  We study the local holomorphic Euler characteristic $\chi(x,\mathcal{F})$ of
sheaves near a surface singularity obtained from contracting a line $\ell$
inside a smooth surface $Z$. We prove non-existence of sheaves with certain
prescribed numerical invariants. Non-existence of instantons on $Z$ with
certain charges follows, and we conclude that $\ell^2$ poses an obstruction to
instanton decay. A Macaulay 2 algorithm to compute $\chi$ is made available at
http://www.maths.ed.ac.uk/~s0571100/Instanton/
