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.
@article{BUMI_2009_9_2_3_537_0,
author = {Fabrizio Catanese},
title = {Algebraic Surfaces and Their Moduli Spaces: Real, Differentiable and Symplectic Structures},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {2},
year = {2009},
pages = {537-558},
zbl = {1184.14061},
mrnumber = {2569289},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2009_9_2_3_537_0}
}
Catanese, Fabrizio. Algebraic Surfaces and Their Moduli Spaces: Real, Differentiable and Symplectic Structures. Bollettino dell'Unione Matematica Italiana, Tome 2 (2009) pp. 537-558. http://gdmltest.u-ga.fr/item/BUMI_2009_9_2_3_537_0/
[1] - - - , Symplectic Lefschetz fibrations with arbitrary fundamental group, With an appendix by Ivan Smith, J. Differential Geom., 54, no. 3 (2000), 489-545. | MR 1823313 | Zbl 1031.57021
[2] - , The second Lefschetz theorem on hyperplane sections, in `Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo, Press, Tokyo (1969), 1-20. | MR 271106
[3] , Theorie der Zöpfe, Hamburg Univ. Math. Seminar Abhandlungen, 4-5 (1926), 47-72. | MR 3069440
[4] , The collected papers of Emil Artin, edited by and , Addison-Wesley Publishing Co., Inc., Reading, Mass.-London (1965). | MR 176888 | Zbl 0146.00101
[5] - , Branched coverings of and invariants of symplectic 4-manifolds, Inv. Math., 142 (2000), 631-673. | MR 1804164 | Zbl 0961.57019
[6] , Fiber sums of genus 2 Lefschetz fibrations, Turkish J. Math.27, no. 1 (2003), 1-10. | MR 1975329 | Zbl 1075.53087
[7] - - , Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves. Math. Ann., 326, no. 1 (2003), 185-203. | MR 1981618 | Zbl 1026.57020
[8] - - - , Fundamental groups of complements of plane curves and symplectic invariants, Topology, 43, no. 6 (2004), 1285-1318. | MR 2081427 | Zbl 1067.53069
[9] - , 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. | MR 2400878 | Zbl 1153.57018
[10] - - , Compact complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin (1984). | Zbl 0718.14023
[11] - - , Beauville surfaces without real structures. In: Geometric methods in algebra and number theory, Progr. Math., 235, Birkhäuser (2005), 1-42. | MR 2159375 | Zbl 1094.14508
[12] - - , Chebycheff and Belyi polynomials, dessins d'enfants, Beauville surfaces and group theory. Mediterranean J. Math. 3, no.2 (2006) 119-143. | MR 2241319
[13] - - , The classification of surfaces with isogenous to a product of curves, math.AG/0610267, Pure Appl. Math. Q. 4, no. 2, part 1 (2008), 547-586. | MR 2400886 | Zbl 1151.14027
[14] - - , The absolute Galois group acts faithfully on the connected components of the moduli spaces of surfaces of general type? preliminary version 2007.
[15] , 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 375281
[16] , Canonical models of surfaces of general type, Publ. Math. I.H.E.S., 42 (1973), 173-219. | MR 318163
[17] , The Mordell conjecture revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 17, no. 4 (1990), 615-640. | MR 1093712 | Zbl 0722.14010
[18] , 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 714742
[19] , On the Moduli Spaces of Surfaces of General Type, J. Differential Geom., 19 (1984), 483-515. | MR 755236 | Zbl 0549.14012
[20] , On a problem of Chisini, Duke Math. J., 53 (1986), 33-42. | MR 835794 | Zbl 0609.14031
[21] , Automorphisms of Rational Double Points and Moduli Spaces of Surfaces of General Type, Comp. Math., 61 (1987), 81-102. | MR 879190 | Zbl 0615.14021
[22] , Connected Components of Moduli Spaces, J. Differential Geom., 24 (1986), 395-399. | MR 868977 | Zbl 0621.14014
[23] , Moduli of algebraic surfaces, Theory of moduli (Montecatini Terme, 1985), Lecture Notes in Math., 1337, Springer, Berlin (1988), 1-83. | MR 963062
[24] , Moduli and classification of irregular Kähler manifolds (and algebraic varieties) with Albanese general type fibrations, Invent. math., 104 (1991), 263-289. | MR 1098610 | Zbl 0743.32025
[25] , Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math., 122, no. 1 (2000), 1-44. | MR 1737256 | Zbl 0983.14013
[26] , Moduli Spaces of Surfaces and Real Structures, Ann. Math. (2), 158, no. 2 (2003), 577-592. | MR 2018929 | Zbl 1042.14011
[27] , Symplectic structures of algebraic surfaces and deformation, 14 pages, math.AG/0207254.
