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!!