When can you reverse the orientation of a complex manifold and still get a complex manifold? I'm told that $\overline{\mathbb{C}P^2}$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold.  But for example, $\overline{\mathbb{C}}$ is still a complex manifold and biholomorphic to $\mathbb{C}$.
This makes me wonder, if $X$ is complex manifold is there a general criterion for when $\overline{X}$ also has a complex structure?  For example, it seems that if $X$ is an affine variety than simply replacing $i$ with $-i$ gives $\overline{X}$ a complex structure and $X, \overline{X}$ are biholomorphic.  
EDIT: the last claim is wrong; see BCnrd's comments below and Dmitri's example.  Also, as explained by Dmitri and BCnrd, $X$ should be taken to have even complex dimension.
Another question: if $X$ and $\overline{X}$ both have complex structures, are they necessarily biholomorphic? 
Edit: No per Dmitri's answer below.
 A: 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:

*

*$X$ is geometrically ruled, or

*the Chern numbers $c_1^2$ and $c_2$ of $X$ vanish, or

*$X$ 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)

A: If you take an odd dimensional complex manifold $X$ with holomorphic structure $J$ then $-J$ defines on $X$ a holomorphic structure as well. And, of course, $J$ and $-J$ induce on $X$ opposite orientations. In general it is not true that these two complex manifolds are biholomorphic. Indeed, if $X$ is a complex curve, then $(X,J)$ is biholomorphic to $(X,-J)$ only if $X$ admits an anti-holomorphic involution (this will be the case for example if $X$ is given by an equation with real coefficients). 
Starting from this example on can construct a (singular) affine variety $Y$ of dimension $3$, such that $(Y,J)$ is not biholomorphic to $(Y,-J)$. Namely, let $C$ be a compact complex curve that does not admit an anti-holomorphic involution say of genus $g=2$. Consider the rank two bundle over it, equal to the sum $TC\oplus TC$ ($TC$ is the tangent bundle to $C$). Contract the zero section of the total space of this bundle, this gives you desired singular $Y$.
