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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.)

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.

These two holomorphic structures are not isomorphic in general: In the case of a line bundle $L=E$ this holomorphic structure on of degree $f^* L$ is isomorphic to the 0$one on$L.$One might see this as followsfor degree$deg L=0$(if$deg L\neq 0$one should take a$f$-invariant metric of volumne 1 compatible with the Riemann surface structure and unitary connections with (constant) curvature$2\pi deg L vol$): 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 nitary unitary flat connection corresponding to the holomorphic structure on$f^* L$is the connection$f^* \nabla.$This But this connection is not gauge equivalent to$\nabla.$Thus \nabla$ in general: For example, on a square torus with f given by $z\mapsto \bar z$ the holomorphic structures are isomorphic. connection $d+c idx$ is not gauge equivalent to $d+ci dy$ for $c\in R\setminus 2\pi Z.$

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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.)

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.

In the case of a line bundle $L=E$ this holomorphic structure on $f^* L$ is isomorphic to the one on $L.$ One might see this as follows for degree $deg L=0$ (if $deg L\neq 0$ one should take a $f$-invariant metric of volumne 1 compatible with the Riemann surface structure and unitary connections with (constant) curvature $2\pi deg L vol$): 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 nitary flat connection corresponding to the holomorphic structure on $f^* L$ is the connection $f^* \nabla.$ This connection is gauge equivalent to $\nabla.$ Thus the holomorphic structures are isomorphic.