Partially Logarithmic Characteristic Cycles of Rank One Sheaves

We discuss the computation of the Euler characteristics and the characteristic cycles for rank one sheaves on a smooth variety in positive characteristic in terms of ramification theory. For this purpose, we introduce algebraic cycles for rank one sheaves, which are called the \(\log \) - \(D'\) -characteristic cycles, on the logarithmic cotangent bundle with logarithmic poles along a subdivisor \(D'\) of the boundary by using both the logarithmic and non-logarithmic ramification theories. We compare them with the characteristic cycles and prove the index formula computing the Euler characteristics as intersection numbers for them.