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Smoothings of Singularities and Deformation Types of Surfaces

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Symplectic 4-Manifolds and Algebraic Surfaces

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

The aim of these lectures is to illustrate some major development in the understanding of the connectedness properties of the moduli space of surfaces of general type. We are especially interested in the results obtained after the last C.I.M.E. course on this topic, course made by F. Catanese in 1985 [24]; the reading of the C.I.M.E. notes [24] provides also an ideal starting point for this chapter.

In order to introduce and to give motivation, we begin by recalling the most important notion and some results.

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References

  1. V. Alexeev: Moduli spaces Mg,n (W ) for surfaces. In: (M. Andreatta and T. Peternell eds.) Proceedings of the international conference, Trento 1994, Walter de Gruyter (1996).

    Google Scholar 

  2. A. Andreotti: Sopra le superficie algebriche che posseggono trasformazioni bi-razionali in sé. Rendiconti di Matematica e delle sue applicazioni V, IX Roma (1950).

    Google Scholar 

  3. E. Arbarello, M. Cornalba, P. Griffiths, J. Harris: Geometry of algebraic curves, I. Springer-Verlag (1984).

    Google Scholar 

  4. M. Artin: Deformations of singularities. Tata Institute of Fundamental Research, Bombay (1976).

    MATH  Google Scholar 

  5. M. Artin: Some numerical criteria for contractibility of curves on algebraic surfaces. Am. J. Math. 84 (1962) 485-496.

    MATH  MathSciNet  Google Scholar 

  6. M. Artin: On isolated rational singularities of surfaces. Am. J. Math. 88 (1966) 129-136.

    MATH  MathSciNet  Google Scholar 

  7. M. Artin: Algebraic approximation of structures over complete local ring. Publ. Math. IHES 36 (1969) 23-58.

    MATH  MathSciNet  Google Scholar 

  8. M. Artin: Algebraic construction of Brieskorn’s resolutions. J. Algebra 29 (1974) 330-348.

    MATH  MathSciNet  Google Scholar 

  9. M. Atiyah: On analytic surfaces with double points. Proc. Roy. Soc. A 247 (1958) 237-244.

    MATH  MathSciNet  Google Scholar 

  10. L. Badescu: Suprafete Algebrice. Editura Academiei Republicii Socialiste Romania (1981). English translation Algebraic surfaces. Springer-Verlag Universitext (2001).

    Google Scholar 

  11. C. Banica, O. Stanasila: Méthodes algébrique dans la théorie globale des espaces complexes. Gauthier-Villars (1977).

    Google Scholar 

  12. W. Barth, C. Peters, A. van de Ven: Compact complex surfaces. Springer- Verlag, Berlin (1984).

    MATH  Google Scholar 

  13. W. Barth, K. Hulek, C. Peters, A. van de Ven: Compact complex surfaces. Second edition. Springer-Verlag, Berlin (2004).

    MATH  Google Scholar 

  14. A. Beauville: Complex algebraic surfaces. London Math. Soc. LNS 68 (1983).

    Google Scholar 

  15. E. Bombieri: Canonical models of surfaces of general type. IHES Publ. Sci. 42 (1973) 171-219.

    MathSciNet  Google Scholar 

  16. F. Bonahon: Difféotopies des espaces lenticulaires. Topology 22 (1983) 305-314.

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  18. Burns, J. Wahl: Local contribution to global deformations of surfaces. Invent. Math. 26 (1974) 67-88.

    MATH  MathSciNet  Google Scholar 

  19. H. Cartan: Quotient d’un espace analytique par un groupe d’automorphismes. Algebraic geometry and topology: A symposium in honour of S. Lefschetz. Princeton Math. Series 12 (1957) 90-102.

    MATH  MathSciNet  Google Scholar 

  20. F. Catanese: Moduli of surfaces of general type. In: Algebraic geometry: open problems. Proc. Ravello 1982 Springer L.N.M. 997 (1983) 90-112.

