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Let $X$ be a complex-analytic manifold, not necessarily compact.

Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by the norms of derivatives on compact sets?

This question is related to Serre's paper about Serre duality but does not seem to answered there. If $X$ is Stein, the answer is yes.

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The range of $\bar\partial$ is closed iff $H^{0,1}(X)$ is separable with the induced topology. According to the paper "On the compactification of concave ends" by M. Brumberg and J. Leiterer, the concave end of a $1$-corona $X$ can be compactified iff $H^{0,1}(X)$ is separable. Since there are $2$-dimensional coronas whose concave ends cannot be compactified (these examples are due to Grauert, Andreott, Siu and Rossi), it follows that there are examples where the range of $\bar\partial$ is not closed.

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