Algebraic Surfaces and Their Moduli Spaces: Real, Differentiable and Symplectic Structures

Catanese, Fabrizio

Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 3, pp. 537-558.

Résumé

The theory of algebraic surfaces, according to Federigo Enriques, revealed `riposte armonie' (hidden harmonies) who the mathematicians to undertook their investigation. Purpose of this article is to show that this holds still nowadays; and point out, while reviewing recent progress and unexpected new results, the many faceted connections of the theory, among others, with algebra (Galois group of the rational numbers), with real geometry, and with differential and symplectic topology of 4 manifolds.

Bibliographie

[1] J. Amoros - F. Bogomolov - L. Katzarkov - T. Pantev, Symplectic Lefschetz fibrations with arbitrary fundamental group, With an appendix by Ivan Smith, J. Differential Geom., 54, no. 3 (2000), 489-545. | MR | Zbl

[2] A. Andreotti - T. Frankel, The second Lefschetz theorem on hyperplane sections, in `Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo, Press, Tokyo (1969), 1-20. | MR

[3] E. Artin, Theorie der Zöpfe, Hamburg Univ. Math. Seminar Abhandlungen, 4-5 (1926), 47-72. | DOI | MR

[4] E. Artin, The collected papers of Emil Artin, edited by Serge Lang and John T. Tate, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London (1965). | MR | Zbl

[5] D. Auroux - L. Katzarkov, Branched coverings of $CP^{2}$ and invariants of symplectic 4-manifolds, Inv. Math., 142 (2000), 631-673. | DOI | MR | Zbl

[6] D. Auroux, Fiber sums of genus 2 Lefschetz fibrations, Turkish J. Math. 27, no. 1 (2003), 1-10. | MR | Zbl

[7] D. Auroux - S. Donaldson - L. Katzarkov, Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves. Math. Ann., 326, no. 1 (2003), 185-203. | DOI | MR | Zbl

[8] D. Auroux - S. Donaldson - L. Katzarkov - M. Yotov, Fundamental groups of complements of plane curves and symplectic invariants, Topology, 43, no. 6 (2004), 1285-1318. | DOI | MR | Zbl

[9] D. Auroux - L. Katzarkov, A degree doubling formula for braid monodromies and Lefschetz pencils, arXiv:math/0605001, Pure Appl. Math. Q., 4, no. 2, part 1 (2008) 237-318. | DOI | MR | Zbl

[10] W. Barth - C. Peters - A. Van De Ven, Compact complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin (1984). | Zbl

[11] I. Bauer - F. Catanese - F. Grunewald, Beauville surfaces without real structures. In: Geometric methods in algebra and number theory, Progr. Math., 235, Birkhäuser (2005), 1-42. | DOI | MR | Zbl

[12] I. Bauer - F. Catanese - F. Grunewald, Chebycheff and Belyi polynomials, dessins d'enfants, Beauville surfaces and group theory. Mediterranean J. Math. 3, no.2 (2006) 119-143. | DOI | MR

[13] I. Bauer - F. Catanese - F. Grunewald, The classification of surfaces with $p_{g} = q = 0$ isogenous to a product of curves, math.AG/0610267, Pure Appl. Math. Q. 4, no. 2, part 1 (2008), 547-586. | DOI | MR | Zbl

[14] I. Bauer - F. Catanese - F. Grunewald, The absolute Galois group acts faithfully on the connected components of the moduli spaces of surfaces of general type? preliminary version 2007.

[15] J. S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo (1974). | MR

[16] E. Bombieri, Canonical models of surfaces of general type, Publ. Math. I.H.E.S., 42 (1973), 173-219. | fulltext EuDML | MR

[17] E. Bombieri, The Mordell conjecture revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 17, no. 4 (1990), 615-640. | fulltext EuDML | MR | Zbl

[18] F. Catanese, Moduli of surfaces of general type. in 'Algebraic geometry-open problems' (Ravello, 1982), Lecture Notes in Math., 997, Springer, Berlin-New York (1983), 90-112. | MR

[19] F. Catanese, On the Moduli Spaces of Surfaces of General Type, J. Differential Geom., 19 (1984), 483-515. | MR | Zbl

[20] F. Catanese, On a problem of Chisini, Duke Math. J., 53 (1986), 33-42. | DOI | MR | Zbl

[21] F. Catanese, Automorphisms of Rational Double Points and Moduli Spaces of Surfaces of General Type, Comp. Math., 61 (1987), 81-102. | fulltext EuDML | MR | Zbl

[22] F. Catanese, Connected Components of Moduli Spaces, J. Differential Geom., 24 (1986), 395-399. | MR | Zbl

