##
**Algebraic surfaces and 4-manifolds: Some conjectures and speculations.**
*(English)*
Zbl 0662.57016

The authors survey recent developments in the differential topology of complex algebraic surfaces, and propose a number of wide-ranging conjectures as goals for future research.

The signal feature of these recent developments is the close interaction between the complex geometry of surfaces and their differential topology. For general classes of 4-manifolds new information has been obtained by exploiting the moduli spaces of anti-self-dual Yang-Mills connections, defined with respect to some Riemannian metric. When the 4-manifold is a complex surface and the metric is Kähler the moduli spaces can be identified with those of stable holomorphic bundles, and this brings the complex geometry to the fore. Friedman and Morgan explain that this relation between topology and geometry is analogous to that, stemming from the uniformization theorem, for Riemann surfaces in two dimensions.

In the Enriques-Kodaira rough classification there are just 4 classes of simply connected surfaces: rational, elliptic, K3 and general type. Two complex surfaces are said to be deformation equivalent if they occur as members of a holomorphic family of surfaces over a connected base space. This is an equivalence relation which is a priori finer than that of diffeomorphism. Any K3 surface is deformation equivalent to an elliptic surface, so the rough classification leads to just three categories, in which the rational case is very well understood.

The authors’ conjectures are all in the general direction that the diffeomorphism and deformation classifications of surfaces should be very similar. This is in line with the classical case of Riemann surfaces, but quite opposite to what happens in higher dimensions. One of their main conjectures is that the forgetful map taking deformation classes of simply connected surfaces to diffeomorphism classes is finite-to-one. This conjecture has recently been proved by Friedman and Morgan. (It is possible that the map is one-to-one, so that surfaces are diffeomorphic if and only if they are deformation equivalent, but this seems to lie well beyond present techniques.)

In a similar vein the authors discuss the \(C^{\infty}\)-invariance of the canonical class \(c_ 1(S)\) of a minimal complex surface S. They conjecture that if f: \(S\to S\) is an orientation-preserving diffeomorphism then \(f^*(c_ 1(S))=\pm c_ 1(S).\) This conjecture has been proved for many surfaces. For a non-minimal surface there is a distinguished set of homology classes corresponding to the exceptional curves: Friedman and Morgan conjecture that, if the surface is irrational, the subspace spanned by this set is invariant under diffeomorphisms. They note that this would imply that the only non-zero classes in \(H_ 2(S)\) which can be realized by embedded 2-spheres are those of the exceptional curves.

Moving to the diffeomorphism classification of arbitrary smooth, closed, simply connected oriented, 4-manifolds: an extreme possibility is that any such manifold is diffeomorphic to a connected sum of algebraic surfaces, each with either the standard orientation or the reverse. A weaker conjecture is that any such manifold is homotopy equivalent to a connected sum of this kind. The authors point out that this second conjecture is equivalent to the “eleven-eights” conjecture - that if the intersection form of such a manifold M is even then the second Betti number of M is at least 11/8 of the absolute value of the signature. This reduction is made using the Bogomolov-Miyaoka-Yau inequality for complex surfaces.

The signal feature of these recent developments is the close interaction between the complex geometry of surfaces and their differential topology. For general classes of 4-manifolds new information has been obtained by exploiting the moduli spaces of anti-self-dual Yang-Mills connections, defined with respect to some Riemannian metric. When the 4-manifold is a complex surface and the metric is Kähler the moduli spaces can be identified with those of stable holomorphic bundles, and this brings the complex geometry to the fore. Friedman and Morgan explain that this relation between topology and geometry is analogous to that, stemming from the uniformization theorem, for Riemann surfaces in two dimensions.

In the Enriques-Kodaira rough classification there are just 4 classes of simply connected surfaces: rational, elliptic, K3 and general type. Two complex surfaces are said to be deformation equivalent if they occur as members of a holomorphic family of surfaces over a connected base space. This is an equivalence relation which is a priori finer than that of diffeomorphism. Any K3 surface is deformation equivalent to an elliptic surface, so the rough classification leads to just three categories, in which the rational case is very well understood.

The authors’ conjectures are all in the general direction that the diffeomorphism and deformation classifications of surfaces should be very similar. This is in line with the classical case of Riemann surfaces, but quite opposite to what happens in higher dimensions. One of their main conjectures is that the forgetful map taking deformation classes of simply connected surfaces to diffeomorphism classes is finite-to-one. This conjecture has recently been proved by Friedman and Morgan. (It is possible that the map is one-to-one, so that surfaces are diffeomorphic if and only if they are deformation equivalent, but this seems to lie well beyond present techniques.)

