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This paper deals with complexity classes ${\cal P}_{\Bbb C}$ and ${\cal NP}_{\Bbb C}$ as they were introduced over the complex numbers by Blum, Shub, and Smale [Bull. Amer. Math. Soc., 21 (1989), p. 1]. Under the assumption ${\cal P}_{\Bbb C} \ne {\cal NP}_{\Bbb C}$ the existence of noncomplete problems in ${\cal NP}_{\Bbb C}$ not belonging to ${\cal P}_{\Bbb C}$ is established.