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Differentiable and Deformation Type of Algebraic Surfaces, Real and Symplectic Structures

  • Chapter
Symplectic 4-Manifolds and Algebraic Surfaces

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1938))

As already announced in Cetraro at the beginning of the C.I.M.E. course, we deflected from the broader target ‘Classification and deformation types of complex and real manifolds’, planned and announced originally.

In lecture three we show first that deformation equivalence implies diffeomorphism, and then, using a result concerning symplectic approximations of projective varieties with isolated singularities and Moser's theorem, we show that a surfaces of general type has a ‘canonical symplectic structure’, i.e., a symplectic structure whose class is the class of the canonical divisor, and which is unique up to symplectomorphism.

In lecture three and the following ones we thus enter ‘in medias res’, since one of the main problems that we discuss in these Lecture Notes is the comparison of differentiable and deformation type of minimal surfaces of general type, keeping also in consideration the canonical symplectic structure (unique up to symplectomorphism and invariant for smooth deformation) which these surfaces possess.

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References

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

    MATH  MathSciNet  Google Scholar 

  2. A. Andreotti, T. Frankel, The second Lefschetz theorem on hyper-plane sections, in ‘Global Analysis (Papers in Honor of K. Kodaira)’ Univ. Tokyo Press, Tokyo (1969), 1-20.

    Google Scholar 

  3. A. Andreotti, Nine lectures on complex analysis, in ‘Complex analysis(Centro Internaz. Mat. Estivo C.I.M.E., I Ciclo, Bressanone, 1973)’, Edizioni Cremonese, Roma(1974), 1-175.

    Google Scholar 

  4. D. Arapura, Geometry of cohomology support loci for local systems I, J. Algebraic Geom. 6 no. 3 (1997), 563-597.

    MATH  MathSciNet  Google Scholar 

  5. E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris, Geometry of algebraic curves Volume I, Grundlehren der mathematischen Wissenschaften 267, Springer-Verlag, New York. (1985), XVI + 386 p.

    Google Scholar 

  6. E. Artin, Theorie der Zöpfe, Hamburg Univ. Math. Seminar Abhandlungen 4-5 (1926), 47-72.

    Google Scholar 

  7. 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).

    Google Scholar 

  8. M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129-136.

    MATH  MathSciNet  Google Scholar 

  9. D. Auroux, L. Katzarkov , Branched coverings of CP2 and invariants of symplectic 4-manifolds Inv. Math. 142(2000),631-673.

    MATH  MathSciNet  Google Scholar 

  10. D. Auroux, Fiber sums of genus 2 Lefschetz fibrations. Turkish J. Math. 27, no. 1 (2003), 1-10 .

    MATH  MathSciNet  Google Scholar 

  11. 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.

    MATH  MathSciNet  Google Scholar 

  12. D. Auroux, S. Donaldson, L. Katzarkov and M. Yotov, Fundamental groups of complements of plane curves and symplectic invariants Topology 43, no. 6 (2004), 1285-1318.

    MATH  MathSciNet  Google Scholar 

  13. W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin,(1984).

    Google Scholar 

  14. G. Barthel, F. Hirzebruch, T. Höfer, Geradenkonfigurationen und Algebraische Flächen. Aspekte der Math. D4, Vieweg (1987).

    Google Scholar 

  15. I. Bauer, Irrational pencils on non-compact algebraic manifolds, Internat. J. Math. 8 , no. 4 (1997), 441-450.

    MATH  MathSciNet  Google Scholar 

  16. I. Bauer, F. Catanese, Some new surfaces with pg = q = 0, The Fano Conference, Univ. Torino, Torino, (2004), 123-142.

    Google Scholar 

  17. 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.

    Google Scholar 

  18. 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.

    MathSciNet  Google Scholar 

  19. I. Bauer, F. Catanese, F. Grunewald, The classification of surfaces with pg= q= 0 isogenous to a product of curves, math.AG/0610267.

    Google Scholar 

  20. I. Bauer, F. Catanese, A volume maximizing canonical surface in 3-space, math.AG/0608020, to appear in Comm. Math. Helvetici.

