As is well known , Hodge theorem tells us
Let $(X, g)$ be a compact hermitian manifold. Then the canoni. cal projection $\mathcal{H}_{\bar{\partial}}^{p, q}(X, g) \rightarrow H^{p, q}(X)$ is an isomorphism. i.e., Dolbeault cohomology groups is isomorphic with the corresponding harmonic spaces.
My question is:
1) Is there the Hodge decomposition on noncompact manifolds?
2) Are Dolbeault cohomology groups isomorphic with the corresponding harmonic spaces. ?
