Measuring wild ramification and Euler characteristic -- Liang Xiao, January 30, 2009

Let X be a smooth proper variety over a field of characteristic p. Let D be a divisor of X with simple normal crossings. Let F be a lisse l-adic sheaf or an overconvergent F-isocrystal on U = X - D. We want to understand the Euler characteristic of F, i.e. the alternating sum of the dimensions of the cohomology groups of F, in terms of the local ramification information. When X is a curve, this is exactly the Grothendieck-Ogg-Shafarevich formula. When dim X > 1, we will encounter many problems. For example, defining the Swan conductor along a divisor is already not easy. For another instance, when X is a surface, we need to understand the ramification at the intersection of two irreducible components of D. We will propose a conjectural formula (under some conditions) at the end.