    Google Scholar 

  21. A. Beauville, Surfaces algébriques complexes, Asterisque 54, Soc. Math. France (1978).

    Google Scholar 

  22. 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).

    Google Scholar 

  23. E. Bombieri, Canonical models of surfaces of general type, Publ. Math. I.H.E.S., 42 (1973), 173-219.

    Google Scholar 

  24. E. Bombieri, D. Husemoller, Classification and embeddings of surfaces, in ‘Algebraic geometry, Humboldt State Univ., Arcata, Calif., 1974’, Proc. Sympos. Pure Math., Vol. 29, Amer. Math. Soc., Providence, R.I. (1975), 329-420.

    Google Scholar 

  25. F. Bonahon, Difféotopies des espaces lenticulaires, Topology 22(1983),305-314.

    MATH  MathSciNet  Google Scholar 

  26. E. Brieskorn, Rationale Singularitäten komplexer Flächen, Invent. Math. 4 (1967/1968), 336-358.

    MathSciNet  Google Scholar 

  27. E. Brieskorn, Die Auflösung der rationalen Singularitäten holomorpher Abbildungen, Math. Ann. 178 (1968), 255-270.

    MATH  MathSciNet  Google Scholar 

  28. E. Brieskorn, Singular elements of semi-simple algebraic groups, ‘Actes du Congrés International des Mathématiciens’ (Nice, 1970), Tome 2, Gauthier-Villars, Paris (1971), 279-284.

    Google Scholar 

  29. T. Bröcker, K. Jänich Einführung in die Differentialtopologie , Heidelberger Taschenbücher Band 143, Springer Verlag (1990).

    Google Scholar 

  30. D. Burns, J. Wahl, Local contributions to global deformations of surfaces, Invent. Math. 26(1974), 67-88.

    MATH  MathSciNet  Google Scholar 

  31. H. Cartan, Quotient d’un espace analytique par un groupe d’automorphismes, in ‘A symposium in honor of S. Lefschetz, Algebraic geometry and topology’ Princeton University Press, Princeton, N. J. (1957), 90-102.

    Google Scholar 

  32. 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.

    Google Scholar 

  33. F. Catanese, On the Moduli Spaces of Surfaces of General Type, J. Differential Geom. 19 (1984), 483-515.

    MathSciNet  Google Scholar 

  34. F. Catanese, On a problem of Chisini, Duke Math. J. 53(1986), 33-42.

    MATH  MathSciNet  Google Scholar 

  35. F. Catanese, Automorphisms of Rational Double Points and Moduli Spaces of Surfaces of General Type, Comp. Math. 61(1987), 81-102.

    MATH  MathSciNet  Google Scholar 

  36. F. Catanese, Connected Components of Moduli Spaces, J. Differential Geom. 24 (1986), 395-399.

    MATH  MathSciNet  Google Scholar 

  37. F. Catanese, Moduli of algebraic surfaces. Theory of moduli (Montecatini Terme, 1985), Lecture Notes in Math. 1337, Springer, Berlin (1988), 1-83.

    Google Scholar 

  38. F. Catanese, Everywhere non reduced Moduli Spaces , Inv. Math. 98,293-310 (1989).

    MATH  MathSciNet  Google Scholar 

  39. F. Catanese, Moduli and classification of irregular Kähler manifolds (and algebraic varieties) with Albanese general type fibrations. Invent. math. 104, (1991), 263-289.

    MathSciNet  Google Scholar 

  40. F. Catanese, Fundamental groups with few relations, in ‘Higher dimensional complex varieties. Proceedings of the international conference, Trento, Italy, June 15-24, 1994’ Walter de Gruyter, Berlin (1996),163-165.

    Google Scholar 

  41. F. Catanese, M. Franciosi, Divisors of small genus on algebraic surfaces and projective embeddings, in ‘Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993)’, Israel Math. Conf. Proc., 9, BarIlan Univ., Ramat Gan, (1996), 109-140.

    Google Scholar 

  42. F. Catanese, M. Franciosi, K. Hulek, M. Reid, Embeddings of curves and surfaces, Nagoya Math. J. 154 (1999), 185-220.

