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

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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.