In a similar vein the authors discuss the \(C^{\infty}\)-invariance of the canonical class \(c_ 1(S)\) of a minimal complex surface S. They conjecture that if f: \(S\to S\) is an orientation-preserving diffeomorphism then \(f^*(c_ 1(S))=\pm c_ 1(S).\) This conjecture has been proved for many surfaces. For a non-minimal surface there is a distinguished set of homology classes corresponding to the exceptional curves: Friedman and Morgan conjecture that, if the surface is irrational, the subspace spanned by this set is invariant under diffeomorphisms. They note that this would imply that the only non-zero classes in \(H_ 2(S)\) which can be realized by embedded 2-spheres are those of the exceptional curves.

Moving to the diffeomorphism classification of arbitrary smooth, closed, simply connected oriented, 4-manifolds: an extreme possibility is that any such manifold is diffeomorphic to a connected sum of algebraic surfaces, each with either the standard orientation or the reverse. A weaker conjecture is that any such manifold is homotopy equivalent to a connected sum of this kind. The authors point out that this second conjecture is equivalent to the “eleven-eights” conjecture - that if the intersection form of such a manifold M is even then the second Betti number of M is at least 11/8 of the absolute value of the signature. This reduction is made using the Bogomolov-Miyaoka-Yau inequality for complex surfaces.

Reviewer: S.Donaldson

### MSC:

57R55 | Differentiable structures in differential topology |

14J15 | Moduli, classification: analytic theory; relations with modular forms |

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

### Keywords:

\(C^{\infty}\)-invariance of the canonical class of a minimal complex surface; Kähler manifold; irrational surface; differential topology of complex algebraic surfaces; moduli spaces of anti-self-dual Yang-Mills connections; holomorphic bundles; deformation equivalent; K3 surface; elliptic surface; intersection form; second Betti number; signature; Bogomolov-Miyaoka-Yau inequality for complex surfaces
PDF
BibTeX
XML
Cite

\textit{R. Friedman} and \textit{J. W. Morgan}, Bull. Am. Math. Soc., New Ser. 18, No. 1, 1--19 (1988; Zbl 0662.57016)

Full Text:
DOI

### References:

[1] | Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385 – 404. · Zbl 0104.29902 |

[2] | M. F. Atiyah, Geometrical aspects of gauge theories, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 881 – 885. |

[3] | M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425 – 461. · Zbl 0389.53011 |

[4] | Rebecca Barlow, A simply connected surface of general type with \?_{\?}=0, Invent. Math. 79 (1985), no. 2, 293 – 301. · Zbl 0561.14015 |

[5] | W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. · Zbl 0718.14023 |

[6] | Lipman Bers, Quasiconformal mappings and Teichmüller’s theorem, Analytic functions, Princeton Univ. Press, Princeton, N.J., 1960, pp. 89 – 119. · Zbl 0100.28904 |

[7] | F. Bogomolov, Surfaces of class VIIoand affine geometry, Math. USSR-Izv. 21 (1983), 31-73. · Zbl 0527.14029 |

[8] | E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171 – 219. · Zbl 0259.14005 |

[9] | Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245 – 274. · Zbl 0312.55011 |

[10] | I. Dolgachev, Algebraic surfaces with pg = q = 0, Algebraic Surfaces, C.I.M.E. 1977, Cortona, Liguori, Napoli, 1981, pp. 97-215. |

[11] | S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), no. 2, 279 – 315. · Zbl 0507.57010 |

[12] | S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. (3) 50 (1985), no. 1, 1 – 26. · Zbl 0529.53018 |

[13] | S. K. Donaldson, Connections, cohomology and the intersection forms of 4-manifolds, J. Differential Geom. 24 (1986), no. 3, 275 – 341. · Zbl 0635.57007 |

[14] | S. K. Donaldson, Irrationality and the \?-cobordism conjecture, J. Differential Geom. 26 (1987), no. 1, 141 – 168. · Zbl 0631.57010 |

[15] | S. Donaldson, Polynomial invariants for smooth 4-manifolds (in preparation). · Zbl 0715.57007 |

[16] | Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357 – 453. · Zbl 0528.57011 |

[17] | Robert Friedman and John W. Morgan, On the diffeomorphism types of certain algebraic surfaces. I, J. Differential Geom. 27 (1988), no. 2, 297 – 369. · Zbl 0669.57016 |

