On rational homology disk smoothings of valency 4 surface singularities

Wahl, Jonathan¹

¹ Department of Mathematics, The University of North Carolina, Chapel Hill NC 27599-3250, USA

Geometry & topology, Tome 15 (2011) no. 2, pp. 1125-1156.

Résumé

Thanks to recent work of Stipsicz, Szabó and the author and of Bhupal and Stipsicz, one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk (“$\mathbb{Q}HD$”) smoothing, that is, one with Milnor number $0$. They fall into several classes, the most interesting of which are the $3$ classes whose resolution dual graph has central vertex with valency $4$. We give a uniform “quotient construction” of the $\mathbb{Q}HD$ smoothings for those classes; it is an explicit $\mathbb{Q}$–Gorenstein smoothing, yielding a precise description of the Milnor fibre and its non-abelian fundamental group. This had already been done for two of these classes; what is new here is the construction of the third class, which is far more difficult. In addition, we explain the existence of two different $\mathbb{Q}HD$ smoothings for the first class.

We also prove a general formula for the dimension of a $\mathbb{Q}HD$ smoothing component for a rational surface singularity. A corollary is that for the valency $4$ cases, such a component has dimension $1$ and is smooth. Another corollary is that “most” $H$–shaped resolution graphs cannot be the graph of a singularity with a $\mathbb{Q}HD$ smoothing. This result, plus recent work of Bhupal and Stipsicz, is evidence for a general conjecture:

Conjecture The only complex surface singularities admitting a $\mathbb{Q}HD$ smoothing are the (known) weighted homogeneous examples.

Détail

DOI : 10.2140/gt.2011.15.1125

Classification : 14B07, 14J17, 32S30
Keywords: surface singularity, rational homology disk fillings, smoothing surface singularities, Milnor fibre

Bibliographie

[1] M Bhupal, A Stipsicz, Weighted homogeneous singularities and rational homology disk smoothings, Amer. J. Math. (to appear)

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

[3] D Gay, A Stipsicz, On symplectic caps, from: "Perspectives in Analysis, Geometry and Topology", Progress in Mathematics, Birkhäuser (to appear)

[4] G M Greuel, H A Hamm, Invarianten quasihomogener vollständiger Durchschnitte, Invent. Math. 49 (1978) 67

[5] G M Greuel, E Looijenga, The dimension of smoothing components, Duke Math. J. 52 (1985) 263

[6] G M Greuel, J Steenbrink, On the topology of smoothable singularities, from: "Singularities, Part 1 (Arcata, Calif., 1981)", Proc. Sympos. Pure Math. 40, Amer. Math. Soc. (1983) 535

[7] H A Hamm, Exotische Sphären als Umgebungsränder in speziellen komplexen Räumen, Math. Ann. 197 (1972) 44

[8] T De Jong, D Van Straten, Deformation theory of sandwiched singularities, Duke Math. J. 95 (1998) 451

[9] J Kollár, Is there a topological Bogomolov–Miyaoka–Yau inequality?, Pure Appl. Math. Q. 4 (2008) 203

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

[11] H B Laufer, Taut two-dimensional singularities, Math. Ann. 205 (1973) 131

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

[13] E Looijenga, Riemann–Roch and smoothings of singularities, Topology 25 (1986) 293

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

[15] W D Neumann, An invariant of plumbed homology spheres, from: "Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979)", Lecture Notes in Math. 788, Springer (1980) 125

[16] W D Neumann, Abelian covers of quasihomogeneous surface singularities, from: "Singularities, Part 2 (Arcata, Calif., 1981)", Proc. Sympos. Pure Math. 40, Amer. Math. Soc. (1983) 233

[17] P Orlik, P Wagreich, Isolated singularities of algebraic surfaces with C$^{\ast}$ action, Ann. of Math. $(2)$ 93 (1971) 205

[18] H Pinkham, Normal surface singularities with $C^*$ action, Math. Ann. 227 (1977) 183

[19] H Pinkham, Deformations of normal surface singularities with $C^*$ action, Math. Ann. 232 (1978) 65

[20] J Stevens, Deformations of singularities, Lecture Notes in Mathematics 1811, Springer (2003)

[21] A Stipsicz, On the $\bar\mu$–invariant of rational surface singularities, Proc. Amer. Math. Soc. 136 (2008) 3815

[22] A Stipsicz, Z Szabó, J Wahl, Rational blowdowns and smoothings of surface singularities, J. Topol. 1 (2008) 477

[23] J Wahl, Equisingular deformations of normal surface singularities. I, Ann. of Math. $(2)$ 104 (1976) 325

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

[25] J A Wolf, Spaces of constant curvature, McGraw-Hill Book Co. (1967)