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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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]**J. W. Alexander,*Note on Riemann spaces*, Bull. Amer. Math. Soc.**26**(1920), 370-377. MR**1560318****[E]**C. J. Earle and J. Eells,*The diffeomorphism group of a compact Riemann surface*, Bull. Amer. Math. Soc.**73**(1967), 557-560. MR**0212840 (35:3705)****[Gf]**P. A. Griffiths,*Periods of integrals on algebraic manifolds*, Bull. Amer. Math. Soc.**76**(1970), 228-296. MR**0258824 (41:3470)****[GR]**R. Gunning and H. Rossi,*Analytic functions of several complex variables*, Prentice-Hall, Englewood Cliffs, N. J., 1965. MR**0180696 (31:4927)****[H]**R. Hartshorne,*Ample subvarieties of algebraic varieties*, Lecture Notes in Math., vol. 156, Springer-Verlag, 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), 357-387; Invent. Math.**37**(1976), 121-155. 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), 137-156. MR**517689 (80e:57023)****[MM]**R. Mandelbaum and B. Moishezon,*On the topological structure of non-singular algebraic surfaces in*, Topology**1**(1976). MR**0405458 (53:9251)****[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,*Algebraic geometry*, Clarendon Press, Oxford, 1949. MR**0034048 (11:535d)****[S]**S. Smale,*Diffeomorphisms of the*2-*sphere*, Proc. Amer. Math. Soc.**10**(1959), 621-626. MR**0112149 (22:3004)****[Sf1]**I. R. Shaferevich,*Basic algebraic geometry*, Springer-Verlag, 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), 17-86. MR**0061823 (15:890a)****[W1]**C. T. C. Wall,*Diffeomorphisms of*4-*manifolds*, J. London Math. Soc.**39**(1964), 131-140. MR**0163323 (29:626)****[W2]**-,*On simply-connected*4-*manifolds*, J. London Math. Soc.**39**(1964), 141-149. MR**0163324 (29:627)****[Wv]**J. J. Wavrik,*Deformations of Banach coverings of complex manifolds*, Amer. J. Math.**90**(1968), 926-960. 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)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
14J99,
57R99

Retrieve articles in all journals with MSC: 14J99, 57R99

Additional Information

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

Article copyright:
© Copyright 1980
American Mathematical Society