Intersection Pairing and Poincaré Duality
If $V$ is the closed submanifold of dimension $k$, I would think of $\eta_V$ as the unique closed $n-k$-form with the property that, for all $\gamma \in H_{n-k}(X)$, one has $$ \int_{\gamma} \eta_v = \gamma\cdot V, $$ where $\cdot$ is the intersection product and $\gamma$ is an arbitrary homology class of dimension $n - k$. The integral is well-defined here by Stokes.