Diff Geom in Complex Spaces: Hermitian, Anti-Symmetric & Affine Connections

thehangedman
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Are there any good papers or books that go over our current understanding of differential geometry for 2-dimensional complex spaces? Hermitian vs anti-symmetric metric tensors, dealing with complex conjugates, and defining affine connections?

Yes, I've already hit up Google, so I was hoping an expect on this forum might know a good book or paper to start, perhaps someone who already knows this topic rather well?

I am also interested in how the current formalism of general relativity would need to be modified if it were to deal with complex-valued vectors rather than just real-valued vectors.

Thanks!
 
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