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Exotic analytic structures and Eisenman intrinsic measures. (English) Zbl 0821.14025
Author’s abstract: “Using Eisenman intrinsic measures we prove a cancellation theorem. This theorem allows to find new examples of exotic analytic structures on $$\mathbb{C}^ n$$ under which we understand smooth complex affine algebraic varieties which are diffeomorphic to $$\mathbb{R}^{2n}$$ but not biholomorphic to $$\mathbb{C}^ n$$. We also develop a new method of constructing these structures with a given number of hypersurfaces isomorphic to $$\mathbb{C}^ 2$$ and a family of these structures with a given number of moduli”.

##### MSC:
 14J10 Families, moduli, classification: algebraic theory 14J15 Moduli, classification: analytic theory; relations with modular forms 32G05 Deformations of complex structures
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##### References:
 [1] [D] A. Dimca,Hypersurfaces in C 2n diffeomorphic to R 4n-2 (n), Max-Plank Institute, preprint, 1990. [2] [E] D. A. Eisenman,Intrinsic measures on complex manifolds and holomorphic mappings, Mem. AMS, No. 96, AMS, Providence, R.I., 1970. · Zbl 0197.05901 [3] [FZ] H. Flenner and M. Zaidenberg,Q-acyclic surfaces and their deformations, preprint, 1992. [4] [F] T. Fujita,On the topology of non-complete algebraic surfaces, J. Fac. Sci. Univ. Tokyo (ser. 1A)29 (1982), 503–566. · Zbl 0513.14018 [5] [GW] I. Graham and H. Wu,Some remarks on the intrinsic measures of Eisenman, Trans. AMS288 (1985), 625–660. · Zbl 0582.32034 · doi:10.1090/S0002-9947-1985-0776396-4 [6] [IF] S. Iitaka and T. Fujita,Cancellation theorem for algebraic varieties. J. Fac. Sci. Univ. Tokyo (Sec. 1A)24 (1977), 123–127. · Zbl 0353.14013 [7] [Ka] S. Kaliman,Smooth contractible hypersurfaces in C n and exotic algebraic structures on C 3, Math. Zeitschrift (to appear). [8] [Ko] S. Kobayashi,Intrinsic distances, measures and geometric function theory, Bull. AMS82 (1976), 357–416. · Zbl 0346.32031 · doi:10.1090/S0002-9904-1976-14018-9 [9] [NS] T. Nishino and M. Suzuki,Sur les singularités essentielles et isolées des applications holomorphes á valeuers dans une surface complexe, Publ. RIMS, Kyoto Univ.16 (1980), 461–497. · Zbl 0506.32007 · doi:10.2977/prims/1195187213 [10] [PtD] T. Petrie and T. tom Dieck,Contractible affine surfaces of Kodaira dimension one, Japan. J. Math.16 (1990), no. 1, 147–169. · Zbl 0721.14018 [11] [R] C. P. Ramanujam,A topological characterization of the affine plane as an algebraic variety, Ann. Math.94 (1971), 69–88. · Zbl 0218.14021 · doi:10.2307/1970735 [12] [Ru] P. Russell,On a class of C 3-like threefolds, Preliminary report, 1992. [13] [Sa] F. Sakai,Kodaira dimension of complement of divisor, inComplex Analysis and Algebraic Geometry, Iwanami, Tokyo, 1977, pp. 239–257. [14] [T] R. Tsushima,Rational maps of varieties of hyperbolic type, Proc. Japan Acad. (Ser. A)22 (1979), 95–100. · Zbl 0443.14006 · doi:10.3792/pjaa.55.95 [15] [W] J. Winkelmann,On free holomorphic C *-action on C n and homogeneous Stein manifolds, Math. Ann.286 (1990), 593–612. · Zbl 0708.32004 · doi:10.1007/BF01453590 [16] [Z1] M. Zaidenberg,Analytic cancellation theorem and exotic algebraic structures on C n ,n, Max-Planck-Institut für Mathematik, Bonn, preprint MPI/91-26, 33p, 1991. [17] [Z2] M. Zaidenberg,Ramanujam surfaces and exotic algebraic structures on C n , Dokl. AN SSSR314 (1990), 1303–1307; English transl. in Soviet Math. Doklady42 (1991), 636–640.
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