Gromov-Witten/Pairs correspondence for the quintic 3-fold

Pandharipande, R.; Pixton, A.

doi:10.1090/jams/858

Gromov-Witten/Pairs correspondence for the quintic 3-fold
HTML articles powered by AMS MathViewer

by R. Pandharipande and A. Pixton;

J. Amer. Math. Soc. 30 (2017), 389-449

DOI: https://doi.org/10.1090/jams/858

Published electronically: March 17, 2016

PDF | Request permission

Abstract:

We use the Gromov-Witten/Pairs (GW/P) descendent correspondence for toric 3-folds and degeneration arguments to establish the GW/P correspondence for several compact Calabi-Yau (CY) 3-folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of the GW/P correspondence for descendents in relative geometries. Projective bundles over surfaces relative to a section play a special role. The GW/P correspondence for Calabi-Yau complete intersections provides a structure result for the Gromov-Witten invariants in a fixed curve class. After a change of variables, the Gromov-Witten series is a rational function in the variable $-q=e^{iu}$ invariant under $q \leftrightarrow q^{-1}$.

References

Mina Aganagic, Albrecht Klemm, Marcos Mariño, and Cumrun Vafa, The topological vertex, Comm. Math. Phys. 254 (2005), no. 2, 425–478. MR 2117633, DOI 10.1007/s00220-004-1162-z
K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617. MR 1431140, DOI 10.1007/s002220050132
K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88. MR 1437495, DOI 10.1007/s002220050136
Tom Bridgeland, Hall algebras and curve-counting invariants, J. Amer. Math. Soc. 24 (2011), no. 4, 969–998. MR 2813335, DOI 10.1090/S0894-0347-2011-00701-7
J. Bryan and R. Pandharipande, The local Gromov-Witten theory of curves, J. Amer. Math. Soc. 21 (2008), 101–136., DOI 10.1090/S0894-0347-06-00545-5
R. Gopakumar and C. Vafa, M-theory and topological strings I and II, available at arXiv:hep-th/9809187, arXiv:hep-th/9812127.
T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. MR 1666787, DOI 10.1007/s002220050293
Mark Gross and Bernd Siebert, Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc. 26 (2013), no. 2, 451–510. MR 3011419, DOI 10.1090/S0894-0347-2012-00757-7
Eleny-Nicoleta Ionel and Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96. MR 1954264, DOI 10.4007/annals.2003.157.45
E. Ionel and T. Parker, The Gopakumar-Vafa formula for symplectic manifolds, available at arXiv:1306.1516., DOI 10.4007/annals.2018.187.1.1
A. Klemm and M. Mariño, Counting BPS states on the Enriques Calabi-Yau, available at arXiv:hep-th/0512227., DOI 10.1007/s00220-007-0407-z
An-Min Li and Yongbin Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), no. 1, 151–218. MR 1839289, DOI 10.1007/s002220100146
Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR 1938113
Jun Li and Gang Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119–174. MR 1467172, DOI 10.1090/S0894-0347-98-00250-1
Jun Li and Baosen Wu, Good degeneration of Quot-schemes and coherent systems, Comm. Anal. Geom. 23 (2015), no. 4, 841–921. MR 3385781, DOI 10.4310/CAG.2015.v23.n4.a5
Davesh Maulik, Gromov-Witten theory of $\scr A_n$-resolutions, Geom. Topol. 13 (2009), no. 3, 1729–1773. MR 2496055, DOI 10.2140/gt.2009.13.1729
D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. I, Compos. Math. 142 (2006), no. 5, 1263–1285. MR 2264664, DOI 10.1112/S0010437X06002302
D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. II, Compos. Math. 142 (2006), no. 5, 1286–1304. MR 2264665, DOI 10.1112/S0010437X06002314
