On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities
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Alessandro Oneto
Abstract
In earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X, which is a generating function for Gromov–Witten invariants of X, coincides with the classical period of its mirror partner f.
In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with
Acknowledgement
This project started during the PRAGMATIC 2013 Research School “Topics in Higher Dimensional Algebraic Geometry” held in Catania, Italy, in September 2013. We thank the organisers Alfio Ragusa, Francesco Russo, and Giuseppe Zappalà. We are very grateful to Alessio Corti for introducing us to this subject and supporting us during the last two years and to Tom Coates for countless invaluable comments and suggestions. We thank Alexander Kasprzyk and Andrew Strangeway for many useful conversations. Our computations rely heavily on the use of the computer algebra software Magma [9]; we thank John Cannon and the Magma team at the University of Sydney for providing licences.
References
[1] D. Abramovich, T. Graber, A. Vistoli, Algebraic orbifold quantum products. In: Orbifolds in mathematics and physics (Madison, WI, 2001), volume 310 of Contemp. Math., 1–24, Amer. Math. Soc. 2002. MR1950940 Zbl 1067.1405510.1090/conm/310/05397Search in Google Scholar
[2] D. Abramovich, T. Graber, A. Vistoli, Gromov-Witten theory of Deligne-Mumford stacks. Amer. J. Math. 130 (2008), 1337–1398. MR2450211 Zbl 1193.1407010.1353/ajm.0.0017Search in Google Scholar
[3] D. Abramovich, A. Vistoli, Compactifying the space of stable maps. J. Amer. Math. Soc. 15 (2002), 27–75. MR1862797 Zbl 0991.1400710.1090/S0894-0347-01-00380-0Search in Google Scholar
[4] M. Akhtar, T. Coates, A. Corti, L. Heuberger, A. Kasprzyk, A. Oneto, A. Petracci, T. Prince, K. Tveiten, Mirror symmetry and the classification of orbifold del Pezzo surfaces. Proc. Amer. Math. Soc. 144 (2016), 513–527. MR3430830 Zbl 0654866510.1090/proc/12876Search in Google Scholar
[5] M. Akhtar, T. Coates, S. Galkin, A. M. Kasprzyk, Minkowski polynomials and mutations. SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012), Paper 094, 17. MR3007265 Zbl 1280.5201410.3842/SIGMA.2012.094Search in Google Scholar
[6] M. Akhtar, A. Kasprzyk, Singularity content. Preprint. arXiv:1401.5458 [math.AG]Search in Google Scholar
[7] A. Bertram, I. Ciocan-Fontanine, B. Kim, Gromov-Witten invariants for abelian and nonabelian quotients. J. Algebraic Geom. 17 (2008), 275–294. MR2369087 Zbl 1166.1403510.1090/S1056-3911-07-00456-0Search in Google Scholar
[8] L. A. Borisov, L. Chen, G. G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks. J. Amer. Math. Soc. 18 (2005), 193–215. MR2114820 Zbl 1178.1405710.1090/S0894-0347-04-00471-0Search in Google Scholar
[9] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), 235–265. MR1484478 Zbl 0898.6803910.1006/jsco.1996.0125Search in Google Scholar
[10] W. Chen, Y. Ruan, A new cohomology theory of orbifold. Comm. Math. Phys. 248 (2004), 1–31. MR2104605 Zbl 1063.5309110.1007/s00220-004-1089-4Search in Google Scholar
[11] I. Ciocan-Fontanine, B. Kim, C. Sabbah, The abelian/nonabelian correspondence and Frobenius manifolds. Invent. Math. 171 (2008), 301–343. MR2367022 Zbl 1164.1401210.1007/s00222-007-0082-xSearch in Google Scholar
[12] T. Coates, The quantum Lefschetz principle for vector bundles as a map between Givental cones. Preprint. arXiv:1405.2893 [math.AG]Search in Google Scholar
[13] T. Coates, A. Corti, S. Galkin, V. Golyshev, A. Kasprzyk, Mirror symmetry and Fano manifolds. In: European Congress of Mathematics, 285–300, Eur. Math. Soc., Zürich 2013. MR3469127 Zbl 1364.1403210.4171/120-1/16Search in Google Scholar
[14] T. Coates, A. Corti, S. Galkin, A. Kasprzyk, Quantum periods for 3-dimensional Fano manifolds. Geom. Topol. 20 (2016), 103–256. MR3470714 Zbl 1348.1410510.2140/gt.2016.20.103Search in Google Scholar
[15] T. Coates, A. Corti, H. Iritani, H.-H. Tseng, Some applications of the mirror theorem for toric stacks. Preprint. arXiv:1401.2611 [math.AG]10.4310/ATMP.2019.v23.n3.a4Search in Google Scholar
[16] T. Coates, A. Corti, H. Iritani, H.-H. Tseng, Computing genus-zero twisted Gromov-Witten invariants. Duke Math. J. 147 (2009), 377–438. MR2510741 Zbl 1176.1400910.1215/00127094-2009-015Search in Google Scholar
[17] T. Coates, A. Corti, H. Iritani, H.-H. Tseng, A mirror theorem for toric stacks. Compos. Math. 151 (2015), 1878–1912. MR3414388 Zbl 1330.1409310.1112/S0010437X15007356Search in Google Scholar
[18] T. Coates, A. Corti, H. Iritani, H.-H. Tseng, Personal communication.Search in Google Scholar
[19] T. Coates, A. Gholampour, H. Iritani, Y. Jiang, P. Johnson, C. Manolache, The quantum Lefschetz hyperplane principle can fail for positive orbifold hypersurfaces. Math. Res. Lett. 19 (2012), 997–1005. MR3039825 Zbl 1287.1402710.4310/MRL.2012.v19.n5.a3Search in Google Scholar
