It seems to me that you could be interested in the following (I haven't checked the paper in detail, but I think theorems of this "style" could be helpful for you):

• Dieter Kotschick, Orientations and geometrisations of compact complex surfaces (Bull. London Math. Soc. 29 (1997), no. 2, 145–149.)

Theorem Let $X$ be a compact complex surface admitting a complex structure for $\bar{X}$. Then $X$ (and $\bar{X}$) satisfies one of the following:
(1) $X$ is geometrically ruled, or
(2) the Chern numbers $c_1^2$ and $c_2$ of $X$ vanish, or
(3) $X$ is uniformised by the polydisk.
In particular, the signature of $X$ vanishes.

Other material that could be helpful is:
 

• Dieter Kotschick, Orientation-reversing homeomorphisms in surface geography (Math. Ann. 292 (1992), no. 2, 375–381.)
• Arnaud Beauville, Surfaces complexes et orientation (Astérisque 126  (1985), 41–43.)

It seems to me that you could be interested in the following (I haven't checked the paper in detail, but I think theorems of this "style" could be helpful for you):

• Dieter Kotschick, Orientations and geometrisations of compact complex surfaces (Bull. London Math. Soc. 29 (1997), no. 2, 145–149. Zbl 0896.32014)

Theorem Let $X$ be a compact complex surface admitting a complex structure for $\bar{X}$. Then $X$ (and $\bar{X}$) satisfies one of the following:

1. $X$ is geometrically ruled, or
2. the Chern numbers $c_1^2$ and $c_2$ of $X$ vanish, or
3. $X$ is uniformised by the polydisk.

In particular, the signature of $X$ vanishes.

Other material that could be helpful is:

• Dieter Kotschick, Orientation-reversing homeomorphisms in surface geography (Math. Ann. 292 (1992), no. 2, 375–381. Zbl 0753.14034)
• Arnaud Beauville, Surfaces complexes et orientation (Astérisque 126  (1985), 41–43. Zbl 0574.14032)

It seems to me that you could be interested in the following (I haven't checked the paper in detail, but I think theorems of this "style" could be helpful for you):

Dieter Kotschick, Orientations and geometrisations of compact complex surfaces (Bull. London Math. Soc. 29 (1997), no. 2, 145–149.)

Theorem: Part of [Kotschick97, Thm. 2]
Let X be a compact complex surface admitting a complex structure for −X. Then the signature of X vanishes.

Other material that could be helpful is:

Dieter Kotschick, Orientation-reversing homeomorphisms in surface geography (Math. Ann. 292 (1992), no. 2, 375–381.)

Arnaud Beauville, Surfaces complexes et orientation (Astérisque 126 (1985), 41–43.)

It seems to me that you could be interested in the following (I haven't checked the paper in detail, but I think theorems of this "style" could be helpful for you):

• Dieter Kotschick, Orientations and geometrisations of compact complex surfaces (Bull. London Math. Soc. 29 (1997), no. 2, 145–149.)

Theorem Let $X$ be a compact complex surface admitting a complex structure for $\bar{X}$. Then $X$ (and $\bar{X}$) satisfies one of the following:
(1) $X$ is geometrically ruled, or
(2) the Chern numbers $c_1^2$ and $c_2$ of $X$ vanish, or
(3) $X$ is uniformised by the polydisk.
In particular, the signature of $X$ vanishes.

Other material that could be helpful is:

• Dieter Kotschick, Orientation-reversing homeomorphisms in surface geography (Math. Ann. 292 (1992), no. 2, 375–381.)
• Arnaud Beauville, Surfaces complexes et orientation (Astérisque 126 (1985), 41–43.)