[23] F. Catanese, Moduli of algebraic surfaces, Theory of moduli (Montecatini Terme, 1985), Lecture Notes in Math., 1337, Springer, Berlin (1988), 1-83. | DOI | MR

[24] F. Catanese, Moduli and classification of irregular Kähler manifolds (and algebraic varieties) with Albanese general type fibrations, Invent. math., 104 (1991), 263-289. | fulltext EuDML | DOI | MR | Zbl

[25] F. Catanese, Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math., 122, no. 1 (2000), 1-44. | MR | Zbl

[26] F. Catanese, Moduli Spaces of Surfaces and Real Structures, Ann. Math. (2), 158, no. 2 (2003), 577-592. | DOI | MR | Zbl

[27] F. Catanese, Symplectic structures of algebraic surfaces and deformation, 14 pages, math.AG/0207254.

[28] F. Catanese - B. Wajnryb, Diffeomorphism of simply connected algebraic surfaces, math.AG/0405299, J. Differ. Geom., 76, No. 2 (2007), 177-213. | DOI | MR | Zbl

[29] F. Catanese - B. Wajnryb, The 3-cuspidal quartic and braid monodromy of degree 4 coverings, Ciliberto, C. (ed.) et al., Projective varieties with unexpected properties. A volume in memory of Giuseppe Veronese. Proceedings of the international conference "Varieties with unexpected properties", Siena, Italy, June 8-13, 2004. Berlin: Walter de Gruyter (2005), 113-129. | MR

[30] F. Catanese, Canonical symplectic structures and deformations of algebraic surfaces, math.AG/0608110, Comm. Contemp. Math. (2009). | DOI | MR | Zbl

[31] F. Catanese, Differentiable and deformation type of algebraic surfaces, real and symplectic structures. (English) Catanese, Fabrizio (ed.) et al., Symplectic 4-manifolds and algebraic surfaces. Lectures given at the C.I.M.E. summer school, Cetraro, Italy, September 2-10, 2003. Berlin: Springer; Florence: Fondazione C.I.M.E. Lecture Notes in Mathematics 1938 (2008), 55-167. | DOI | MR | Zbl

[32] F. Catanese - M. Lönne - B. Wajnryb, Moduli spaces and braid monodromy types of bidouble covers of the quadric, preliminary version 2008.

[33] O. Chisini, Sulla identità birazionale delle funzioni algebriche di due variabili dotate di una medesima curva di diramazione, Ist. Lombardo Sci. Lett. Cl. Sci. Mat. Nat. Rend. (3), 8 (77) (1944), 339-356. | MR

[34] O. Chisini, Il teorema di esistenza delle trecce algebriche. I-III, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8), 17-18 (1954), 143-149; (1955), 307-311; (1955), 8-13. | MR

[35] M. Dehn, Die Gruppe der Abbildungsklassen. (Das arithmetische Feld auf Flächen.), Acta Math., 69 (1938), 135-206. | DOI | MR | Zbl

[36] S. K. Donaldson, An Application of Gauge Theory to Four-Dimensional Topology, J. Differential Geom., 18 (1983), 279-315. | MR | Zbl

[37] S. K. Donaldson, Connections, cohomology and the intersection forms of 4-manifolds, J. Differential Geom., 24 (1986), 275-341. | MR | Zbl

[38] S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology, 29, 3 (1990), 257-315. | DOI | MR | Zbl

[39] S. K. Donaldson, Gauge theory and four-manifold topology. [CA] Joseph, A. (ed.) et al., First European congress of mathematics (ECM), Paris, France, July 6-10, 1992. Volume I: Invited lectures (Part 1). Basel: Birkhäuser, Prog. Math., 119 (1994), 121-151. | MR | Zbl

[40] S. K. Donaldson, The Seiberg-Witten Equations and 4-manifold topology. Bull. Am. Math. Soc. (N S), 33, 1 (1996), 45-70. | DOI | MR | Zbl

[41] S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom., 44, 4 (1996), 666-705. | MR | Zbl

[42] S. K. Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom., 53, no. 2 (1999), 205-236. | MR | Zbl

[43] S. K. Donaldson - P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs. Oxford: Clarendon Press (1990), ix-440. | MR | Zbl

[44] C. Ehresmann, Sur les espaces fibrés différentiables, C.R. Acad. Sci. Paris, 224 (1947), 1611-1612. | MR | Zbl

[45] F. Enriques, Le Superficie Algebriche, Zanichelli, Bologna (1949). | MR

[46] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73, no. 3 (1983), 349-366. | fulltext EuDML | DOI | MR

