Complex symplectic geometry is the study of complex manifolds admitting a global closed nondegenerate holomorphic two‐form. Compact, simply connected complex symplectic manifolds are of interest for various reasons; for example, they always admit a Ricci‐flat metric. In this work, the close relationship between the complex geometry of such manifolds and their Riemannian structures is exploited in order to obtain results in both directions: Riemannian techniques are used to obtain results on the complex automorphism group, and the problem of constructing examples of compact locally hyper‐Kählerian manifolds with prescribed holonomy groups is discussed.

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