The universal central extension of the holomorphic current algebra

We identify the universal differential module ¹(A) for the Fréchet algebra A of holomorphic functions on a complex Stein manifold X, and more generally on a Riemannian domain R over X and for the algebra of germs of holomorphic functions on a compact subset K ⁿ . It turns out to be isomorphic to the Fréchet space of holomorphic 1-forms on X, resp. R, resp. to the space ¹(K) of germs of holomorphic 1-forms in K. This determines the center of the universal central extension of the Lie algebra (R, of holomorphic maps from R to a finite-dimensional simple complex Lie algebra .