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Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.

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Real-holomorphic Hamiltonian vector fields

Consider a Kähler manifold with complex structure $J$. Is there a characterization of real-valued functions $H$ for which the corresponding Hamiltonian vector field $X_H$ is real-holomorphic, that is, …
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