On the topology of simply connected algebraic surfaces

Mandelbaum, Richard; Moishezon, Boris

doi:10.1090/S0002-9947-1980-0570786-6

On the topology of simply connected algebraic surfaces
HTML articles powered by AMS MathViewer

by Richard Mandelbaum and Boris Moishezon PDF

Trans. Amer. Math. Soc. 260 (1980), 195-222 Request permission

Abstract:

Suppose x is a smooth simply-connected compact 4-manifold. Let $p = {\textbf {C}}{P^2}$ and $Q = - {\textbf {C}}{P^2}$ 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 $aP {\text {\# }} bQ$. By a result of Wall [W1] there always exists an integer k such that $X \# (k + 1)P \# kQ$ is completely decomposable. If $X \# P$ is completely decomposable we shall say that X is almost completely decomposable. In [MM] we demonstrated that any nonsingular hypersurface of ${\textbf {C}}{P^3}$ 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 ${m_0} \geqslant 1$ such that any hypersurface section ${V_m}$ of W of degree $m \geqslant {m_0}$ 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.

References

James W. Alexander, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920), no. 8, 370–372. MR 1560318, DOI 10.1090/S0002-9904-1920-03319-7
C. J. Earle and J. Eells, The diffeomorphism group of a compact Riemann surface, Bull. Amer. Math. Soc. 73 (1967), 557–559. MR 212840, DOI 10.1090/S0002-9904-1967-11746-4
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 258824, DOI 10.1090/S0002-9904-1970-12444-2
Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
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, DOI 10.1007/BFb0067839
Eiji Horikawa, Algebraic surfaces of general type with small $c^{2}_{1}$. II, Invent. Math. 37 (1976), no. 2, 121–155. MR 460340, DOI 10.1007/BF01418966
James Morrow and Kunihiko Kodaira, Complex manifolds, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971. MR 0302937
Richard Mandelbaum, Irrational connected sums and the topology of algebraic surfaces, Trans. Amer. Math. Soc. 247 (1979), 137–156. MR 517689, DOI 10.1090/S0002-9947-1979-0517689-2
R. Mandelbaum and B. Moishezon, On the topological structure of non-singular algebraic surfaces in $CP^{3}$, Topology 15 (1976), no. 1, 23–40. MR 405458, DOI 10.1016/0040-9383(76)90047-1

Complex surfaces

J. G. Semple and L. Roth, Introduction to Algebraic Geometry, Oxford, at the Clarendon Press, 1949. MR 0034048
Stephen Smale, Diffeomorphisms of the $2$-sphere, Proc. Amer. Math. Soc. 10 (1959), 621–626. MR 112149, DOI 10.1090/S0002-9939-1959-0112149-8
I. R. Shafarevich, Basic algebraic geometry, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch. MR 0366917, DOI 10.1007/978-3-642-96200-4
Algebraic surfaces, Proceedings of the Steklov Institute of Mathematics, No. 75 (1965), American Mathematical Society, Providence, R.I., 1965. By the members of the seminar of I. R. afarevič; Translated from the Russian by Susan Walker; Translation edited, with supplementary material, by K. Kodaira and D. C. Spencer. MR 0215850
René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 61823, DOI 10.1007/BF02566923
C. T. C. Wall, Diffeomorphisms of $4$-manifolds, J. London Math. Soc. 39 (1964), 131–140. MR 163323, DOI 10.1112/jlms/s1-39.1.131
C. T. C. Wall, On simply-connected $4$-manifolds, J. London Math. Soc. 39 (1964), 141–149. MR 163324, DOI 10.1112/jlms/s1-39.1.141
John J. Wavrik, Deformations of Banach [branched] coverings of complex manifolds, Amer. J. Math. 90 (1968), 926–960. MR 233384, DOI 10.2307/2373491
Oscar Zariski, Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, vol. 4, Mathematical Society of Japan, Tokyo, 1958. MR 0097403