    Google Scholar 

  21. F. Catanese: On the moduli spaces of surfaces of general type. J. Differential Geom. 19 (1984) 483-515.

    MathSciNet  Google Scholar 

  22. F. Catanese: Connected components of moduli spaces. J. Differ. Geom. 24 (1986) 395-399.

    MATH  MathSciNet  Google Scholar 

  23. F. Catanese: Automorphisms of rational double points and moduli spaces of surfaces of general type. Compositio Math. 61 (1987) 81-102.

    MATH  MathSciNet  Google Scholar 

  24. F. Catanese: Moduli of algebraic surfaces. In C.I.M.E. course Theory of muduli, Montecatini Terme, 1985 Springer-Verlag LNM 1337 (1988) 1-83.

    Google Scholar 

  25. F. Catanese: Chow varieties, Hilbert schemes and moduli spaces of surfaces of general type. J. Alg. Geometry 1 (1992) 561-595.

    MATH  MathSciNet  Google Scholar 

  26. F. Catanese: private communication (1992).

    Google Scholar 

  27. F. Catanese: (Some) Old and new results on algebraic surfaces. Proc. I Euro-pean congress of Math. Paris 1992, Birkhauser (1994).

    Google Scholar 

  28. F. Catanese: Singular bidouble covers and the construction of interesting alge-braic surfaces. in Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 97-120, Contemp. Math. 241, Am. Math. Soc., Providence, RI, (1999).

    Google Scholar 

  29. F. Catanese: Moduli spaces of surfaces and real structures. Ann. of Math. 158 (2003) 577-592.

    MATH  MathSciNet  Google Scholar 

  30. F. Catanese: Symplectic structures of algebraic surfaces and deformation. arXiv:math/0207254v1 [math.AG].

    Google Scholar 

  31. F. Catanese, B. Wajnryb: Diffeomorphism of simply connected algebraic surfaces. J. Differential Geom. 76 (2007), no. 2, 177-213.

    MATH  MathSciNet  Google Scholar 

  32. C. Chevalley: Invariants of finite groups generated by reflections. Am. J. Math. 77 (1955), 778-782.

    MATH  MathSciNet  Google Scholar 

  33. C. Ciliberto, E. Sernesi: Families of varieties and the Hilbert scheme. Lecture Notes (1990).

    Google Scholar 

  34. A. Corti: Polynomial bounds for automorphisms of surfaces of general type. Ann. E.N.S. IV, 24 (1991) 113-137.

    MathSciNet  Google Scholar 

  35. S.K. Donaldson: Gauge theory and four-manifold topology. Proc. I Eureopean congress of Math. Paris 1992, Birkhauser (1994).

    Google Scholar 

  36. S.K. Donaldson, P.B. Kronheimer: The geometry of four-manifolds. Oxford University Press (1990).

    Google Scholar 

  37. S. Donaldson: The Seiberg-Witten equations and 4-manifold topology. Bull. A.M.S. 33 (1996) 45-70.

    MATH  MathSciNet  Google Scholar 

  38. A. Douady, J.L. Verdier: Séminaire de géometrie analytique. Astérisque 16-17 Soc. Math. France (1976).

    Google Scholar 

  39. A. Durfee: Fifteen characterizations of rational double points and simple critical points. Ens. Math. 25 (1979) 131-163.

    MathSciNet  Google Scholar 

  40. P. Du Val: On isolated singularities of surfaces which do not affect the conditions of adjunction. Proc. Camb. Phil. Soc. 30 453-459.

    Google Scholar 

  41. C. Ehresmann: Sur les espaces fibrés differentiables. C. R. Acad. Sci. 224 (1947) 1611-1612.

    MATH  MathSciNet  Google Scholar 

  42. R. Elkik: Singularites rationelles et deformations. Invent. Math. 47 (1978) 139-147.

    MATH  MathSciNet  Google Scholar 

  43. H. Esnault, E. Viehweg: Two dimensional quotient singularities deform to quotient singularities. Math. Ann. 271 (1985) 439-449.

    MATH  MathSciNet  Google Scholar 

  44. B. Fantechi, M. Manetti: Obstruction calculus for functors of Artin rings, I. J. Algebra 202 (1998) 541-576.

    MATH  MathSciNet  Google Scholar 

  45. B. Fantechi, R. Pardini: Automorphism and moduli spaces of varieties with ample canonical class via deformations of abelian covers. Comm. in Algebra 25 (1997) 1413-1441.

