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,  <a href="http://blms.oxfordjournals.org/content/29/2/145.abstract">Orientations and geometrisations of compact complex surfaces</a>  (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:<br>
(1) $X$ is geometrically ruled, or<br>
(2) the Chern numbers $c_1^2$ and $c_2$ of $X$ vanish, or<br>
(3) $X$ is uniformised by the polydisk.<br>
In particular, the signature of $X$ vanishes.


Other material that could be helpful is:<br>

 - Dieter Kotschick, <a href="http://www.springerlink.com/content/q5326v7710055u12/">Orientation-reversing homeomorphisms in surface
geography </a>  (Math. Ann. 292 (1992), no. 2, 375–381.)
 - Arnaud Beauville, Surfaces complexes et orientation (Astérisque 126
(1985), 41–43.)