    MATH  MathSciNet  Google Scholar 

  43. F. Catanese, Singular bidouble covers and the construction of interesting algebraic surfaces. AMS Cont. Math. bf 241 (1999), 97-120.

    MathSciNet  Google Scholar 

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

    Google Scholar 

  45. F. Catanese, Moduli Spaces of Surfaces and Real Structures, Ann. Math. (2) 158, no.2 (2003), 577-592.

    MATH  MathSciNet  Google Scholar 

  46. F. Catanese, Fibred Kähler and quasi projective groups, Advances in Geometry, suppl., Special Issue dedicated to A. Barlotti’s 80-th birthday (2003), Adv. Geom. Suppl. (2003) S13-S27.

    Google Scholar 

  47. F. Catanese, J. Keum, K. Oguiso, Some remarks on the universal cover of an open K3 surface. Math. Ann. 325, No. 2, (2003), 279-286.

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  49. F. Catanese, B. Wajnryb, Diffeomorphism of simply connected algebraic surfaces. math.AG/0405299, J. Differential Geom. 76 no. 2 (2007), 177-213.

    MATH  MathSciNet  Google Scholar 

  50. F. Catanese, Canonical symplectic structures and deformations of algebraic surfaces, 11 pages, math.AG/0608110 to appear in Communications in Contemporary Mathematics.

    Google Scholar 

  51. O. Chisini, Sulla identità birazionale delle funzioni algebriche di due variabili dotate di una medesima curva di diramazione, Ist. Lom-bardo Sci. Lett. Cl. Sci. Mat. Nat. Rend. (3) 8(77) (1944), 339-356.

    MathSciNet  Google Scholar 

  52. 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.

    MathSciNet  Google Scholar 

  53. M. Cornalba, Complex tori and Jacobians. Complex analysis and its applications (Lectures, Internat. Sem., Trieste, 1975), Vol. II, Internat. Atomic Energy Agency, Vienna, (1976), 39-100.

    Google Scholar 

  54. M. Dehn, Die Gruppe der Abbildungsklassen. (Das arithmetische Feld auf Flächen.) Acta Math. 69 (1938), 135-206.

    MATH  MathSciNet  Google Scholar 

  55. P. Deligne,Equations différentiellesá points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163. Springer-Verlag, Berlin-New York(1970), iii+133 pp.

    Google Scholar 

  56. P. Deligne, G.D. Mostow, Commensurabilities among lattices in PU(1, n), Annals of Mathematics Studies, 132, Princeton University Press, Princeton, NJ (1993), viii+183 pp.

    Google Scholar 

  57. I. Dolgachev, Weighted projective varieties, in ‘Group actions and vector fields (Vancouver, B.C., 1981)’, Lecture Notes in Math., 956, Springer, Berlin, (1982), 34-71.

    Google Scholar 

  58. S.K. Donaldson, An Application of Gauge Theory to Four-Dimensional Topology, J. Differential Geom. 18 (1983), 279-315.

    MATH  MathSciNet  Google Scholar 

  59. S.K. Donaldson, Connections, cohomology and the intersection forms of 4-manifolds. J. Differential Geom. 24 (1986), 275-341.

    MATH  MathSciNet  Google Scholar 

  60. S.K. Donaldson, Polynomial invariants for smooth four-manifolds. Topology 29, 3 (1990), 257-315.

    MATH  MathSciNet  Google Scholar 

  61. 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.

    Google Scholar 

  62. S.K. Donaldson, The Seiberg-Witten Equations and 4-manifold topology. Bull. Am. Math. Soc., (N S) 33, 1 (1996) 45-70.

    MATH  MathSciNet  Google Scholar 

  63. S.K. Donaldson, Symplectic submanifolds and almost-complex geometry. J. Differential Geom. 44, 4 (1996), 666-705.

    MATH  MathSciNet  Google Scholar 

  64. S.K. Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53, no. 2(1999), 205-236.

    MATH  MathSciNet  Google Scholar 

  65. S.K. Donaldson, P.B. Kronheimer The geometry of four-manifolds. Oxford Mathematical Monographs. Oxford: Clarendon Press, (1990) ix, 440 p.

