Abstract
We study abelian covers of rational surfaces branched over line arrangements. We use these covers to address the geography problem for closed simply connected nonspin irreducible symplectic 4-manifolds with positive signature.
Funding statement: The third author was partially supported by a Discovery Grant from the NSERC of Canada. The research of the fourth author was supported in part by a SERB MATRICS Grant: MTR/2019/001613.
Acknowledgements
The authors thank Krishna Hanumanthu for helpful email correspondence.
Communicated by: P. Eberlein
References
[1] T. Abe, Double points of free projective line arrangements. Int. Math. Res. Not. IMRN no. 3 (2022), 1811–1824. MR4373226 Zbl 1483.1409610.1093/imrn/rnaa145Search in Google Scholar
[2] A. Akhmedov, S. Baldridge, R. I. Baykur, P. Kirk, B. D. Park, Simply connected minimal symplectic 4-manifolds with signature less than −1. J. Eur. Math. Soc. (JEMS) 12 (2010), 133–161. MR2578606 Zbl 1185.5702310.4171/JEMS/192Search in Google Scholar
[3] A. Akhmedov, M. C. Hughes, B. D. Park, Geography of simply connected nonspin symplectic 4-manifolds with positive signature. Pacific J. Math. 261 (2013), 257–282. MR3037567 Zbl 1270.5706710.2140/pjm.2013.261.257Search in Google Scholar
[4] A. Akhmedov, B. D. Park, Exotic smooth structures on small 4-manifolds with odd signatures. Invent. Math. 181 (2010), 577–603. MR2660453 Zbl 1206.5702910.1007/s00222-010-0254-ySearch in Google Scholar
[5] A. Akhmedov, B. D. Park, Geography of simply connected nonspin symplectic 4-manifolds with positive signature. II. Canad. Math. Bull. 64 (2021), 418–428. MR4273209 Zbl 1470.5704110.4153/S0008439520000533Search in Google Scholar
[6] A. Akhmedov, S. Sakallı, On the geography of simply connected nonspin symplectic 4-manifolds with nonnegative signature. Topology Appl. 206 (2016), 24–45. MR3494431 Zbl 1344.5701010.1016/j.topol.2016.03.026Search in Google Scholar
[7] T. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Lundman, P. Pokora, T. Szemberg, Bounded negativity and arrangements of lines. Int. Math. Res. Not. IMRN no. 19 (2015), 9456–9471. MR3431599 Zbl 1330.1400710.1093/imrn/rnu236Search in Google Scholar
[8] R. Fintushel, R. J. Stern, Knots, links, and 4-manifolds. Invent. Math. 134 (1998), 363–400. MR1650308 Zbl 0914.5701510.1007/s002220050268Search in Google Scholar
[9] A. T. Fomenko, S. V. Matveev, Algorithmic and computer methods for three-manifolds, volume 425 of Mathematics and its Applications. Kluwer 1997. MR1486574 Zbl 0885.5700910.1007/978-94-017-0699-5Search in Google Scholar
[10] M. H. Freedman, The topology of four-dimensional manifolds. J. Differential Geometry 17 (1982), 357–453. MR679066 Zbl 0528.5701110.4310/jdg/1214437136Search in Google Scholar
[11] R. E. Gompf, A new construction of symplectic manifolds. Ann. of Math. (2) 142 (1995), 527–595. MR1356781 Zbl 0849.5302710.2307/2118554Search in Google Scholar
[12] M. J. D. Hamilton, D. Kotschick, Minimality and irreducibility of symplectic four-manifolds. Int. Math. Res. Not. (2006), Art. ID 35032, 13. MR2211144 Zbl 1101.5305210.1155/IMRN/2006/35032Search in Google Scholar
[13] K. Hanumanthu, B. Harbourne, Real and complex supersolvable line arrangements in the projective plane. J. Algebraic Combin. 54 (2021), 767–785. MR4330411 Zbl 1475.1410710.1007/s10801-020-00987-8Search in Google Scholar
[14] E. Hironaka, Abelian coverings of the complex projective plane branched along configurations of real lines. Mem. Amer. Math. Soc. 105 (1993), vi+85 pages. MR1164128 Zbl 0788.1405410.1090/memo/0502Search in Google Scholar
[15] F. Hirzebruch, Arrangements of lines and algebraic surfaces. In: Arithmetic and geometry, Vol. II, 113–140, Birkhäuser, Boston, Mass. 1983. MR717609 Zbl 0527.1403310.1007/978-1-4757-9286-7_7Search in Google Scholar
[16] D. Kotschick, The Seiberg-Witten invariants of symplectic four-manifolds (after C. H. Taubes). Séminaire Bourbaki, Vol. 1995/96. Astérisque 241 (1997), 195–220, Exp. No. 812. MR1472540 Zbl 0882.57026Search in Google Scholar
[17] V. S. Kulikov, Old examples and a new example of surfaces of general type with pg = 0. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), 123–170. English translation: Izv. Math. 68 (2004), no. 5, 965–1008 MR2104852 Zbl 1073.1405510.1070/IM2004v068n05ABEH000505Search in Google Scholar
[18] J. D. McCarthy, J. G. Wolfson, Symplectic normal connect sum. Topology 33 (1994), 729–764. MR1293308 Zbl 0812.5303310.1016/0040-9383(94)90006-XSearch in Google Scholar
[19] M. V. Nori, Zariski’s conjecture and related problems. Ann. Sci. École Norm. Sup. (4) 16 (1983), 305–344. MR732347 Zbl 0527.1401610.24033/asens.1450Search in Google Scholar
[20] M. Usher, Minimality and symplectic sums. Int. Math. Res. Not. (2006), Art. ID 49857, 17 pages. MR2250015 Zbl 1110.57017Search in Google Scholar
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