  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.