[28] - , Diffeomorphism of simply connected algebraic surfaces, math.AG/0405299, J. Differ. Geom., 76, No. 2 (2007), 177-213. | MR 2330412 | Zbl 1127.14039
[29] - , The 3-cuspidal quartic and braid monodromy of degree 4 coverings, (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 2202250
[30] , Canonical symplectic structures and deformations of algebraic surfaces, math.AG/0608110, Comm. Contemp. Math. (2009). | MR 2538209 | Zbl 1183.14046
[31] , Differentiable and deformation type of algebraic surfaces, real and symplectic structures. (English) (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. | MR 2441412 | Zbl 1145.14001
[32] - - , Moduli spaces and braid monodromy types of bidouble covers of the quadric, preliminary version 2008.
[33] , 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 19351
[34] , 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 71103
[35] , Die Gruppe der Abbildungsklassen. (Das arithmetische Feld auf Flächen.), Acta Math., 69 (1938), 135-206. | MR 1555438 | Zbl 0019.25301
[36] , An Application of Gauge Theory to Four-Dimensional Topology, J. Differential Geom., 18 (1983), 279-315. | MR 710056 | Zbl 0507.57010
[37] , Connections, cohomology and the intersection forms of 4-manifolds, J. Differential Geom., 24 (1986), 275-341. | MR 868974 | Zbl 0635.57007
[38] , Polynomial invariants for smooth four-manifolds, Topology, 29, 3 (1990), 257-315. | MR 1066174 | Zbl 0715.57007
[39] , Gauge theory and four-manifold topology. [CA] (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 1341822 | Zbl 0855.57002
[40] , The Seiberg-Witten Equations and 4-manifold topology. Bull. Am. Math. Soc. (N S), 33, 1 (1996), 45-70. | MR 1339810 | Zbl 0872.57023
[41] , Symplectic submanifolds and almost-complex geometry, J. Differential Geom., 44, 4 (1996), 666-705. | MR 1438190 | Zbl 0883.53032
[42] , Lefschetz pencils on symplectic manifolds, J. Differential Geom., 53, no. 2 (1999), 205-236. | MR 1802722 | Zbl 1040.53094
[43] - , The geometry of four-manifolds, Oxford Mathematical Monographs. Oxford: Clarendon Press (1990), ix-440. | MR 1079726 | Zbl 0820.57002
[44] , Sur les espaces fibrés différentiables, C.R. Acad. Sci. Paris, 224 (1947), 1611-1612. | MR 20774 | Zbl 0029.42001
[45] , Le Superficie Algebriche, Zanichelli, Bologna (1949). | MR 31770
[46] , Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73, no. 3 (1983), 349-366. | MR 718935
[47] , Finiteness theorems for abelian varieties over number fields', Invent. Math., 75, no. 2 (1984), 381. | MR 732554 | Zbl 0602.14044
[48] , The topology of four-dimensional manifolds, J. Differential Geom., 17, n. 3 (1982), 357-454. | MR 679066 | Zbl 0528.57011
[49] - , Topology of 4-manifolds, Princeton Math. Series, 39 (1990). | MR 1201584 | Zbl 0705.57001
[50] - - , On the invariance of the canonical classes of certain algebraic surfaces, Bull. Amer. Math. Soc., 17 (1987), 283-286. | MR 903733 | Zbl 0627.57014
[51] - , Algebraic surfaces and four-manifolds: some conjectures and speculations, Bull. Amer. Math. Soc., 18 (1988) 1-19. | MR 919651 | Zbl 0662.57016
[52] - , Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 27, Springer-Verlag, Berlin, (1994), x+520. | MR 1288304 | Zbl 0817.14017
