If $(E,h)$ is a holomorphic vector bundle over a compact complex manifold $M$ (which is a projective variety), and $S$ is a coherent subsheaf of $M$, then $S$ is secretly a holomorphic vector bundle outside a codimension $2$ subvariety $Y$. On $X-Y$, the vector bundle $S$ inherits the Chern connection from $(E,h)$. So we can form Chern-Weil forms $c_k$ on $X-Y$.
1) Are the forms $c_k \wedge \alpha_{n-k}$ (where $\alpha_{n-k}$ is a smooth $(n-k,n-k)$ closed form on $X$) integrable on $X-Y$ ?
2) If so, if we integrate $c_k \wedge \alpha_{n-k}$ (where $\alpha_{n-k}$ is a smooth $(n-k,n-k)$ closed form on $X$) over $X-Y$, then do we get $[c_k(S)].[\alpha_{n-k}]$ ? (where the Chern classes of coherent sheaves are defined using a free resolution and the cup product formula)
If so, can you please point me to a reference ?
