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For compact complex surfaces (i.e., when $\dim X =2$) the map is always surjective. See Theorem 2.10, p.141 in

Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 4. Berlin: Springer (ISBN 3-540-00832-2/hbk). xii, 436 p. (2004). ZBL1036.14016.

In general, the $\partial \bar{\partial}$-lemma (Lemma 13.6 p. 44 of the reference above) says that, if $X$ is a compact complex manifold such that

1. the Frölicher spectral sequence degenerates at $E_1$ and
2. there is a formal Hodge decomposition,

then $H^{p, \, q}(X)$ coincides with the subspace of $H^{p+q}(X)$ representable by closed forms of type $(p, \, q)$.

For compact complex surfaces (i.e., when $\dim X =2$) the map is always surjective. See Theorem 2.10, p.141 in

Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 4. Berlin: Springer (ISBN 3-540-00832-2/hbk). xii, 436 p. (2004). ZBL1036.14016.

In general, the $\partial \bar{\partial}$-lemma (Lemma 13.6 p. 44 of the reference above) says that, if $X$ is a compact complex manifold such that

1. the Frölicher spectral sequence degenerates at $E_1$ and
2. there is a formal Hodge decomposition,

then $H^{p, \, q}(X)$ coincides with the subspace of $H^{p+q}(X)$ representable by $d$-closed forms of type $(p, \, q)$.

For compact complex surfaces (i.e., when $\dim X =2$) the map is always surjective. See Theorem 2.10, p.141 in

Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 4. Berlin: Springer (ISBN 3-540-00832-2/hbk). xii, 436 p. (2004). ZBL1036.14016.

For compact complex surfaces (i.e., when $\dim X =2$) the map is always surjective. See Theorem 2.10, p.141 in

Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 4. Berlin: Springer (ISBN 3-540-00832-2/hbk). xii, 436 p. (2004). ZBL1036.14016.

In general, the $\partial \bar{\partial}$-lemma (Lemma 13.6 p. 44 of the reference above) says that, if $X$ is a compact complex manifold such that

1. the Frölicher spectral sequence degenerates at $E_1$ and
2. there is a formal Hodge decomposition,

then $H^{p, \, q}(X)$ coincides with the subspace of $H^{p+q}(X)$ representable by closed forms of type $(p, \, q)$.

For compact complex surfaces (i.e., when $\dim X =2$) this is indeed always true. See Theorem 2.10, p.141 in

Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 4. Berlin: Springer (ISBN 3-540-00832-2/hbk). xii, 436 p. (2004). ZBL1036.14016.

For compact complex surfaces (i.e., when $\dim X =2$) the map is always surjective. See Theorem 2.10, p.141 in

Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 4. Berlin: Springer (ISBN 3-540-00832-2/hbk). xii, 436 p. (2004). ZBL1036.14016.