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In this paper we show that a Kähler K3 surface containing 16 nonsingular rational curves which do not intersect one another is a Kummer surface. We also give a direct proof of the global Torelli theorem for Kummer surfaces and develop a criterion for a surface to be Kummer which refines the criterion in the paper A Torelli theorem for algebraic K3 surfaces by I. I. Pjateckii-Šapiro and I. R. Šafarevič (Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530-572 = Math. USSR-Izv. 5 (1971), 547-588; MR 44 #1666). Bibliography: 8 items.
Vadim V. Nikulin (Wed,) studied this question.