Sheaf quantization and intersection of rational Lagrangian immersions (Q6114289)

The microlocal theory of sheaves due to \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)] has been effectively applied to symplectic geometry for a decade (see [\textit{D. Nadler} and \textit{E. Zaslow}, J. Am. Math. Soc. 22, No. 1, 233--286 (2009; Zbl 1227.32019); \textit{D. Nadler}, Sel. Math., New Ser. 15, No. 4, 563--619 (2009; Zbl 1197.53116); \textit{D. Tamarkin}, Springer Proc. Math. Stat. 269, 99--223 (2018; Zbl 1416.35019)]). Now the theory is considered to be a powerful tool other than Floer theory for the study of symplectic geometry. The authors [J. Symplectic Geom. 18, No. 3, 613--649 (2020; Zbl 1473.18012)] gave a purely sheaf-theoretic bound for the displacement energy of compact subsets in a cotangent bundle, which asserts that if we find sheaves associated with given compact subsets, we can estimate their displacement energy using these sheaves, though saying nothing about the existence of such objects. For a certain class of Lagrangian submanifolds, this paper constructs such objects that give an explicit bound of the displacement energy, based on sheaf quantization. This paper constructs a sheaf quantization of a general compact Legendrian submanifold of \(ST^{\ast}\left( M\times\mathbb{R}/\theta\mathbb{Z}\right) \) with some \(\theta\in\mathbb{R}_{\geqslant0}\). Such a Legendrian is a conification of a strongly rational Lagrangian immersion. The construction follows the idea of \textit{S. Guillermon} [``Sheaves and symplectic geometry of cotangent bundles'', Preprint, \url{arXiv:1905.07341}]. Based on the main result of the authors [loc. cit.] and the above sheaf quantization, they give here an explicit bound for the displacement energy of a rational Lagrangian immersion with a purely sheaf-theoretic method, giving estimates of the number of intersection points against the total Betti number or the cup-length of the Lagrangian.