[18] | S. Glashow, Quarks with color and flavor, Sci. Amer. 223 no. 4 (1975), 38-50. |

[19] | Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. · Zbl 0408.14001 |

[20] | John Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983), no. 2, 221 – 239. · Zbl 0533.57003 |

[21] | Joe Harris and David Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23 – 88. With an appendix by William Fulton. · Zbl 0506.14016 |

[22] | Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001 |

[23] | Masahisa Inoue, On surfaces of Class \?\?\?\(_{0}\), Invent. Math. 24 (1974), 269 – 310. · Zbl 0283.32019 |

[24] | K. Kodaira, On compact analytic surfaces. II, Ann of Math (2) 77 (1963), 563-626. · Zbl 0118.15802 |

[25] | K. Kodaira, On the structure of compact complex analytic surfaces. II, Amer. J. Math. 88 (1966), 682 – 721. · Zbl 0193.37701 |

[26] | K. Kodaira, Complex structures on \?\textonesuperior \times \?³, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 240 – 243. · Zbl 0141.27402 |

[27] | Martin Lübke and Christian Okonek, Differentiable structures of elliptic surfaces with cyclic fundamental group, Compositio Math. 63 (1987), no. 2, 217 – 222. · Zbl 0663.14027 |

[28] | F. Maier, Tulane Univ. thesis, 1987. |

[29] | Richard Mandelbaum, Four-dimensional topology: an introduction, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 1 – 159. · Zbl 0476.57005 |

[30] | R. Mandelbaum and B. Moishezon, On the topological structure of non-singular algebraic surfaces in \?\?³, Topology 15 (1976), no. 1, 23 – 40. · Zbl 0323.57005 |

[31] | John Milnor, On simply connected 4-manifolds, Symposium internacional de topología algebraica International symposi um on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 122 – 128. · Zbl 0105.17204 |

[32] | Boris Moishezon, Complex surfaces and connected sums of complex projective planes, Lecture Notes in Mathematics, Vol. 603, Springer-Verlag, Berlin-New York, 1977. With an appendix by R. Livne. · Zbl 0392.32015 |

[33] | C. Okonek and A. Van de Ven, Stable bundles and differentiable structures on certain elliptic surfaces, Invent. Math. 86 (1986), no. 2, 357 – 370. · Zbl 0613.14018 |

[34] | Roger Penrose, The complex geometry of the natural world, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 189 – 194. |

[35] | V. A. Rohlin, New results in the theory of four-dimensional manifolds, Doklady Akad. Nauk SSSR (N.S.) 84 (1952), 221 – 224 (Russian). |

[36] | Jean-Pierre Serre, Cours d’arithmétique, Collection SUP: ”Le Mathématicien”, vol. 2, Presses Universitaires de France, Paris, 1970 (French). · Zbl 0376.12001 |

[37] | Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269 – 331 (1978). · Zbl 0374.57002 |

[38] | Clifford Henry Taubes, Self-dual Yang-Mills connections on non-self-dual 4-manifolds, J. Differential Geom. 17 (1982), no. 1, 139 – 170. · Zbl 0484.53026 |

[39] | O. Teichmüller, Extremale quasikonforme Abbildungen und quadratische differentiale, Preuss. Akad. 22 (1939). · JFM 66.1252.01 |

[40] | W. Thurston, On the geometry and dynamics of homeomorphisms of surfaces, preprint. · Zbl 0674.57008 |

[41] | Masaaki Ue, On the diffeomorphism types of elliptic surfaces with multiple fibers, Invent. Math. 84 (1986), no. 3, 633 – 643. · Zbl 0595.14028 |

[42] | Karen K. Uhlenbeck, Removable singularities in Yang-Mills fields, Comm. Math. Phys. 83 (1982), no. 1, 11 – 29. · Zbl 0491.58032 |

[43] | C. T. C. Wall, On simply-connected 4-manifolds, J. London Math. Soc. 39 (1964), 141 – 149. · Zbl 0131.20701 |

[44] | S. Weinberg, Unified theory of elementary particle interaction, Sci. Amer. 231, no. 1, (1974), 50-59. |

[45] | J. H. C. Whitehead, On simply connected, 4-dimensional polyhedra, Comment. Math. Helv. 22 (1949), 48 – 92. · Zbl 0036.12704 |

[46] | C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Physical Rev. (2) 96 (1954), 191 – 195. · Zbl 1378.81075 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.