    MATH  MathSciNet  Google Scholar 

  46. R. Fintushel, R. Stern: Rational Blowdowns of smooth 4-manifolds. J. Differ. Geom. 46 (1997) 181-235.

    MATH  MathSciNet  Google Scholar 

  47. G. Fischer: Complex analytic geometry. Springer-Verlag LNM 538 (1976).

    Google Scholar 

  48. H. Flenner:Uber Deformationen holomorpher Abbildungen. Habilitationss-chrift, Osnabruck 1978.

    Google Scholar 

  49. O. Forster, K. Knorr: Konstruction verseller Familien kompakter komplexer Raüme. Springer Lect. Notes Math. 705 (1979).

    Google Scholar 

  50. M. Freedman: The topology of four-dimensional manifolds. J. Diff. Geometry 17 (1982) 357-454.

    MATH  Google Scholar 

  51. R. Friedman: Donaldson and Seiberg-Witten invariants of algebraic surfaces. Algebraic geometry-Santa Cruz 1995, 85-100, Proc. Sympos. Pure Math., 62, Part 1, Am. Math. Soc., Providence, RI, (1997).

    Google Scholar 

  52. R. Friedman, B. Moishezon, J.W. Morgan: On the C ∞ invariance of the canon-ical classes of certain algebraic surfaces. Bull. Am. Math. Soc. (N.S.) 17 (1987) 283-286.

    MATH  MathSciNet  Google Scholar 

  53. R. Friedman, J.W. Morgan: Algebraic surfaces and four-manifolds: some con-jectures and speculations. Bull. Am. Math. Soc. (N.S.) 18 (1988) 1-19.

    MATH  MathSciNet  Google Scholar 

  54. R. Friedman, J.W. Morgan: Smooth four-manifolds and complex surfaces. Ergebnisse der Mathematik 27 Springer-Verlag (1994).

    Google Scholar 

  55. R. Friedman, J.W. Morgan: Algebraic surfaces and Seiberg-Witten invariants. J. Algebraic Geom. 6 (1997), no. 3, 445-479.

    MATH  MathSciNet  Google Scholar 

  56. D. Gieseker: Global moduli for surfaces of general type. Invent. Math. 43 (1977) 233-282.

    MATH  MathSciNet  Google Scholar 

  57. D. Gieseker: Geometric invariant theory and applications to moduli problems. Springer lectures notes 996, Ed. Gherardelli (1982) 45-73.

    Google Scholar 

  58. H. Grauert: Der satz von Kuranishi für kompakte komplexe Raüme. Invent. Math. 25 (1974) 107-142.

    MATH  MathSciNet  Google Scholar 

  59. G.M. Greuel, U. Karras: Families of varieties with prescribed singularities. Comp. Math. 69 (1989) 83-110.

    MATH  MathSciNet  Google Scholar 

  60. G.M. Greuel, J. Steenbrink: On the topology of smoothable singularities. Proc. Symp. Pure Math. 40 part 1 (1983) A.M.S. 535-545.

    MathSciNet  Google Scholar 

  61. A. Grothendieck: Local Cohomology. Springer L.N.M. 41 (1967).

    Google Scholar 

  62. R. Hartshorne: Algebraic geometry. Springer Verlag GTM 52 (1977).

    Google Scholar 

  63. M.W. Hirsch: Differential Topology. Springer-Verlag GTM 33 (1976).

    Google Scholar 

  64. E. Horikawa: Deformations of holomorphic maps I. J. Math. Soc. Japan 25 (1973) 372-396; II J. Math. Soc. Japan 26 (1974) 647-667; III Math. Ann. 222 (1976) 275-282.

    Google Scholar 

  65. D. Iacono: Local structure of abelian covers. J. Algebra 301 (2006), no. 2, 601-615.

    MATH  MathSciNet  Google Scholar 

  66. D. Iacono: Differential Graded Lie Algebras and Deformations of Holomorphic Maps. Tesi di Dottorato, Roma (2006). arXiv:math/0701091v1 [math.AG].

