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On entire rational maps of real surfaces

In this paper, we define for a component X-0 of a nonsingular compact real algebraic surface X the complex genus of X-0, denoted by g(C)(X-0), and use this to prove the nonexistence of nonzero degree entire rational maps f : X-0 --> Y provided that g(C)(Y) > g(C)(X-0), analogously to the topological category. We construct connected real surfaces of arbitrary topological genus with zero complex genus.