Let $X$ be a compact complex manifold, and $V$ be an irreducible analytic subvariety of $X$. Denote $V_{reg}$ the set of regular points of $V$, or the smooth locus of $V$, suppose $V_{reg}$ is of dimension $k$, and $\alpha$ is a smooth real $(k,k)$ form on $X$, then I was told that we can integrate $\alpha$ with respect to $V$.
The logic here is clear. $V_{reg}$ is well-known to be a complex submanifold, and $\alpha$ is a top form on $V_{reg}$. However, it seems to me that $\alpha$ is not necessarily compactly supported on $V_{reg}$, how is the integration well-defined then?