[53] - , Algebraic surfaces and Seiberg-Witten invariants, J. Algebr. Geom., 6, No. 3 (1997) 445-479. | MR 1487223 | Zbl 0896.14015
[54] , Global moduli for surfaces of general type, Invent. Math., 43, no. 3 (1977), 233-282. | MR 498596 | Zbl 0389.14006
[55] , A new construction of symplectic manifolds, Ann. of Math., 142, 3 (1995), 527-595. | MR 1356781 | Zbl 0849.53027
[56] , The topology of symplectic manifolds, Turkish J. Math., 25, no. 1 (2001), 43-59. | MR 1829078 | Zbl 0989.53054
[57] - , 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, 20. American Mathematical Society, Providence, RI (1999) xvi+558. | MR 1707327
[58] , Über die Deformation isolierter Singularitäten analytischer Mengen, Invent. Math., 15 (1972), 171-198. | MR 293127 | Zbl 0237.32011
[59] , Der Satz von Kuranishi für kompakte komplexe Räume, Invent. Math., 25 (1974), 107-142. | MR 346194
[60] , Techniques de construction et théoremes d'existence en géométrie algébrique. IV, Les schemas de Hilbert, Sem. Bourbaki, Vol. 13, (1960-61), 1-28. | MR 1611170 | Zbl 0236.14003
[61] - , A presentation for the mapping class group of a closed orientable surface, Topology, 19, no. 3 (1980), 221-237. | MR 579573 | Zbl 0447.57005
[62] , 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 199184 | Zbl 0122.38603
[63] - , A strong rigidity theorem for a certain class of compact complex analytic surfaces, Math. Ann., 271 (1985), 143-152. | MR 779612 | Zbl 0554.32021
[64] , On the handlebody decomposition associated to a Lefschetz fibration, Pacific J. Math., 89, no. 1 (1980), 89-104. | MR 596919 | Zbl 0457.14011
[65] Open problems: Classification of algebraic and analytic manifolds. Classification of algebraic and analytic manifolds, Proc. Symp. Katata/Jap. 1982. Edited by . Progress in Mathematics, 39. Birkhäuser, Boston, Mass. (1983), 591-630. | MR 728620
[66] - , 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. | MR 1917540 | Zbl 1055.14060
[67] - , Deformation inequivalent complex conjugated complex structures and applications, Turkish J. Math., 26, no. 1 (2002), 1-25. | MR 1892797 | Zbl 1047.14020
[68] - , On deformations of complex analytic structures. I, II, Ann. of Math. (2), 67 (1958), 328-466. | MR 112154 | Zbl 0128.16901
[69] , On compact complex analytic surfaces, I, Ann. of Math., 71 (1960), 111-152. | MR 132556 | Zbl 0098.13004
[70] , On compact analytic surfaces, III, Ann. Math. (2), 78 (1963), 1-40. | MR 184257 | Zbl 0171.19601
[71] , On the structure of compact complex analytic surfaces. I, Amer. J. Math., 86 (1964), 751-798. | MR 187255 | Zbl 0137.17501
[72] , On the structure of compact complex analytic surfaces. II, Amer. J. Math., 88 (1966), 682-721. | MR 205280 | Zbl 0193.37701
[73] - , Threefolds and deformations of surface singularities, Inv. Math., 91 (1988) 299-338. | MR 922803 | Zbl 0642.14008
[74] , 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. | MR 1748562
[75] , On Chisini's conjecture. II, math. AG/0610356v1. | Zbl 1153.14012
[76] , On the locally complete families of complex analytic structures, Ann. Math. (2), 75 (1962), 536-577. | MR 141139 | Zbl 0106.15303
