It is wellknown that the space of Sequivalence classes of rank 2 semistable holomorphic vector bundles with trivial determinant on a genus 2 surface M is $CP^3$ (more concretely $PH^0(Jac(M),L(2\theta)$). Especially, the points corresponding to semistable (and not stable) bundles are smooth points. On the open dense subspace consisting of points corresponding to stable bundles, there is a natural symplectic structure, compatible with the natural Riemannian metric. It defines a Kaehler structure. It should be true that this Kaehler structure extends to the semistable points (and therefore $CP^3$ is equipped with its natural (only) Kaehler structure). Does anyone know a reference, or a proof of this? Thank you.
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