We study the local holomorphic Euler characteristic χ(x,F) of sheaves near a surface singularity x. We find lower bounds for χ(x,F) in terms of the multiplicity of the singularity and prove non-existence of certain sheaves with small values of χ(x,F). A similar result follows for the local charge of instantons on a neighborhood Zk of a −k line inside a smooth surface. We conclude that the self-intersection number of the line poses an obstruction to arbitrary local lowering of the instanton charge (instanton decay). We discuss the Kobayashi–Hitchin correspondence on Zk and calculate dimensions of local moduli.