On the topology of simply connected algebraic surfaces
Authors:
Richard Mandelbaum and Boris Moishezon
Journal:
Trans. Amer. Math. Soc. 260 (1980), 195222
MSC:
Primary 14J99; Secondary 57R99
MathSciNet review:
570786
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Abstract: Suppose x is a smooth simplyconnected compact 4manifold. Let and be the complex projective plane with orientation opposite to the usual. We shall say that X is completely decomposable if there exist integers a, b such that X is diffeomorphic to . By a result of Wall [W1] there always exists an integer k such that is completely decomposable. If is completely decomposable we shall say that X is almost completely decomposable. In [MM] we demonstrated that any nonsingular hypersurface of is almost completely decomposable. In this paper we generalize this result in two directions as follows: Theorem 3.5. Suppose W is a simplyconnected nonsingular complex projective 3fold. Then there exists an integer such that any hypersurface section of W of degree which is nonsingular will be almost completely decomposable. Theorem 5.3. Let V be a nonsingular complex algebraic surface which is a complete intersection. Then V is almost completely decomposable.
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 [E]
 C. J. Earle and J. Eells, The diffeomorphism group of a compact Riemann surface, Bull. Amer. Math. Soc. 73 (1967), 557560. MR 0212840 (35:3705)
 [Gf]
 P. A. Griffiths, Periods of integrals on algebraic manifolds, Bull. Amer. Math. Soc. 76 (1970), 228296. MR 0258824 (41:3470)
 [GR]
 R. Gunning and H. Rossi, Analytic functions of several complex variables, PrenticeHall, Englewood Cliffs, N. J., 1965. MR 0180696 (31:4927)
 [H]
 R. Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Math., vol. 156, SpringerVerlag, Berlin and New York, 1970. MR 0282977 (44:211)
 [Hor]
 E. Horikawa, Algebraic surfaces of general type. I, II, Ann. of Math. (2) 104 (1976), 357387; Invent. Math. 37 (1976), 121155. MR 0460340 (57:334)
 [KM]
 K. Kodaira and J. Morrow, Complex manifolds, Holt Rinehart and Winston, New York, 1971. MR 0302937 (46:2080)
 [M]
 R. Mandelbaum, Irrational connected sums and the topology of algebraic surfaces, Trans. Amer. Math. Soc. 247 (1979), 137156. MR 517689 (80e:57023)
 [MM]
 R. Mandelbaum and B. Moishezon, On the topological structure of nonsingular algebraic surfaces in , Topology 1 (1976). MR 0405458 (53:9251)
 [Msh]
 B. Moishezon, Complex surfaces, Lecture Notes in Math., vol. 603, SpringerVerlag, Berlin and New York, 1977.
 [SR]
 J. G. Semple and L. Roth, Algebraic geometry, Clarendon Press, Oxford, 1949. MR 0034048 (11:535d)
 [S]
 S. Smale, Diffeomorphisms of the 2sphere, Proc. Amer. Math. Soc. 10 (1959), 621626. MR 0112149 (22:3004)
 [Sf1]
 I. R. Shaferevich, Basic algebraic geometry, SpringerVerlag, Berlin and New York, 1974. MR 0366917 (51:3163)
 [Sf2]
 , ed., Algebraic surfaces, Proc. Steklov Inst. Math. 75 (1965). MR 0215850 (35:6685)
 [T]
 R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 1786. MR 0061823 (15:890a)
 [W1]
 C. T. C. Wall, Diffeomorphisms of 4manifolds, J. London Math. Soc. 39 (1964), 131140. MR 0163323 (29:626)
 [W2]
 , On simplyconnected 4manifolds, J. London Math. Soc. 39 (1964), 141149. MR 0163324 (29:627)
 [Wv]
 J. J. Wavrik, Deformations of Banach coverings of complex manifolds, Amer. J. Math. 90 (1968), 926960. MR 0233384 (38:1706)
 [Z]
 O. Zariski, Introduction to the problem of minimal models in the theory of algebraic surfaces, Publ. Math. Soc. Japan, no. 4, Math. Soc. Japan, Tokyo, 1958. MR 0097403 (20:3872)
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DOI:
http://dx.doi.org/10.1090/S00029947198005707866
PII:
S 00029947(1980)05707866
Article copyright:
© Copyright 1980
American Mathematical Society
