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Does anyone know a good resource to read up on differential geometry for 2 complex dimensions with an anti-symmetric metric tensor?

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When you say anti-symmetric do you mean Hermitian? –  Dylan Moreland Jul 21 '12 at 4:01
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I would recommend Clifford Taube's book on differential geometry. You should be able to find a book called "foundations on differential geometry" which treats complex geometry very well. There are others books tangentially related if you search several complex variables, complex deformation theory, or even just complex geometry.

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