Definition of a complex structure on a vector bundle - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:16:23Z http://mathoverflow.net/feeds/question/17629 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17629/definition-of-a-complex-structure-on-a-vector-bundle Definition of a complex structure on a vector bundle TonyS 2010-03-09T17:39:52Z 2010-03-17T16:59:59Z <p>Given a Riemann surface $S$, e.g. $\mathbb{P}^1(\mathbb{C})$, with complex conjugation on the coordinates and a holomorphic vector bundle $E$ over $S$.</p> <p>The complex conjugation $f$ is not holomorphic, but $C^{\infty}$. So we can look at $f^{*}E$ as a smooth real vector bundle. Since $f$ is an involution $E$ and $f^{*}E$ are isomorphic as smooth real bundles (this is most likely wrong). But somehow there should be more structure on $f^{*}E$. I mean $f$ is antiholomorphic, that should be useful somehow.</p> <p>Is there a way to define a (canonical?) complex structure on $f^{*}E$, such that for any point $p$ we have an antilinear isomorphism <code>$E_{p} \rightarrow f^{*}E_{p}=E_{\overline{p}}$</code>? </p> <p>I think we don't get a linear iso, since the complex conjugation itself is only antilinear.</p> http://mathoverflow.net/questions/17629/definition-of-a-complex-structure-on-a-vector-bundle/18495#18495 Answer by Sebastian for Definition of a complex structure on a vector bundle Sebastian 2010-03-17T15:10:25Z 2010-03-17T16:59:59Z <p>For any antiholomorphic Diffeomorphism $f\colon S\to S$ we get a canonical identification $f^\star\bar K=K,$ $K$ and $\bar K$ being the canonical and anticanonical bundle of the Riemann surface. A holomorphic structure on a complex vectorbundle $E$ is the same as an complex operator $D\colon\Gamma(E)\to\Gamma(\bar KE)$ satisfying the (Cauchy Riemann) Leibnitz rule. (holomorphic sections are exactly the one in the kernel of D). (For higher dimensions this is not true.) </p> <p>Now, $f^* E$ has a natural complex structure (it's just i). Therefore one gets an anti-holomorphic structure $\bar D\colon\Gamma(f^\star E)\to\Gamma(Kf^* E)$ satisfying the antiholomorphic Cauchy Riemann equation. But the complex conjugate bundle $\bar E$ also has a anti-holomorphic structure, since $\overline{\bar K E}=K\bar E.$ Therefore, $f^* \bar E$ has a natural holomorphic structure.</p> <p>These two holomorphic structures are not isomorphic in general: In the case of a line bundle $L=E$ of degree $0$ one might see this as follows: every holomorphic structure $D$ gives rise to an unique unitary flat connection $\nabla$ such that $D=1/2(\nabla+i*\nabla).$ Then the anti-holomorphic structure on $\bar L$ is given by $1/2(\nabla-i*\nabla)$ and, the unitary flat connection corresponding to the holomorphic structure on $f^* L$ is the connection $f^* \nabla.$ But this connection is not gauge equivalent to $\nabla$ in general: For example, on a square torus with f given by $z\mapsto \bar z$ the connection $d+c idx$ is not gauge equivalent to $d+ci dy$ for $c\in R\setminus 2\pi Z.$</p>