On the characteristic cycle of an étale sheaf

For an étale sheaf on a smooth variety over a perfect field of positive characteristic, the characteristic cycle is expected to be defined. In this series of lectures, we give a conditional definition and prove some of basic properties assuming the existence of a singular support satisfying certain local acyclicity conditions for families of morphisms to curves and to surfaces. For a sheaf on a surface, the ramification theory implies that the assumption is satisfied and we obtain unconditional results consequently. Deligne describes, in unpublished notes [4], a theory of characteristic cycles of an étale sheaf assuming the existence of a closed subset of the cotangent bundle or a jet bundle satisfying a certain local acyclicity condition on morphisms to curves. He sketches or indicates proofs of the statements and formulates some conjectures. A crucial ingredient of his arguments lies in the continuity of the Swan conductor [9]. The main purpose of this lectures is to give proofs of his statements and conjectures, assuming the existence of a closed subset of the cotangent bundle, called a singular support, satisfying certain local acyclicity conditions for families of morphisms to curves and to surfaces. The statements include the Milnor formula (1.2) for an isolated characteristic point of a morphism to a curve and the Euler-Poincaré formula (3.4). The ramification theory developed in [12] provides a closed subset of the cotangent bundle satisfying the required local acyclicity condition, on the complement of a closed subset of codimension ≧ 2. Using this, we obtain unconditional results for surfaces [13] including the Euler-Poincaré formula (3.4) without any assumption on ramification cf. [10], [3]. The two definitions of the characteristic cycle one by the Milnor formula and the other by the ramification theory are shown to be the same by reduction to the case of surfaces. The proofs have two sides. A geometric side is the use of the universal family of hyperplane sections and its variant cf. [8]. The more abstract side including the continuity of the Swan conductor is based on the theory of vanishing cycles over a general base scheme developed in [7], [11]. In this course, the focus will be put on the geometric side. After formulating the defining property of a singular support using local acyclicity of a family of morphisms to curves, we state the existence of the characteristic cycle characterized by the Milnor formula for the total dimension of the space of vanishing cycles at isolated characteristic points. We construct the characteristic cycle using the universal family of morphisms defined by pencils by taking an embedding to a projective space and prove the Milnor formula using the continuity of the Swan conductor. Finally, we state fundamental properties of the characteristic cycles including the Euler-Poincaré formula.