Singular supports, characteristic cycles, and microlocalization of constructible sheaves

Abstract

For a constructible sheaf $\mathcal{F}$ on a smooth scheme $X$ over a perfect field, the singular support $SS(\mathcal{F})$ and the characteristic cycle $CC(\mathcal{F})$ are defined on the cotangent bundle. In analogy with the construction of Kashiwara--Schapira via microlocalization in the transcendental setting, we construct the microlocal versions ${SS}_{\mu }(\mathcal{F})$ and $C{C}_{\mu }(\mathcal{F})$ of $SS(\mathcal{F})$ and $CC(\mathcal{F})$ respectively, and prove \[ SS_μ(\mathcal{F})= SS(\mathcal{F}),\qquad CC_μ(\mathcal{F})= {\rm cl}CC(\mathcal{F}), \] where $\mathrm{cl}$ is the cycle class map. As a consequence of the second equality, we show that the cohomological characteristic classes are the cycle classes associated with the corresponding characteristic cycles, as conjectured in (\emph{Invent.~Math.~207: 597--695, 2017}). As further applications, we give new proofs of the index formula and Bloch's conductor formula in the proper case, independently of Abe's homotopical approach via $\infty$-categories.

arXiv commentThis version merges the previous version of this paper with arXiv:2607.17026, and supersedes both earlier manuscripts