A simply connected surface of general type with pg = 0 and K2 = 3

Park, Heesang¹ ; Park, Jongil¹ ; Shin, Dongsoo²

¹ Department of Mathematical Sciences, Seoul National University, San 56-1, Sillim-dong, Gwanak-gu, Seoul 151-747, Korea
² Department of Mathematics, Pohang University of Science and Technology, San 31, Hyoja-dong, Nam-gu, Pohang, Gyungbuk 790-784, Korea

Geometry & topology, Tome 13 (2009) no. 2, pp. 743-767.

Résumé

Motivated by a recent result of Y Lee and the second author [Invent. Math. 170 (2007) 483-505], we construct a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=3$ using a rational blow-down surgery and $\mathbb{Q}$–Gorenstein smoothing theory. In a similar fashion, we also construct a new simply connected symplectic $4$–manifold with $b_2^{+}=1$ and $K^2=4$.

Détail

DOI : 10.2140/gt.2009.13.743

Keywords: $\mathbb{Q}$-Gorenstein smoothing, rational blow-down, surface of general type

Bibliographie

[1] A Akhmedov, Small exotic $4$–manifolds

[2] M Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962) 485

[3] S Baldridge, P Kirk, Constructions of small symplectic $4$–manifolds using Luttinger surgery

[4] R Barlow, A simply connected surface of general type with $p_g=0$, Invent. Math. 79 (1985) 293

[5] W P Barth, K Hulek, C A M Peters, A Van De Ven, Compact complex surfaces, Ergebnisse series 3: Modern Surveys in Math. 4, Springer (2004)

[6] R Fintushel, R J Stern, Rational blowdowns of smooth $4$–manifolds, J. Differential Geom. 46 (1997) 181

[7] H Flenner, M Zaidenberg, $\mathbf Q$–acyclic surfaces and their deformations, from: "Classification of algebraic varieties (L'Aquila, 1992)" (editors C Ciliberto, E L Livorni, A J Sommese), Contemp. Math. 162, Amer. Math. Soc. (1994) 143

[8] J Kollár, N I Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988) 299

[9] Y Lee, J Park, A simply connected surface of general type with $p_g=0$ and $K^2=2$, Invent. Math. 170 (2007) 483

[10] S Lichtenbaum, M Schlessinger, The cotangent complex of a morphism, Trans. Amer. Math. Soc. 128 (1967) 41

[11] E Looijenga, J Wahl, Quadratic functions and smoothing surface singularities, Topology 25 (1986) 261

[12] M Manetti, Normal degenerations of the complex projective plane, J. Reine Angew. Math. 419 (1991) 89

[13] D Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. (1961) 5

[14] N Nakayama, Zariski-decomposition and abundance, MSJ Memoirs 14, Math. Soc. of Japan (2004)

[15] P Ozsváth, Z Szabó, On Park's exotic smooth four-manifolds, from: "Geometry and topology of manifolds" (editors H U Boden, I Hambleton, A J Nicas, B D Park), Fields Inst. Commun. 47, Amer. Math. Soc. (2005) 253

[16] V P Palamodov, Deformations of complex spaces, Uspehi Mat. Nauk 31 (1976) 129

[17] J Park, Seiberg–Witten invariants of generalised rational blow-downs, Bull. Austral. Math. Soc. 56 (1997) 363

[18] J Park, Simply connected symplectic $4$–manifolds with $b^+_2=1$ and $c^2_1=2$, Invent. Math. 159 (2005) 657

[19] U Persson, Configurations of Kodaira fibers on rational elliptic surfaces, Math. Z. 205 (1990) 1

[20] A I Stipsicz, Z Szabó, An exotic smooth structure on $\mathbb C\mathbb P^2\#6\overline\mathbb{C\mathbb P^2}$, Geom. Topol. 9 (2005) 813

[21] M Symington, Symplectic rational blowdowns, J. Differential Geom. 50 (1998) 505

[22] M Symington, Generalized symplectic rational blowdowns, Algebr. Geom. Topol. 1 (2001) 503

[23] J Wahl, Elliptic deformations of minimally elliptic singularities, Math. Ann. 253 (1980) 241

[24] J Wahl, Smoothings of normal surface singularities, Topology 20 (1981) 219