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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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vanishing theorem for Nakano k-positive vector bundles ?

I recommend you to read Bernard Shiffman and Andrew John Sammese, Vanishing Theorems on Complex manifolds. In Chapter III, they dealt the k-positivity for line bundles and in Chapter VI they treated t …
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