Let ${\cal J}\subset{\cal O}_{X}$ be a coherent ideal sheaf over a compact complex manifold.
A) A modification of $X$ is a bimeromorphic morphism $\mu:\tilde{X}\rightarrow X$ of compact complex manifolds with connected fibers.
B) A log resolution of the sheaf ${\cal J}$ is a modification $\mu:\tilde{X}\rightarrow X$ such that $\mu^{*}{\cal J}={\cal O}(-D)$, with $D$ an effective divisor with normal crossing and such that the restriction $\tilde{X}\smallsetminus|D|\rightarrow X\smallsetminus V({\cal J})$ is a biholomorphism.