The characteristic cycle and the singular support of a constructible sheaf

Abstract

We define the characteristic cycle of anétale sheaf as a cycle on the cotangent bundle of a smooth variety in positive characteristic using the singular support recently defined by Beilinson. We prove a formulaà la Milnor for the total dimension of the space of vanishing cycles and an index formula computing the Euler-Poincaré characteristic, generalizing the Grothendieck-Ogg-Shafarevich formula to higher dimension.An essential ingredient of the construction and the proof is a partial generalization to higher dimension of the semi-continuity of the Swan conductor due to Deligne-Laumon. We prove the index formula by establishing certain functorial properties of characteristic cycles.

Wild ramification and $$K(\pi , 1)$$ K ( π , 1 ) spaces

“…Proposition 3.4). In the general case, we use the recent work of Takeshi Saito on the characteristic cycle associated to a locally constant F -sheaf [38,39]. It turns out that we can take the automorphism ϕ to be quadratic.…”

“…The recent work of Beilinson [10] and Saito [39] provides an analogue of the classical theory of the singular support and the characteristic cycle [28,Chapter IX] for constructible étale sheaves, fulfilling an expectation of Deligne. Let us review the relevant points briefly, following [39].…”

“…Saito [39] employed Deligne's ideas to define the characteristic cycle CCF . It is an integral combination of the irreducible components of SSF .…”

“…Here we follow the convention that P(T X) parametrizes lines in the tangent bundle T X (this is consistent with [39]). Thus points D × X P(T X) are identified with pairs (x, L) of a point x ∈ D and a tangent direction L ⊆ T x X .…”

Relative singular support and the semi-continuity of characteristic cycles for étale sheaves

“…Its degree is independent of the choice of the base θ P To A 1 k and we denote this intersection number by pA, df q T˚X,v . Sa16,Theorem 5.9]) Let X be a smooth k-scheme of equidimension n, F an object of D b c pX, Λq and tC α u αPI the set of irreducible components of SSpF q. Then, there exists a unique n-cycle CCpF q " ř αPI m α rC α s pm α P Zq of T˚X supported on SSpF q, satisfying the following Milnor type formula (6.2.1):…”