The characteristic cycle and the singular support of a constructible sheaf

We define the characteristic cycle of an etale 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 formula a la Milnor for the total dimension of the space of vanishing cycles and an index formula computing the Euler-Poincare 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.