Singular support and Characteristic cycle of a rank one sheaf in codimension two

We compute the singular support and the characteristic cycle of a rank 1 sheaf on a smooth variety in codimension 2 using ramification theory, when the ramification of the sheaf is clean. We develop a general theory, called the partially logarithmic ramification theory, and define an algebraic cycle on a logarithmic cotangent bundle with partial logarithmic poles along the boundary. We prove that the inverse image of the support of the cycle and the pull-back of the cycle to the cotangent bundle are equal to the singular support and the characteristic cycle, respectively, outside a closed subset of the variety of codimension greater than 2 under a mild assumption.