New topological recursion relations
Authors:
Xiaobo Liu and Rahul Pandharipande
Journal:
J. Algebraic Geom. 20 (2011), 479-494
DOI:
https://doi.org/10.1090/S1056-3911-2010-00559-0
Published electronically:
June 9, 2010
MathSciNet review:
2786663
Full-text PDF
Abstract |
References |
Additional Information
Abstract: Simple boundary expressions for the $k^{th}$ power of the cotangent line class $\psi _1$ on $\overline {M}_{g,1}$ are found for $k\geq 2g$. The method is by virtual localization on the moduli space of maps to $\mathbb {P}^1$. As a consequence, nontrivial tautological classes in the kernel of the boundary push-forward map \[ \iota _*:A^*( \overline {M}_{g,2}) \rightarrow A^*(\overline {M}_{g+1})\] are constructed. The geometry of genus $g+1$ curves, then provides universal equations in genus $g$ Gromov-Witten theory. As an application, we prove all the Gromov-Witten identities conjectured recently by K. Liu and H. Xu.
References
- D. Arcara and F. Sato, Recursive formula for $\psi ^g-\lambda _1\psi ^{g-1}+\dots +(-1)^g\lambda _g$ in $\overline {\scr M}_{g,1}$, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4077–4081. MR 2538568, DOI https://doi.org/10.1090/S0002-9939-09-10018-7
- Pavel Belorousski and Rahul Pandharipande, A descendent relation in genus 2, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 1, 171–191. MR 1765541
- C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173–199. MR 1728879, DOI https://doi.org/10.1007/s002229900028
- C. Faber and R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring, Michigan Math. J. 48 (2000), 215–252. With an appendix by Don Zagier; Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786488, DOI https://doi.org/10.1307/mmj/1030132716
- C. Faber and R. Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 13–49. MR 2120989, DOI https://doi.org/10.4171/JEMS/20
- B. Fantechi and R. Pandharipande, Stable maps and branch divisors, Compositio Math. 130 (2002), no. 3, 345–364. MR 1887119, DOI https://doi.org/10.1023/A%3A1014347115536
- E. Getzler, Intersection theory on $\overline {\scr M}_{1,4}$ and elliptic Gromov-Witten invariants, J. Amer. Math. Soc. 10 (1997), no. 4, 973–998. MR 1451505, DOI https://doi.org/10.1090/S0894-0347-97-00246-4
- E. Getzler, Topological recursion relations in genus $2$, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 73–106. MR 1672112
- T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. MR 1666787, DOI https://doi.org/10.1007/s002220050293
- Eleny-Nicoleta Ionel, Topological recursive relations in $H^{2g}(\scr M_{g,n})$, Invent. Math. 148 (2002), no. 3, 627–658. MR 1908062, DOI https://doi.org/10.1007/s002220100205
- Sean Keel, Intersection theory of moduli space of stable $n$-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574. MR 1034665, DOI https://doi.org/10.1090/S0002-9947-1992-1034665-0
- Takashi Kimura and Xiaobo Liu, A genus-3 topological recursion relation, Comm. Math. Phys. 262 (2006), no. 3, 645–661. MR 2202306, DOI https://doi.org/10.1007/s00220-005-1481-8
- Kefeng Liu and Hao Xu, A proof of the Faber intersection number conjecture, J. Differential Geom. 83 (2009), no. 2, 313–335. MR 2577471
- K. Liu and H. Xu, The n-point functions for intersection numbers on moduli spaces of curves, math.AG/0701319.
- Xiaobo Liu, Quantum product on the big phase space and the Virasoro conjecture, Adv. Math. 169 (2002), no. 2, 313–375. MR 1926225, DOI https://doi.org/10.1006/aima.2001.2062
- X. Liu, On certain vanishing identities for Gromov-Witten invariants, arXiv:0805. 0800.
- D. Maulik, Gromov-Witten theory of $A_n$-resolutions, arXiv:0802.2681.
References
- D. Arcara and F. Sato, Recursive formula for $\psi ^g-\lambda _1\psi ^{g-1} + \cdots + (-1)^g\lambda _g$ in $\overline {M}_{g,1}$, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4077–4081. MR 2538568
- P. Belorousski and R. Pandharipande, A descendent relation in genus 2, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), 171–191. MR 1765541 (2001g:14044)
- C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), 173–199. MR 1728879 (2000m:14057)
- C. Faber and R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring. With an appendix by D. Zagier. Michigan Math. J. 48 (2000), 215–252. MR 1786488 (2002e:14041)
- C. Faber and R. Pandharipande, Relative maps and tautological classes, JEMS 7 (2005), 13–49. MR 2120989 (2005m:14046)
- B. Fantechi and R. Pandharipande, Stable maps and branch divisors, Compositio Math. 130 (2002), 345–364. MR 1887119 (2003a:14039)
- E. Getzler, Intersection theory on $\overline {M}_{1,4}$ and elliptic Gromov-Witten invariants, JAMS 10 (1997), 973–998. MR 1451505 (98f:14018)
- E. Getzler, Topological recursion relations in genus 2, in Integrable systems and algebraic geometry (Kobe/Kyoto 1997), World Scientific Publishing: River Edge, NJ 1998, 73–106. MR 1672112 (2000b:14028)
- T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), 487–518. MR 1666787 (2000h:14005)
- E. Ionel, Topological recursive relations in $H^{2g}(M_{g,n})$, Invent. Math. 148 (2002), 627–658. MR 1908062 (2003d:14065)
- S. Keel, Intersection theory of moduli space of $n$-pointed curves of genus 0, Trans. Amer. Math. Soc. 330 (1992), 545–574. MR 1034665 (92f:14003)
- T. Kimura and X. Liu, A genus 3 topological recursion relation, Comm. Math. Phys. 262 (2006), 645–661. MR 2202306 (2006i:14060)
- K. Liu and H. Xu, A proof of the Faber intersection number conjecture, arXiv: 0803.2204. MR 2577471
- K. Liu and H. Xu, The n-point functions for intersection numbers on moduli spaces of curves, math.AG/0701319.
- X. Liu, Quantum product on the big phase space and Virasoro conjecture, Advances in Mathematics 169 (2002), 313–375. MR 1926225 (2003j:14075)
- X. Liu, On certain vanishing identities for Gromov-Witten invariants, arXiv:0805. 0800.
- D. Maulik, Gromov-Witten theory of $A_n$-resolutions, arXiv:0802.2681.
Additional Information
Xiaobo Liu
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email:
xliu3@nd.edu
Rahul Pandharipande
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000
MR Author ID:
357813
Email:
rahulp@math.princeton.edu
Received by editor(s):
September 10, 2008
Received by editor(s) in revised form:
February 24, 2010
Published electronically:
June 9, 2010
Additional Notes:
The first author was partially supported by NSF grants DMS-0505835 and DMS-0905227. The second author was partially supported by NSF grant DMS-0500187.