    Google Scholar 

  67. S. Iitaka: Deformations of compact complex surfaces II. J. Math. Soc. Jpn. 22 (1970) 247-261.

    MATH  MathSciNet  Google Scholar 

  68. S. Iitaka: On D-dimension of algebraic varieties. J. Math. Soc. Jpn. 23 (1971).

    Google Scholar 

  69. 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.

    MATH  MathSciNet  Google Scholar 

  70. A. Kas: Ordinary double points and obstructed surfaces. Topology 16 (1977) 51-64.

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  72. K. Kodaira: Complex manifold and deformation of complex structures. Springer-Verlag (1986).

    Google Scholar 

  73. K. Kodaira, L. Nirenberg, D.C. Spencer: On the existence of deformations of complex analytic structures. Ann. Math. 68 (1958) 450-459.

    MathSciNet  Google Scholar 

  74. K. Kodaira, D.C. Spencer: On the variation of almost complex structures. In Algebraic geometry and topology, Princeton Univ. Press (1957) 139-150.

    Google Scholar 

  75. K. Kodaira, D.C. Spencer: A theorem of completeness for complex analytic fibre spaces. Acta Math. 100 (1958) 281-294.

    MATH  MathSciNet  Google Scholar 

  76. K. Kodaira, D.C. Spencer: On deformations of complex analytic structures, I-II, III. Ann. Math. 67 (1958) 328-466; 71 (1960) 43-76.

    Google Scholar 

  77. J. Kollár, N.I. Shepherd-Barron: Threefolds and deformations of surface sin-gularities. Invent. Math. 91 (1988) 299-338.

    MATH  MathSciNet  Google Scholar 

  78. M. Kuranishi: New Proof for the existence of locally complete families of com-plex structures. In: Proc. Conf. Complex Analysis (Minneapolis 1964) Springer-Verlag (1965) 142-154.

    Google Scholar 

  79. H.B. Laufer: Taut Two-Dimensional Singularities. Math. Ann. 205 (1973) 131-164.

    MATH  MathSciNet  Google Scholar 

  80. H.B. Laufer: Normal twodimensional singularities. Princeton University press (1971).

    Google Scholar 

  81. F. Lazzeri: Analytic singularities. C.I.M.E. 1974, editore Cremonese, Roma.

    Google Scholar 

  82. Y. Lee, J. Park: A simply connected surface of general type with pg = 0 and K 2 = 2. Invent. Math. 170 (2007), no. 3, 483-505.

    MATH  MathSciNet  Google Scholar 

  83. A.S. Libgober, J.W. Wood: Differentiable structures on complete intersections I,II. I Topology 21 (1982) 469-482. II Proc. Symp. Pure Math. 40 part 2, (1983)123-133.

    MATH  MathSciNet  Google Scholar 

  84. J. Lipman: Introduction to resolution of singularities. Proc. Symp. Pure Math. 29 (Arcata 1974) 187-230.

    Google Scholar 

  85. J. Lipman: Double point resolutions of deformations of rational singularities. Comp. Math. 38 (1079) 37-43.

    MathSciNet  Google Scholar 

  86. E.J.N. Looijenga: Riemann-Roch and smoothings of singularities. Topology 25 (1986) 293-302.

    MATH  MathSciNet  Google Scholar 

  87. E.J.N. Looijenga: Isolated singular points on complete intersection. London Math. Soc. L.N.S. 77 (1983).

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  89. M. Manetti: Q-Gorenstein smoothings of quotient singularities. Preprint Scuola Normale Superiore Pisa (1990). Electronic version available at author’s URL.

    Google Scholar 

  90. M. Manetti: Normal degenerations of the complex projective plane. J. Reine Angew. Math. 419 (1991) 89-118.

    MATH  MathSciNet  Google Scholar 

  91. M. Manetti: On Some Components of Moduli Space of Surfaces of General Type. Comp. Math. 92 (1994) 285-297.

    MATH  MathSciNet  Google Scholar 

  92. M. Manetti: Degenerations of algebraic surfaces and applications to moduli problems. Tesi di Perfezionamento, Scuola Normale Superiore, Pisa (1995).