    Google Scholar 

  66. A.H. Durfee, Fifteen characterizations of rational double points and simple critical points, Enseign. Math. (2) 25, no. 1-2 (1979), 131-163.

    MathSciNet  Google Scholar 

  67. C. Ehresmann, Sur les espaces fibrés différentiables. C.R. Acad. Sci. Paris 224 (1947), 1611-1612.

    MATH  MathSciNet  Google Scholar 

  68. T. Ekedahl, Canonical models of surfaces of general type in positive characteristic, Inst. Hautes Études Sci. Publ. Math. No. 67 (1988),97-144.

    MATH  MathSciNet  Google Scholar 

  69. F. Enriques, Le Superficie Algebriche. Zanichelli, Bologna (1949).

    MATH  Google Scholar 

  70. M. Freedman, The topology of four-dimensional manifolds., J. Differential Geom. 17, n. 3 (1982), 357-454.

    MATH  MathSciNet  Google Scholar 

  71. M. Freedman and F.Quinn, Topology of 4-manifolds., Princeton Math. Series 39 (1990).

    Google Scholar 

  72. R. Friedman, B.G. Moishezon, J.W.Morgan, On the C ∞ invariance of the canonical classes of certain algebraic surfaces., Bull. Amer. Math. Soc. 17 (1987), 283-286.

    MATH  MathSciNet  Google Scholar 

  73. R. Friedman and J.W.Morgan, Algebraic surfaces and four-manifolds: some conjectures and speculations., Bull. Amer. Math. Soc. 18 (1988) 1-19.

    MATH  MathSciNet  Google Scholar 

  74. 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 pp.

    Google Scholar 

  75. R. Friedman and J.W.Morgan, Algebraic surfaces and Seiberg-Witten invariants. J. Algebr. Geom. 6, No.3 (1997) 445-479. Differentiable and Deformation Type of Algebraic Surfaces163

    MATH  MathSciNet  Google Scholar 

  76. R. Friedman. Letter to the author, August 9 2005.

    Google Scholar 

  77. D. Gieseker, Global moduli for surfaces of general type, Invent. Math. 43, no. 3(1977), 233-282.

    MATH  MathSciNet  Google Scholar 

  78. R.E. Gompf, A new construction of symplectic manifolds Ann. of Math. 142 3 (1995), 527-595.

    MATH  MathSciNet  Google Scholar 

  79. R.E. Gompf, The topology of symplectic manifolds. Turkish J. Math. 25 no. 1, (2001), 43-59.

    MATH  MathSciNet  Google Scholar 

  80. R.E. Gompf, A. Stipsicz, 4-manifolds and Kirby calculus. Graduate Studies in Mathematics, 20. American Mathematical Society, Providence, RI, (1999) xvi+558 pp.

    Google Scholar 

  81. H. Grauert, R. Remmert, Komplexe Räume, Math. Ann. 136 (1958),245-318.

    MATH  MathSciNet  Google Scholar 

  82. H. Grauert, Über die Deformation isolierter Singularitäten analytis-cher Mengen, Invent. Math. 15 (1972), 171-198.

    MATH  MathSciNet  Google Scholar 

  83. H. Grauert, Der Satz von Kuranishi für kompakte komplexe Räume, Invent. Math. 25(1974), 107-142.

    MATH  MathSciNet  Google Scholar 

  84. M. Gromov, Sur le groupe fondamental d’une variété kählérienne, C. R. Acad. Sci., Paris, Sér. I 308, No. 3(1989), 67-70.

    MATH  MathSciNet  Google Scholar 

  85. A. Grothendieck, Techniques de construction et théoremes d’existence en géométrie algébrique. IV, Les schemas de Hilbert, Sem. Bourbaki, Vol. 13, (1960-1961), 1-28.

    Google Scholar 

  86. R. Hartshorne, Algebraic geometry, Springer GTM 52 (1977).

    Google Scholar 

  87. R. Hartshorne, Connectedness of Hilbert scheme, Publ. Math., Inst. Hautes Étud. Sci. 29(1966), 5-48.

    MATH  MathSciNet  Google Scholar 

  88. A. Hatcher, W.Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19, no. 3 (1980), 221-237.

    MATH  MathSciNet  Google Scholar 

  89. 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.

