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 $X$ be a compact complex surface admitting a complex structure for −X. $\bar{X}$. 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 $X$ 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.)