    Google Scholar 

  93. M. Manetti: Iterated double covers and connected components of moduli spaces. Topology 36 (1996) 745-764.

    MathSciNet  Google Scholar 

  94. M. Manetti: Automorphisms of generic cyclic covers. Revista Matemática de la Universidad Complutense de Madrid 10 (1997) 149-156.

    MATH  MathSciNet  Google Scholar 

  95. M. Manetti: Degenerate double covers of the projective plane. In: (Catanese, Hulek, Peters and Reid eds.) New trends in algebraic geometry. Cambridge Univ. Press (1999).

    Google Scholar 

  96. M. Manetti: On the moduli space of diffeomorphic algebraic surfaces. Invent. Math. 143 (2001) 29-76.

    MATH  MathSciNet  Google Scholar 

  97. H. Matsumura: Commutative Ring Theory. Cambridge University Press (1986).

    Google Scholar 

  98. H. Matsumura: On algebraic groups of birational transformations. Rend. Accad. Lincei Ser. 8 34 (1963) 151-155.

    MathSciNet  Google Scholar 

  99. J. Milnor: Morse theory. Ann. Math. Studies 51, Princeton Univ. Press, Princeton (1963).

    Google Scholar 

  100. J. Milnor: Singular points of complex hypersurfaces. Ann. Math. Studies 61, Princeton Univ. Press, Princeton (1968).

    Google Scholar 

  101. J. Milnor, J. Stasheff: Characteristic classes. Ann. Math. Studies 76, Princeton Univ. Press, Princeton (1974).

    Google Scholar 

  102. B.G. Moishezon: Analogs of Lefschetz Theorems for linear systems with isolated singularities. J. Diff. Geometry 31 (1990) 47-72.

    MATH  MathSciNet  Google Scholar 

  103. D. Mumford: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publ. Math. IHES 9 (1961) 5-22.

    MATH  MathSciNet  Google Scholar 

  104. D. Mumford: Abelian varieties. Oxford Univ. Press (1970).

    Google Scholar 

  105. M. Namba: Families of meromorphic functions on compact Riemann surfaces. Lecture Notes in Mathematics 767 Springer-Verlag, New York/Berlin, (1979).

    Google Scholar 

  106. P.E. Newstead: Introduction to moduli problems and orbit spaces. Tata Institute Fundamenta Research (1978).

    Google Scholar 

  107. C. Okonek, A. Teleman: Les invariants de Seiberg-Witten et la conjecture de van de Ven. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 4, 457-461.

    MATH  MathSciNet  Google Scholar 

  108. V.P. Palamodov: Deformations of complex spaces. Uspekhi Mat. Nauk. 31:3 (1976)129-194. Transl. Russian Math. Surveys 31:3 (1976) 129-197.

    MathSciNet  Google Scholar 

  109. V.P. Palamodov: Deformations of complex spaces. In: Several complex vari-ables IV. Encyclopaedia of Mathematical Sciences 10, Springer-Verlag (1986) 105-194.

    Google Scholar 

  110. R. Pardini: Abelian covers of algebraic varieties. J. Reine Angew. Math. 417 (1991) 191-213.

    MATH  MathSciNet  Google Scholar 

  111. U. Persson: An introduction to the geography of surfaces of general type. Proc. Sympos. Pure Math. 46 (1987) 195-218.

    MathSciNet  Google Scholar 

  112. H. Pinkham: Singularités rationelles des surfaces (avec un appendice). Lectures Notes in Math. 777 Springer Verlag (1980) 147-178.

    MathSciNet  Google Scholar 

  113. H. Pinkham: Some local obstructions to deforming global surfaces. Nova acta Leopoldina NF 52 Nr. 240 (1981) 173-178.

    MathSciNet  Google Scholar 

  114. D. Prill: Local classification of quotients of complex manifolds by discontinuous groups. Duke Math. J. 34 (1967) 375-386.

