We define the characteristic cycle of a constructible sheaf on a smooth surface in the cotangent bundle. We prove that the intersection number with the 0-section equals the Euler number and that the total dimension of vanishing cycles at an isolated characteristic point is also computed as an intersection number.
Notes on the characteristic cycle of a constructible sheaf
On the proper push-forward of the characteristic cycle of a constructible sheaf
Singular support and Characteristic cycle of a rank one sheaf in codimension two
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