Davesh Maulik and Alexei Oblomkov, Quantum cohomology of the Hilbert scheme of points on $\scr A_n$-resolutions, J. Amer. Math. Soc. 22 (2009), no. 4, 1055–1091. MR 2525779, DOI 10.1090/S0894-0347-09-00632-8
Davesh Maulik and Alexei Oblomkov, Donaldson-Thomas theory of ${\scr A}_n\times P^1$, Compos. Math. 145 (2009), no. 5, 1249–1276. MR 2551996, DOI 10.1112/S0010437X09003972
D. Maulik, A. Oblomkov, A. Okounkov, and R. Pandharipande, Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds, Invent. Math. 186 (2011), no. 2, 435–479. MR 2845622, DOI 10.1007/s00222-011-0322-y
D. Maulik and R. Pandharipande, A topological view of Gromov-Witten theory, Topology 45 (2006), no. 5, 887–918. MR 2248516, DOI 10.1016/j.top.2006.06.002
D. Maulik and R. Pandharipande, New calculations in Gromov-Witten theory, Pure Appl. Math. Q. 4 (2008), no. 2, Special Issue: In honor of Fedor Bogomolov., 469–500. MR 2400883, DOI 10.4310/PAMQ.2008.v4.n2.a7
D. Maulik, R. Pandharipande, and R. P. Thomas, Curves on $K3$ surfaces and modular forms, J. Topol. 3 (2010), no. 4, 937–996. With an appendix by A. Pixton. MR 2746343, DOI 10.1112/jtopol/jtq030
A. Oblomkov, A. Okounkov, and R. Pandharipande. In preparation.
A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2) 163 (2006), no. 2, 517–560. MR 2199225, DOI 10.4007/annals.2006.163.517
A. Okounkov and R. Pandharipande, Virasoro constraints for target curves, Invent. Math. 163 (2006), no. 1, 47–108. MR 2208418, DOI 10.1007/s00222-005-0455-y
A. Okounkov and R. Pandharipande, Quantum cohomology of the Hilbert scheme of points in the plane, Invent. Math. 179 (2010), no. 3, 523–557. MR 2587340, DOI 10.1007/s00222-009-0223-5
A. Okounkov and R. Pandharipande, The local Donaldson-Thomas theory of curves, Geom. Topol. 14 (2010), no. 3, 1503–1567. MR 2679579, DOI 10.2140/gt.2010.14.1503
R. Pandharipande and A. Pixton, Descendents on local curves: rationality, Compos. Math. 149 (2013), no. 1, 81–124. MR 3011879, DOI 10.1112/S0010437X12000498
Rahul Pandharipande and Aaron Pixton, Descendent theory for stable pairs on toric 3-folds, J. Math. Soc. Japan 65 (2013), no. 4, 1337–1372. MR 3127827, DOI 10.2969/jmsj/06541337
R. Pandharipande and A. Pixton, Descendents on local curves: stationary theory, Geometry and arithmetic, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2012, pp. 283–307. MR 2987666, DOI 10.4171/119-1/17
Rahul Pandharipande and Aaron Pixton, Gromov-Witten/pairs descendent correspondence for toric 3-folds, Geom. Topol. 18 (2014), no. 5, 2747–2821. MR 3285224, DOI 10.2140/gt.2014.18.2747
R. Pandharipande and R. P. Thomas, Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009), no. 2, 407–447. MR 2545686, DOI 10.1007/s00222-009-0203-9
Rahul Pandharipande and Richard P. Thomas, The 3-fold vertex via stable pairs, Geom. Topol. 13 (2009), no. 4, 1835–1876. MR 2497313, DOI 10.2140/gt.2009.13.1835
R. Pandharipande and R. P. Thomas, Stable pairs and BPS invariants, J. Amer. Math. Soc. 23 (2010), no. 1, 267–297. MR 2552254, DOI 10.1090/S0894-0347-09-00646-8
R. Pandharipande and R. P. Thomas, The Katz-Klemm-Vafa conjecture for $K3$ surfaces, available at arXiv:1407.3181.
R. Pandharipande and R. P. Thomas, 13/2 ways of counting curves, Moduli spaces, London Math. Soc. Lecture Note Ser., vol. 411, Cambridge Univ. Press, Cambridge, 2014, pp. 282–333. MR 3221298
Iman Setayesh, Relative Hilbert scheme of points, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–Princeton University. MR 2912104
Yukinobu Toda, Curve counting theories via stable objects I. DT/PT correspondence, J. Amer. Math. Soc. 23 (2010), no. 4, 1119–1157. MR 2669709, DOI 10.1090/S0894-0347-10-00670-3