    MathSciNet  Google Scholar 

  90. F. Hirzebruch, Arrangements of lines and algebraic surfaces,in ‘Arithmetic and geometry, vol. II’, Progr. math 36, Birkhäuser (1983),113-140.

    Google Scholar 

  91. E. Horikawa, On deformations of quintic surfaces, Invent. Math. 31(1975),43-85.

    MATH  Google Scholar 

  92. J. E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics 21, Springer-Verlag New York - Heidelberg - Berlin, (1975), XV, 247 p.

    Google Scholar 

  93. D. Huybrechts, Complex geometry. An introduction. Universitext. Springer-Verlag, Berlin, (2005), xii+309 pp.

    Google Scholar 

  94. J. Jost, S.T. Yau A strong rigidity theorem for a certain class of compact complex analytic surfaces, Math. Ann. 271(1985),143-152.

    MATH  MathSciNet  Google Scholar 

  95. A. Kas, On the handlebody decomposition associated to a Lefschetz fibration. Pacific J. Math. 89 no. 1, (1980), 89-104.

    MATH  MathSciNet  Google Scholar 

  96. 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.

    Google Scholar 

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

    Google Scholar 

  98. V. Kharlamov, V. Kulikov, Deformation inequivalent complex conjugated complex structures and applications, Turkish J. Math. 26, no. 1(2002), 1-25.

    MATH  MathSciNet  Google Scholar 

  99. F. Klein, Lectures on the icosahedron and the solution of equations of the fifth degree, second and revised edition, Dover Publications, Inc., New York, N.Y. (1956), xvi+289 pp.

    Google Scholar 

  100. K. Kodaira, D. C. Spencer, On deformations of complex analytic structures. I, II, Ann. of Math. (2) 67 (1958), 328-466.

    MathSciNet  Google Scholar 

  101. K. Kodaira, On compact complex analytic surfaces, I, Ann. of Math. 71 (1960), 111-152.

    MathSciNet  Google Scholar 

  102. K. Kodaira, On compact analytic surfaces, III, Ann. Math. (2) 78 (1963),1-40.

    MathSciNet  Google Scholar 

  103. K. Kodaira, On stability of compact submanifolds of complex manifolds, Am. J. Math. 85(1963), 79-94.

    MATH  MathSciNet  Google Scholar 

  104. K. Kodaira, On the structure of compact complex analytic surfaces. I., Amer. J. Math. 86 (1964), 751-798.

    MATH  MathSciNet  Google Scholar 

  105. K. Kodaira, On the structure of compact complex analytic surfaces. II. Amer. J. Math. 88(1966),682-721.

    MATH  MathSciNet  Google Scholar 

  106. J. Kollár and N.I. Shepherd Barron, Threefolds and deformations of surface singularities. Inv. Math. 91 (1988) 299-338.

    MATH  Google Scholar 

  107. 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.

    MathSciNet  Google Scholar 

  108. V.S. Kulikov, On Chisini’s conjecture. II math. AG/0610356v1.

    Google Scholar 

  109. M. Kuranishi, On the locally complete families of complex analytic structures, Ann. Math. (2) 75 (1962), 536-577.

    MathSciNet  Google Scholar 

  110. M. Kuranishi, New proof for the existence of locally complete families of complex structures, Proc. Conf. Complex Analysis, Minneapolis 1964(1965),142-154.

    MathSciNet  Google Scholar 

  111. E. Looijenga and J. Wahl, Quadratic functions and smoothing surface singularities Topology 25 (1986), 261-291.

    MATH  MathSciNet  Google Scholar 

  112. R. Mandelbaum, B. Moishezon, On the topological structure of non-singular algebraic surfaces in CP 3 , Topology 15(1976), 23-40.

    MATH  MathSciNet  Google Scholar 

  113. R. Mandelbaum, B. Moishezon, On the topology of simply-connected algebraic surfaces, Trans. Am. Math. Soc. 260 (1980), 195-222.

