On the topology of simply connected algebraic surfaces

Authors:
Richard Mandelbaum and Boris Moishezon

Journal:
Trans. Amer. Math. Soc. **260** (1980), 195-222

MSC:
Primary 14J99; Secondary 57R99

DOI:
https://doi.org/10.1090/S0002-9947-1980-0570786-6

MathSciNet review:
570786

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Abstract: Suppose *x* is a smooth simply-connected compact 4-manifold. 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 simply-connected nonsingular complex projective* 3-*fold. 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*.

**[A]**James W. Alexander,*Note on Riemann spaces*, Bull. Amer. Math. Soc.**26**(1920), no. 8, 370–372. MR**1560318**, https://doi.org/10.1090/S0002-9904-1920-03319-7**[E]**C. J. Earle and J. Eells,*The diffeomorphism group of a compact Riemann surface*, Bull. Amer. Math. Soc.**73**(1967), 557–559. MR**0212840**, https://doi.org/10.1090/S0002-9904-1967-11746-4**[Gf]**Phillip A. Griffiths,*Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems*, Bull. Amer. Math. Soc.**76**(1970), 228–296. MR**0258824**, https://doi.org/10.1090/S0002-9904-1970-12444-2**[GR]**Robert C. Gunning and Hugo Rossi,*Analytic functions of several complex variables*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR**0180696****[H]**Robin Hartshorne,*Ample subvarieties of algebraic varieties*, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR**0282977****[Hor]**Eiji Horikawa,*Algebraic surfaces of general type with small 𝑐²₁. II*, Invent. Math.**37**(1976), no. 2, 121–155. MR**0460340**, https://doi.org/10.1007/BF01418966**[KM]**James Morrow and Kunihiko Kodaira,*Complex manifolds*, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971. MR**0302937****[M]**Richard Mandelbaum,*Irrational connected sums and the topology of algebraic surfaces*, Trans. Amer. Math. Soc.**247**(1979), 137–156. MR**517689**, https://doi.org/10.1090/S0002-9947-1979-0517689-2**[MM]**R. Mandelbaum and B. Moishezon,*On the topological structure of non-singular algebraic surfaces in 𝐶𝑃³*, Topology**15**(1976), no. 1, 23–40. MR**0405458**, https://doi.org/10.1016/0040-9383(76)90047-1**[Msh]**B. Moishezon,*Complex surfaces*, Lecture Notes in Math., vol. 603, Springer-Verlag, Berlin and New York, 1977.**[SR]**J. G. Semple and L. Roth,*Introduction to Algebraic Geometry*, Oxford, at the Clarendon Press, 1949. MR**0034048****[S]**Stephen Smale,*Diffeomorphisms of the 2-sphere*, Proc. Amer. Math. Soc.**10**(1959), 621–626. MR**0112149**, https://doi.org/10.1090/S0002-9939-1959-0112149-8**[Sf1]**I. R. Shafarevich,*Basic algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch; Die Grundlehren der mathematischen Wissenschaften, Band 213. MR**0366917****[Sf2]***Algebraic surfaces*, By the members of the seminar of I. R. Šafarevič. Translated from the Russian by Susan Walker. Translation edited, with supplementary material, by K. Kodaira and D. C. Spencer. Proceedings of the Steklov Institute of Mathematics, No. 75 (1965), American Mathematical Society, Providence, R.I., 1965. MR**0215850****[T]**René Thom,*Quelques propriétés globales des variétés différentiables*, Comment. Math. Helv.**28**(1954), 17–86 (French). MR**0061823**, https://doi.org/10.1007/BF02566923**[W1]**C. T. C. Wall,*Diffeomorphisms of 4-manifolds*, J. London Math. Soc.**39**(1964), 131–140. MR**0163323**, https://doi.org/10.1112/jlms/s1-39.1.131**[W2]**C. T. C. Wall,*On simply-connected 4-manifolds*, J. London Math. Soc.**39**(1964), 141–149. MR**0163324**, https://doi.org/10.1112/jlms/s1-39.1.141**[Wv]**John J. Wavrik,*Deformations of Banach [branched] coverings of complex manifolds*, Amer. J. Math.**90**(1968), 926–960. MR**0233384**, https://doi.org/10.2307/2373491**[Z]**Oscar Zariski,*Introduction to the problem of minimal models in the theory of algebraic surfaces*, Publications of the Mathematical Society of Japan, no. 4, The Mathematical Society of Japan, Tokyo, 1958. MR**0097403**

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DOI:
https://doi.org/10.1090/S0002-9947-1980-0570786-6

Article copyright:
© Copyright 1980
American Mathematical Society