    MATH  MathSciNet  Google Scholar 

  115. R. Răsdeaconu, I. Şuvaina: The algebraic rational blow-down. arXiv:math/0601270v1 [math.SG].

    Google Scholar 

  116. M. Reid: Young person’s guide to canonical singularities. Proc. Symp. Pure Math. 46 part 1 (1987) A.M.S. 345-414.

    Google Scholar 

  117. O. Riemenschneider: Deformationen von quotientensingularitäten (nach zyklischen gruppen). Math. Ann. 209 (1974) 211-248.

    MATH  MathSciNet  Google Scholar 

  118. M. Salvetti: On the number of non-equivalent differentiable structures on 4-manifolds. Man. Math. 63 (1989) 157-171.

    MATH  MathSciNet  Google Scholar 

  119. M. Salvetti: A lower bound for the number of differentiable structures on 4-manifolds. Bollettino U.M.I. (7) 5-A (1991) 33-40.

    MathSciNet  Google Scholar 

  120. M. Schlessinger: Functors of Artin rings. Trans. Am. Math. Soc. 130 (1968) 208-222.

    MATH  MathSciNet  Google Scholar 

  121. M. Schlessinger: Rigidity of quotient singularities. Invent. Math. 14 (1971) 17-26.

    MATH  MathSciNet  Google Scholar 

  122. C.S. Seshadri: Quotient spaces modulo reductive algebraic groups. Ann. of Math. (2) 95 (1972), 511-556; errata, ibid. (2) 96 (1972), 599.

    MathSciNet  Google Scholar 

  123. J.P. Serre: Cours d’arithmétique. P.U.F. Paris (1970).

    Google Scholar 

  124. J. Steenbrink: Mixed Hodge structures associated with isolated singularities. Proc. Symp. Pure Math. 40 part 2 (1983) A.M.S. 513-536.

    MathSciNet  Google Scholar 

  125. S.G. Tankeev: On a global theory of moduli of algebraic surfaces of general type. Math. USSR Izvestija 6 (1972) 1200-1216.

    Google Scholar 

  126. B. Teissier: The hunting invariants in the geometry of discriminants. In: (P. Holm ed.) Real and complex singularities. Oslo (1976) 565-678.

    Google Scholar 

  127. A.N. Tyurin: Six lectures on four manifolds. In: C.I.M.E. lectures Transcendental methods in algebraic geometry, 1994. Springer-Verlag LNM 1646 (1996).

    Google Scholar 

  128. G.N. Tyurina: On the tautness of rationally contractible curves on a surface. Math. USSR Izvestija 2 (1968) 907-934.

    Google Scholar 

  129. G.N. Tyurina: Resolutions of singularities of plane deformations of double rational points. Funk. Anal. i. Priloz. 4 (1970) 77-83.

    Google Scholar 

  130. E. Viehweg: Quasi-projective moduli for polarized manifolds. Springer-Verlag Ergebnisse Math. Grenz. 30 (1995).

    Google Scholar 

  131. J. Wahl: Smoothings of normal surface singularities. Topology 20 (1981) 219-246.

    MATH  MathSciNet  Google Scholar 

  132. J. Wahl: Elliptic deformations of minimally elliptic singularities. Math. Ann. 253 (1980) 241-262.

    MATH  MathSciNet  Google Scholar 

  133. J. Wahl: Simultaneous resolution of rational singularities. Comp. Math. 38 (1979) 43-54.

    MATH  MathSciNet  Google Scholar 

  134. J. Wahl: Equisingular deformations of normal surface singularities. Ann. Math. 104 (1976) 325-356.

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  136. J.J. Wavrik: Obstruction to the existence of a space of moduli. In Global Analysis Princeton University Press (1969), 403-414.

    Google Scholar 

  137. E. Witten: Monopoles and four-manifolds. Math. Res. Lett. 1 (1994) 769-796.

    MATH  MathSciNet  Google Scholar 

  138. O. Zariski: Sur la normalité analytique des variétés normales. Ann. Inst. Fourier 2 (1950) 161-164.

    MATH  MathSciNet  Google Scholar 

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  1. Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, 00185, Roma, Italy

    Marco Manetti

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

    Fabrizio Catanese

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

    Gang Tian

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Manetti, M. (2008). Smoothings of Singularities and Deformation Types of Surfaces. 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_3

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