[77] , New proof for the existence of locally complete families of complex structures, Proc. Conf. Complex Analysis, Minneapolis, 1964 (1965), 142-154. | MR 176496 | Zbl 0144.21102
[78] - , On the topological structure of non-singular algebraic surfaces in , Topology, 15 (1976), 23-40. | MR 405458 | Zbl 0323.57005
[79] - , On the topology of simply-connected algebraic surfaces, Trans. Am. Math. Soc., 260 (1980), 195-222. | MR 570786 | Zbl 0465.57014
[80] , On some Components of the Moduli Space of Surfaces of General Type, Comp. Math., 92 (1994), 285-297. | MR 1286128 | Zbl 0849.14016
[81] , Degenerations of Algebraic Surfaces and Applications to Moduli Problems, Tesi di Perfezionamento Scuola Normale Pisa (1996), 1-142.
[82] , Iterated Double Covers and Connected Components of Moduli Spaces, Topology, 36, 3 (1997), 745-764. | MR 1422433 | Zbl 0889.14014
[83] , On the Moduli Space of diffeomorphic algebraic surfaces, Inv. Math., 143 (2001), 29-76. | MR 1802792 | Zbl 1060.14520
[84] , Singular points of complex hypersurfaces, Annals of Mathematics Studies61, Princeton University Press and the University of Tokyo Press (1968), 122. | MR 239612 | Zbl 0184.48405
[85] , Complex surfaces and connected sums of complex projective planes, Lecture Notes in Math., 603. Springer, Berlin (1977). | MR 491730 | Zbl 0392.32015
[86] , Stable branch curves and braid monodromies, Algebraic geometry, Proc. Conf., Chicago Circle 1980, Lect. Notes Math., 862 (1981), 107-192. | MR 644819
[87] , 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 717613 | Zbl 0592.14012
[88] - , Finite fundamental groups, free over , for Galois covers of , Math. Ann., 293, no. 4 (1992), 749-766. | MR 1176029 | Zbl 0739.14003
[89] , 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. | MR 1282761 | Zbl 0837.14020
[90] ,The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes, 44. Princeton Univ. Press vi (1996). | MR 1367507 | Zbl 0846.57001
[91] , On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294. | MR 182927 | Zbl 0141.19407
[92] , Gli automorfismi del corpo complesso, ed un problema di Corrado Segre, Atti Acc. Naz. Lincei (8), 3 (1947), 414-420. | MR 26987
[93] , Exemples de variétés projectives conjuguées non homéomorphes, C. R. Acad. Sci. Paris, 258 (1964), 4194-4196. | MR 166197 | Zbl 0117.38003
[94] , Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc., 55, no. 2 (1976), 467-468. | MR 402764 | Zbl 0324.53031
[95] , A simple presentation for the mapping class group of an orientable surface, Israel J. Math., 45, no. 2-3 (1983), 157-174. | MR 719117 | Zbl 0533.57002
[96] , An elementary approach to the mapping class group of a surface, Geom. Topol., 3 (1999), 405-466. | MR 1726532 | Zbl 0947.57015
[97] , On the orthogonal groups of unimodular quadratic forms, Math. Ann., 147 (1962), 328-338. | MR 138565 | Zbl 0109.03305
[98] , Monopoles and Four-Manifolds, Math. Res. Lett.,1 (1994), 809-822. | MR 1306021
[99] , Calabi's conjecture and some new results in algebraic geometry, Proc. Natl. Acad. Sci. USA, 74 (1977), 1798-1799. | MR 451180 | Zbl 0355.32028
[100] , 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. | MR 480350