Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



New topological recursion relations

Authors: Xiaobo Liu and Rahul Pandharipande
Journal: J. Algebraic Geom. 20 (2011), 479-494
Published electronically: June 9, 2010
MathSciNet review: 2786663
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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

$\displaystyle \iota_*:A^*( \overline{M}_{g,2}) \rightarrow A^*(\overline{M}_{g+1})$

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.

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Additional Information

Xiaobo Liu
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Rahul Pandharipande
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000

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.