On the compactification of strongly pseudoconvex surfaces. III (Q1113338)

Having previously classified the compactifications mentioned in the title [part I, II of this paper, Proc. Am. Math. Soc. 82, 407-410 (1981; Zbl 0477.32020); ibid. 90, 189-194 (1984; Zbl 0555.32017)], the author turns attention to the question of uniqueness, proving the following theorem. Any two algebraic (resp. nonalgebraic) compactifications of a strongly pseudoconvex non-Stein surface are birationally (resp. biholomorphically) equivalent. The algebraic situation breaks into cases according to the logarithmic Kodaira dimension; the proof in the nonalgebraic situation follows an idea of \textit{I. Enoki} [Tôhoku Math. J, II Ser. 33, 453-492 (1981; Zbl 0476.14013)]. On the other hand, examples are given of rationally (resp. bimeromorphically) inequivalent algebraic (resp. nonalgebraic) compactifications of the Stein surface \(({\mathbb{C}}\setminus \{0\})^ 2\). These results were announced by the author in Proc. Japan Acad., Ser. A 62, 189-192 (1986; Zbl 0594.32028). The paper under review concludes by posing questions about the existence of affine structures on compactifiable surfaces.