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Sep 23, 2010 at 7:43 comment added user2529 Originally I had ${\mathbb{C}}$ as a complex manfold and $\bar{{\mathbb{C}}}$ as its conjugate manifold. (That is the transition functions are now antiholomorphic). Then the product manifold ${\mathbb{C}}\times \bar{\mathbb{C}}$ has a sheaf of rings formed ${\mathbb{C}}[x]\otimes \mathbb{C}[\bar{y}]$ by tensoring the original two coordinate sheaves. Restricting along the diagonal $(x,\bar{x})$ gives the ${\mathbb{C}}[x,\bar{x}]$
Sep 23, 2010 at 5:56 history answered S. Carnahan CC BY-SA 2.5