[47] G. Faltings, Finiteness theorems for abelian varieties over number fields', Invent. Math., 75, no. 2 (1984), 381. | DOI | MR | Zbl

[48] M. Freedman, The topology of four-dimensional manifolds, J. Differential Geom., 17, n. 3 (1982), 357-454. | MR | Zbl

[49] M. Freedman - F. Quinn, Topology of 4-manifolds, Princeton Math. Series, 39 (1990). | MR | Zbl

[50] R. Friedman - B. G. Moishezon - J. W. Morgan, On the $C^{\infty}$ invariance of the canonical classes of certain algebraic surfaces, Bull. Amer. Math. Soc., 17 (1987), 283-286. | DOI | MR | Zbl

[51] R. Friedman - J. W. Morgan, Algebraic surfaces and four-manifolds: some conjectures and speculations, Bull. Amer. Math. Soc., 18 (1988) 1-19. | DOI | MR | Zbl

[52] R. Friedman - J. W. Morgan, Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 27, Springer-Verlag, Berlin, (1994), x+520. | DOI | MR | Zbl

[53] R. Friedman - J. W. Morgan, Algebraic surfaces and Seiberg-Witten invariants, J. Algebr. Geom., 6, No. 3 (1997) 445-479. | MR | Zbl

[54] D. Gieseker, Global moduli for surfaces of general type, Invent. Math., 43, no. 3 (1977), 233-282. | fulltext EuDML | DOI | MR | Zbl

[55] R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math., 142, 3 (1995), 527-595. | DOI | MR | Zbl

[56] R. E. Gompf, The topology of symplectic manifolds, Turkish J. Math., 25, no. 1 (2001), 43-59. | MR | Zbl

[57] R. E. Gompf - A. Stipsicz, 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, 20. American Mathematical Society, Providence, RI (1999) xvi+558. | DOI | MR

[58] H. Grauert, Über die Deformation isolierter Singularitäten analytischer Mengen, Invent. Math., 15 (1972), 171-198. | fulltext EuDML | DOI | MR | Zbl

[59] H. Grauert, Der Satz von Kuranishi für kompakte komplexe Räume, Invent. Math., 25 (1974), 107-142. | fulltext EuDML | DOI | MR

[60] A. Grothendieck, Techniques de construction et théoremes d'existence en géométrie algébrique. IV, Les schemas de Hilbert, Sem. Bourbaki, Vol. 13, (1960-61), 1-28. | fulltext EuDML | MR | Zbl

[61] A. Hatcher - W. Thurston, A presentation for the mapping class group of a closed orientable surface, Topology, 19, no. 3 (1980), 221-237. | DOI | MR | Zbl

[62] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109-203; ibid. 205-326. | MR | Zbl

[63] J. Jost - S. T. Yau, A strong rigidity theorem for a certain class of compact complex analytic surfaces, Math. Ann., 271 (1985), 143-152. | fulltext EuDML | DOI | MR | Zbl

[64] A. Kas, On the handlebody decomposition associated to a Lefschetz fibration, Pacific J. Math., 89, no. 1 (1980), 89-104. | MR | Zbl

[65] Open problems: Classification of algebraic and analytic manifolds. Classification of algebraic and analytic manifolds, Proc. Symp. Katata/Jap. 1982. Edited by Kenji Ueno. Progress in Mathematics, 39. Birkhäuser, Boston, Mass. (1983), 591-630. | MR

[66] V. M. Kharlamov - V. S. Kulikov, On real structures of real surfaces. Izv. Ross. Akad. Nauk Ser. Mat., 66, no. 1 (2002), 133-152; translation in Izv. Math. 66, no. 1 (2002), 133-150. | DOI | MR | Zbl

[67] V. Kharlamov - V. Kulikov, Deformation inequivalent complex conjugated complex structures and applications, Turkish J. Math., 26, no. 1 (2002), 1-25. | MR | Zbl

[68] K. Kodaira - D. C. Spencer, On deformations of complex analytic structures. I, II, Ann. of Math. (2), 67 (1958), 328-466. | DOI | MR | Zbl

[69] K. Kodaira, On compact complex analytic surfaces, I, Ann. of Math., 71 (1960), 111-152. | DOI | MR | Zbl

[70] K. Kodaira, On compact analytic surfaces, III, Ann. Math. (2), 78 (1963), 1-40. | fulltext EuDML | MR | Zbl

[71] K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math., 86 (1964), 751-798. | DOI | MR | Zbl

[72] K. Kodaira, On the structure of compact complex analytic surfaces. II, Amer. J. Math., 88 (1966), 682-721. | DOI | MR | Zbl