    MATH  MathSciNet  Google Scholar 

  114. M. Manetti, Q-Gorenstein smoothings of quotient singularities, Preprint Scuola Normale Pisa (1990).

    Google Scholar 

  115. M. Manetti, On some Components of the Moduli Space of Surfaces of General Type, Comp. Math. 92 (1994) 285-297.

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  117. M. Manetti, Iterated Double Covers and Connected Components of Moduli Spaces, Topology 36, 3 (1997) 745-764.

    MATH  MathSciNet  Google Scholar 

  118. M. Manetti, On the Moduli Space of diffeomorphic algebraic surfaces, Inv. Math. 143 (2001), 29-76.

    MATH  MathSciNet  Google Scholar 

  119. J. Mather, Notes on topological stability, Harvard University Math. Notes (1970).

    Google Scholar 

  120. D. McDuff, D. Salamon, Introduction to symplectic topology. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York (1998).

    Google Scholar 

  121. Y. Miyaoka, Algebraic surfaces with positive indices, in ‘Classification of algebraic and analytic manifolds’, Proc. Symp. Katata/Jap. 1982, Prog. Math. 39 (1983), 281-301.

    MathSciNet  Google Scholar 

  122. J. W. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies 61, Princeton University Press and the University of Tokyo Press (1968), 122 p.

    Google Scholar 

  123. B. Moishezon, Complex surfaces and connected sums of complex projective planes, Lecture Notes in Math., 603. Springer, Berlin (1977).

    Google Scholar 

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

    MathSciNet  Google Scholar 

  125. 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.

    MathSciNet  Google Scholar 

  126. B. Moishezon, M. Teicher, Finite fundamental groups, free over Z/cZ, for Galois covers of CP2 . Math. Ann. 293, no. 4(1992), 749-766.

    MathSciNet  Google Scholar 

  127. B. Moishezon, The arithmetics of braids and a statement of Chisini, in‘Geometric Topology, Haifa1992’, Contemp. Math.164, Amer. Math. Soc., Providence, RI (1994), 151-175.

    Google Scholar 

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

    Google Scholar 

  129. S. Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 , no. 1 (1988), 117-253.

    MATH  Google Scholar 

  130. J. Moser, On the volume elements on a manifold. Trans. Amer. Math. Soc. 120(1965),286-294.

    MATH  MathSciNet  Google Scholar 

  131. G.D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, no. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1973).

    Google Scholar 

  132. D. Mumford, The canonical ring of an algebraic surface, Ann. of Math. (2) 76 (1962) 612-615, appendix to The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, by O. Zariski, ibid. 560-611.

    Google Scholar 

  133. D. Mumford, Algebraic geometry. I. Complex projective varieties. Reprint of the 1976 edition. Classics in Mathematics. SpringerVerlag, Berlin (1995), x+186 pp.

    Google Scholar 

  134. D. Mumford, Towards an enumerative geometry of the moduli space of curves, in ‘Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry’, Prog. Math. 36, Birkhäuser (1983), 271-328.

    Google Scholar 

  135. S.Y. Nemirovskii, On Kulikov’s theorem on the Chisini conjecture. Izv. Ross. Akad. Nauk Ser. Mat. 65, no. 1 (2001), 77-80; translation in Izv. Math. 65 , no. 1(2001), 71-74.

    MathSciNet  Google Scholar 

  136. R. Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Mathematik 417 (1991), 191-213.

    MATH  MathSciNet  Google Scholar 

  137. C.P. Ramanujam, Remarks on the Kodaira vanishing theorem, J. Indian Math. Soc. 36 (1972), 41-51; supplement J. Indian Math. Soc. 38 (1974), 121-124.

    MATH  MathSciNet  Google Scholar 

  138. M. Reid, Canonical 3-folds, in ‘Journées de géometrie algébrique, Angers/France 1979’, Sijthoff and Nordhoff (1980), 273-310.

    Google Scholar 

  139. M. Reid, Young person’s guide to canonical singularities, in ‘Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985’, part 1, Proc. Symp. Pure Math. 46 (1987), 345-414.

    Google Scholar 

  140. O. Riemenschneider, Deformationen von Quotientensingularitäten (nach zyklischen Gruppen), Math. Ann. 209 (1974), 211-248.

    MATH  MathSciNet  Google Scholar 

  141. Sernesi, Edoardo Deformations of algebraic schemes. Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences, 334. Springer-Verlag, Berlin, 2006. xii+339 pp.

