Journal of the
Korean Mathematical Society JKMS

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 $g_{\mathbb C}(X_0)$, and use this to prove the nonexistence of nonzero degree entire rational maps $f:X_0\rightarrow Y$ provided that $g_{\mathbb C}(Y)> g_{\mathbb C}(X_0)$, analogously to the topological category. We construct connected real surfaces of arbitrary topological genus with zero complex genus.

Keywords: real algebraic surfaces, algebraic homology, entire rational maps

MSC numbers: Primary 14P25, 14E05, 14E20; Secondary 14K99, 20J99