## The Michigan Mathematical Journal

### Gromov–Witten Theory of Target Curves and the Tautological Ring

Felix Janda

#### Abstract

In the Gromov–Witten theory of a target curve, we consider descendent integrals against the virtual fundamental class relative to the forgetful morphism to the moduli space of curves. We show that cohomology classes obtained in this way lie in the tautological ring.

#### Article information

Source
Michigan Math. J., Volume 66, Issue 4 (2017), 683-698.

Dates
Received: 26 April 2016
Revised: 23 July 2017
First available in Project Euclid: 24 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1508810814

Digital Object Identifier
doi:10.1307/mmj/1508810814

Mathematical Reviews number (MathSciNet)
MR3720320

Zentralblatt MATH identifier
06822182

#### Citation

Janda, Felix. Gromov–Witten Theory of Target Curves and the Tautological Ring. Michigan Math. J. 66 (2017), no. 4, 683--698. doi:10.1307/mmj/1508810814. https://projecteuclid.org/euclid.mmj/1508810814

#### References

• [1] K. Behrend and B. Fantechi,The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88.
• [2] F. Carel and R. Pandharipande,Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 13–49.
• [3] W. Fulton and R. Pandharipande,Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., 62, pp. 45–96, Amer. Math. Soc., Providence, RI, 1997.
• [4] T. Graber and R. Pandharipande,Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518.
• [5] E.-N. Ionel and T. H. Parker,Relative Gromov–Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96.
• [6] A.-M. Li and Y. Ruan,Symplectic surgery and Gromov–Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), no. 1, 151–218.
• [7] J. Li,A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293.
• [8] J. Li and G. Tian,Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119–174.
• [9] A. Okounkov and R. Pandharipande,The equivariant Gromov–Witten theory of $\mathbf{P}^{1}$, Ann. of Math. (2) 163 (2006), no. 2, 561–605.
• [10] A. Okounkov and R. Pandharipande,Gromov–Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2) 163 (2006), no. 2, 517–560.
• [11] A. Okounkov and R. Pandharipande,Virasoro constraints for target curves, Invent. Math. 163 (2006), no. 1, 47–108.