When can you reverse the orientation of a complex manifold and still get a complex manifold? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T02:23:34Z http://mathoverflow.net/feeds/question/47835 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47835/when-can-you-reverse-the-orientation-of-a-complex-manifold-and-still-get-a-comple When can you reverse the orientation of a complex manifold and still get a complex manifold? solbap 2010-11-30T21:41:31Z 2010-12-01T16:41:26Z <p>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 $\mathbb{C}$.</p> <p>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. </p> <p><strong>EDIT</strong>: 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.</p> <p>Another question: if $X$ and $\overline{X}$ both have complex structures, are they necessarily biholomorphic? <strong>Edit</strong>: No per Dmitri's answer below.</p> http://mathoverflow.net/questions/47835/when-can-you-reverse-the-orientation-of-a-complex-manifold-and-still-get-a-comple/47839#47839 Answer by Spinorbundle for When can you reverse the orientation of a complex manifold and still get a complex manifold? Spinorbundle 2010-11-30T22:12:01Z 2010-11-30T22:37:32Z <p>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):</p> <ul> <li>Dieter Kotschick, <a href="http://blms.oxfordjournals.org/content/29/2/145.abstract" rel="nofollow">Orientations and geometrisations of compact complex surfaces</a> (Bull. London Math. Soc. 29 (1997), no. 2, 145–149.)</li> </ul> <blockquote> <p><strong>Theorem</strong> 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.</p> </blockquote> <p>Other material that could be helpful is:<br></p> <ul> <li>Dieter Kotschick, <a href="http://www.springerlink.com/content/q5326v7710055u12/" rel="nofollow">Orientation-reversing homeomorphisms in surface geography </a> (Math. Ann. 292 (1992), no. 2, 375–381.)</li> <li>Arnaud Beauville, Surfaces complexes et orientation (Astérisque 126 (1985), 41–43.)</li> </ul> http://mathoverflow.net/questions/47835/when-can-you-reverse-the-orientation-of-a-complex-manifold-and-still-get-a-comple/47844#47844 Answer by Dmitri for When can you reverse the orientation of a complex manifold and still get a complex manifold? Dmitri 2010-12-01T00:21:09Z 2010-12-01T00:21:09Z <p>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 biholomprphic. Indeed, if $X$ is a complex curve, then $(X,J)$ is biholomorphic to $(X,-J)$ only if $X$ admits and anti-holomorphic involution (this will be the case for example if $X$ is given by an equation with real coefficients). </p> <p>Starting from this example on can construct a (singular) affine variety $Y$ of dimension $3$, such that $(Y,J)$ is not byholomorphic 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 $Y$.</p>