[73] J. Kollár - N. I. Shepherd Barron, Threefolds and deformations of surface singularities, Inv. Math., 91 (1988) 299-338. | fulltext EuDML | DOI | MR | Zbl

[74] V. S. Kulikov, On Chisini's conjecture, Izv. Ross. Akad. Nauk Ser. Mat. 63, no. 6 (1999), 83-116; translation in Izv. Math., 63, no. 6 (1999), 1139-1170. | DOI | MR

[75] V. S. Kulikov, On Chisini's conjecture. II, math. AG/0610356v1. | Zbl

[76] M. Kuranishi, On the locally complete families of complex analytic structures, Ann. Math. (2), 75 (1962), 536-577. | DOI | MR | Zbl

[77] M. Kuranishi, New proof for the existence of locally complete families of complex structures, Proc. Conf. Complex Analysis, Minneapolis, 1964 (1965), 142-154. | MR | Zbl

[78] R. Mandelbaum - B. Moishezon, On the topological structure of non-singular algebraic surfaces in $CP^{3}$, Topology, 15 (1976), 23-40. | DOI | MR | Zbl

[79] R. Mandelbaum - B. Moishezon, On the topology of simply-connected algebraic surfaces, Trans. Am. Math. Soc., 260 (1980), 195-222. | DOI | MR | Zbl

[80] M. Manetti, On some Components of the Moduli Space of Surfaces of General Type, Comp. Math., 92 (1994), 285-297. | fulltext EuDML | MR | Zbl

[81] M. Manetti, Degenerations of Algebraic Surfaces and Applications to Moduli Problems, Tesi di Perfezionamento Scuola Normale Pisa (1996), 1-142.

[82] M. Manetti, Iterated Double Covers and Connected Components of Moduli Spaces, Topology, 36, 3 (1997), 745-764. | DOI | MR | Zbl

[83] M. Manetti, On the Moduli Space of diffeomorphic algebraic surfaces, Inv. Math., 143 (2001), 29-76. | DOI | MR | Zbl

[84] J. W. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies 61, Princeton University Press and the University of Tokyo Press (1968), 122. | MR | Zbl

[85] B. Moishezon, Complex surfaces and connected sums of complex projective planes, Lecture Notes in Math., 603. Springer, Berlin (1977). | MR | Zbl

[86] B. Moishezon, Stable branch curves and braid monodromies, Algebraic geometry, Proc. Conf., Chicago Circle 1980, Lect. Notes Math., 862 (1981), 107-192. | MR

[87] B. Moishezon, Algebraic surfaces and the arithmetic of braids I, in Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry' Prog. Math., 36, Birkhäuser (1983), 199-269. | MR | Zbl

[88] B. Moishezon - M. Teicher, Finite fundamental groups, free over $Z/cZ$, for Galois covers of $CP^{2}$, Math. Ann., 293, no. 4 (1992), 749-766. | fulltext EuDML | DOI | MR | Zbl

[89] B. Moishezon, The arithmetics of braids and a statement of Chisini, in 'Geometric Topology, Haifa 1992', Contemp. Math., 164, Amer. Math. Soc., Providence, RI (1994), 151-175. | DOI | MR | Zbl

[90] J. W. Morgan,The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes, 44. Princeton Univ. Press vi (1996). | MR | Zbl

[91] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294. | DOI | MR | Zbl

[92] B. Segre, Gli automorfismi del corpo complesso, ed un problema di Corrado Segre, Atti Acc. Naz. Lincei (8), 3 (1947), 414-420. | MR

[93] J. P. Serre, Exemples de variétés projectives conjuguées non homéomorphes, C. R. Acad. Sci. Paris, 258 (1964), 4194-4196. | MR | Zbl

[94] W. P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc., 55, no. 2 (1976), 467-468. | DOI | MR | Zbl

[95] B. Wajnryb, A simple presentation for the mapping class group of an orientable surface, Israel J. Math., 45, no. 2-3 (1983), 157-174. | DOI | MR | Zbl

[96] B. Wajnryb, An elementary approach to the mapping class group of a surface, Geom. Topol., 3 (1999), 405-466. | fulltext EuDML | DOI | MR | Zbl

[97] C. T. C. Wall, On the orthogonal groups of unimodular quadratic forms, Math. Ann., 147 (1962), 328-338. | fulltext EuDML | DOI | MR | Zbl

[98] E. Witten, Monopoles and Four-Manifolds, Math. Res. Lett.,1 (1994), 809-822. | DOI | MR

[99] S. T. Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Natl. Acad. Sci. USA, 74 (1977), 1798-1799. | DOI | MR | Zbl

[100] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math., 31, no. 3 (1978), 339-411. | DOI | MR