. Viterbo}, ``On the supports in the Humilière completion and $\gamma$-coisotropic sets'', Preprint, \url{arXiv:2204.04133}] and cone-coisotropicness. Both notions are generalizations of coisotropicness of submanifolds of a symplectic manifold. The \(\gamma\)-coisotropicness is defined using a spectral distance, which is also called interleaving distance in the literature, induced by a 1-homogeneous function. The main result of the paper is that: \N\begin{enumerate}\N\item the microsupport of a sheaf is \(\gamma\)-coisotropic; \N\item A \(\gamma\)-coisotropic set is cone-isotropic. \N\end{enumerate}\NAs a corollary, the authors recover a result of \textit{M. Kashiwara} and \textit{P. Schapira} [Sheaves on manifolds. With a short history ``Les débuts de la théorie des faisceaux'' by Christian Houzel. Berlin etc.: Springer-Verlag (1990; Zbl 0709.18001)]: the microsupport of a sheaf is cone-coisotropic. Also, some important examples of \(\gamma\)-coisotropic sets and interesting questions are provided.\N\NIn the appendix, the authors develop necessary decomposition theorems for persistence modules and barcodes, especially for those in the metric completion of the category of persistence modules, which is new. Besides, certain comparison results for sheafy spectral invariants with generating function spectral invariants and Floer theoretic spectral invariants are given based on a corresponding comparison for filtered cohomologies with the help of a previous result of \textit{C. Viterbo} [Math. Ann. 292, No. 4, 685--710 (1992; Zbl 0735.58019)].