    Google Scholar 

  142. J.P. Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955),197-278.

    MathSciNet  Google Scholar 

  143. J.P. Serre, Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier, Grenoble 6(1955-1956), 1-42.

    MathSciNet  Google Scholar 

  144. J.P. Serre, Groupes algébriques et corps de classes. Publications de l’institut de mathématique de l’université de Nancago, VII. Hermann, Paris (1959), 202 pp.

    MATH  Google Scholar 

  145. J.P. Serre, Exemples de variétés projectives conjuguées non homéomorphes. C. R. Acad. Sci. Paris 258, (1964) 4194-4196.

    MATH  MathSciNet  Google Scholar 

  146. I.R. Shafarevich, I. R. Basic algebraic geometry. Translated from the Russian by K. A. Hirsch. Revised printing of Grundlehren der mathematischen Wissenschaften, Vol. 213, 1974. Springer Study Edition. Springer-Verlag, Berlin-New York, (1977), xv+439 pp.

    Google Scholar 

  147. C.L. Siegel, Topics in complex function theory. Vol. III. Abelian functions and modular functions of several variables. Translated from the German by E. Gottschling and M. Tretkoff. With a preface by Wilhelm Magnus. Reprint of the 1973 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley and Sons, Inc., New York (1989), x+244 pp.

    Google Scholar 

  148. Y.T. Siu, An effective Matsusaka big theorem, Ann. Inst. Fourier 43(1993),1387-1405.

    MATH  MathSciNet  Google Scholar 

  149. I. Smith, R. Thomas, S.T. Yau, Symplectic conifold transitions, J. Diff. Geom. 62 (2002), 209-242.

    MATH  MathSciNet  Google Scholar 

  150. I. Smith, R. Thomas, Symplectic surgeries from singularities, Turkish J. Math. 27(2003),231-250.

    MATH  MathSciNet  Google Scholar 

  151. W.P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55, no. 2(1976),467-468.

    MATH  MathSciNet  Google Scholar 

  152. G.N. Tjurina, Resolution of singularities of plane (= flat) deformations of double rational points, Funk. Anal. i Prilozen 4 (1970), 77-83.

    MathSciNet  Google Scholar 

  153. A. Tromba, Dirichlet’s energy on Teichmüller’s moduli space and the Nielsen realization problem, Math. Z. 222 (1996), 451-464.

    MathSciNet  Google Scholar 

  154. E. Vesentini, Capitoli scelti della teoria delle funzioni olomorfe, Unione Matematica Italiana (1984).

    Google Scholar 

  155. E. Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 30, Springer-Verlag, Berlin (1995), viii + 320 pp.

    Google Scholar 

  156. C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés,10, Société Mathématique de France, Paris, (2002), viii+595 pp.

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  158. B. Wajnryb, An elementary approach to the mapping class group of a surface, Geom. Topol. 3(1999), 405-466.

    MATH  MathSciNet  Google Scholar 

  159. C.T.C. Wall, On the orthogonal groups of unimodular quadratic forms, Math. Ann. 147 (1962), 328-338.

    MATH  Google Scholar 

  160. E. Witten, Monopoles and Four-Manifolds, Math. Res. Lett. 1 (1994),809-822.

    MathSciNet  Google Scholar 

  161. S.T. Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Natl. Acad. Sci. USA 74 (1977), 1798-1799.

    MATH  Google Scholar 

  162. Yau, S. T.: 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.

    MATH  Google Scholar 

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  1. Mathematisches Institut Lehrstuhl Mathematik VIII, Universität Bayreuth, 95447, Bayreuth, Germany

    Fabrizio Catanese

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    Fabrizio Catanese

  2. Department of Mathematics, Princeton University, Washington Road, 08544-1000, Princeton, NJ, USA

    Gang Tian

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Catanese, F. (2008). Differentiable and Deformation Type of Algebraic Surfaces, Real and Symplectic Structures. In: Catanese, F., Tian, G. (eds) Symplectic 4-Manifolds and Algebraic Surfaces. Lecture Notes in Mathematics, vol 1938. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78279-7_2

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