## 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

## Bibliographic Information

**R. Pandharipande**- Affiliation: Departement Mathematik, ETH Zürich, Zürich, Switzerland
- MR Author ID: 357813
- Email: rahul@math.ethz.ch
**A. Pixton**- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Email: apixton@mit.edu
- Received by editor(s): August 25, 2014
- Received by editor(s) in revised form: January 23, 2016, and February 7, 2016
- Published electronically: March 17, 2016
- Additional Notes: The first author was partially supported by NSF Grant DMS-1001154, SNF Grant 200021-143274, SNF Grant 200020-162928, SwissMAP, and ERC Grant AdG-320368-MCSK

The second author was supported by a NDSEG graduate fellowship. - © Copyright 2016 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**30**(2017), 389-449 - MSC (2010): Primary 14N35; Secondary 14H60
- DOI: https://doi.org/10.1090/jams/858
- MathSciNet review: 3600040