[20] T. Coates, A. Givental, Quantum Riemann-Roch, Lefschetz and Serre. Ann. of Math. (2) 165 (2007), 15–53. MR2276766 Zbl 1189.1406310.4007/annals.2007.165.15Search in Google Scholar
[21] A. Corti, L. Heuberger, Del Pezzo surfaces with 1 (1, 1) points. Manuscripta Math. 153 (2017), 71–118. MR3635974 Zbl 0671454510.1007/s00229-016-0870-ySearch in Google Scholar
[22] A. Corti, M. Reid, Weighted Grassmannians. In: Algebraic geometry, 141–163, De Gruyter 2002. MR1954062 Zbl 1060.1407110.1515/9783110198072.141Search in Google Scholar
[23] B. Fantechi, E. Mann, F. Nironi, Smooth toric Deligne-Mumford stacks. J. Reine Angew. Math. 648 (2010), 201–244. MR2774310 Zbl 1211.1400910.1515/crelle.2010.084Search in Google Scholar
[24] K. Fujita, K. Yasutake, Classification of log del Pezzo surfaces of index three. J. Math. Soc. Japan69 (2017), 163–225. MR3597552 Zbl 0670158810.2969/jmsj/06910163Search in Google Scholar
[25] W. Fulton, Introduction to toric varieties, volume 131 of Annals of Mathematics Studies. Princeton Univ. Press 1993. MR1234037 Zbl 0813.1403910.1515/9781400882526Search in Google Scholar
[26] A. Gholampour, H.-H. Tseng, On computations of genus 0 two-point descendant Gromov–Witten invariants. Michigan Math. J. 62 (2013), 753–768. MR3160540 Zbl 1306.5307710.1307/mmj/1387226163Search in Google Scholar
[27] A. B. Givental, Gromov–Witten invariants and quantization of quadratic Hamiltonians. Mosc. Math. J. 1 (2001), 551–568, 645. MR1901075 Zbl 1008.5307210.17323/1609-4514-2001-1-4-551-568Search in Google Scholar
[28] P. Hacking, Compact moduli of plane curves. Duke Math. J. 124 (2004), 213–257. MR2078368 Zbl 1060.1403410.1215/S0012-7094-04-12421-2Search in Google Scholar
[29] A. R. Iano-Fletcher, Working with weighted complete intersections. In: Explicit birational geometry of 3-folds, volume 281 of London Math. Soc. Lecture Note Ser., 101–173, Cambridge Univ. Press 2000. MR1798982 Zbl 0960.1402710.1017/CBO9780511758942.005Search in Google Scholar
[30] H. Iritani, Quantum cohomology and periods. Ann. Inst. Fourier (Grenoble) 61 (2011), 2909–2958. MR3112512 Zbl 1300.1405510.5802/aif.2798Search in Google Scholar
[31] Y. Jiang, The orbifold cohomology ring of simplicial toric stack bundles. Illinois J. Math. 52 (2008), 493–514. MR2524648 Zbl 1231.1400210.1215/ijm/1248355346Search in Google Scholar
[32] A. Kasprzyk, B. Nill, T. Prince, Minimality and mutation-equivalence of polygons. Forum Math. Sigma5 (2017), e18, 48 pp. MR368676610.1017/fms.2017.10Search in Google Scholar
[33] A. Kasprzyk, K. Tveiten, Maximally mutable Laurent polynomials. In preparation.Search in Google Scholar
[34] B. Kim, Quantum hyperplane section theorem for homogeneous spaces. Acta Math. 183 (1999), 71–99. MR1719555 Zbl 1023.1402810.1007/BF02392947Search in Google Scholar
[35] J. Kollár, Singularities of the minimal model program, volume 200 of Cambridge Tracts in Mathematics. Cambridge Univ. Press 2013. MR3057950 Zbl 1282.1402810.1017/CBO9781139547895Search in Google Scholar
[36] J. Kollár, N. I. Shepherd-Barron, Threefolds and deformations of surface singularities. Invent. Math. 91 (1988), 299–338. MR922803 Zbl 0642.1400810.1007/BF01389370Search in Google Scholar
[37] Y.-P. Lee, Quantum Lefschetz hyperplane theorem. Invent. Math. 145 (2001), 121–149. MR1839288 Zbl 1082.1405610.1007/s002220100145Search in Google Scholar
[38] H.-H. Tseng, Orbifold quantum Riemann-Roch, Lefschetz and Serre. Geom. Topol. 14 (2010), 1–81. MR2578300 Zbl 1178.1405810.2140/gt.2010.14.1Search in Google Scholar
[39] A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math. 97 (1989), 613–670. MR1005008 Zbl 0694.1400110.1007/BF01388892Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Graphs and metric 2-step nilpotent Lie algebras
- Stein manifolds of nonnegative curvature
- Homogeneous spin Riemannian manifolds with the simplest Dirac operator
- On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities
- Harmonicity of vector fields on a class of Lorentzian solvable Lie groups
- Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies
- Criteria for Hattori–Masuda multi-polytopes via Duistermaat–Heckman functions and winding numbers
- On complex Berwald metrics which are not conformal changes of complex Minkowski metrics
- Continuous space-time transformations
Articles in the same Issue
- Frontmatter
- Graphs and metric 2-step nilpotent Lie algebras
- Stein manifolds of nonnegative curvature
- Homogeneous spin Riemannian manifolds with the simplest Dirac operator
- On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities
- Harmonicity of vector fields on a class of Lorentzian solvable Lie groups
- Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies
- Criteria for Hattori–Masuda multi-polytopes via Duistermaat–Heckman functions and winding numbers
- On complex Berwald metrics which are not conformal changes of complex Minkowski metrics
- Continuous space-time transformations
