Chen–Ruan Cohomology of ℳ 1 , n and ℳ ¯ 1 , n

Nicola Pagani[1]

  • [1] Institut fur Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 4, page 1469-1509
  • ISSN: 0373-0956

Abstract

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In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable n -pointed curves of genus 1 . In the first part of the paper we study and describe stack theoretically the twisted sectors of ℳ 1 , n and ℳ ¯ 1 , n . In the second part, we study the orbifold intersection theory of ℳ ¯ 1 , n . We suggest a definition for an orbifold tautological ring in genus 1 , which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.

How to cite

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Pagani, Nicola. "Chen–Ruan Cohomology of $\mathcal{M}_{1,n}$ and $\overline{\mathcal{M}}_{1,n}$." Annales de l’institut Fourier 63.4 (2013): 1469-1509. <http://eudml.org/doc/275455>.

@article{Pagani2013,
abstract = {In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable $n$-pointed curves of genus $1$. In the first part of the paper we study and describe stack theoretically the twisted sectors of $\mathcal\{M\}_\{1,n\}$ and $\overline\{\mathcal\{M\}\}_\{1,n\}$. In the second part, we study the orbifold intersection theory of $\overline\{\mathcal\{M\}\}_\{1,n\}$. We suggest a definition for an orbifold tautological ring in genus $1$, which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.},
affiliation = {Institut fur Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany},
author = {Pagani, Nicola},
journal = {Annales de l’institut Fourier},
keywords = {moduli spaces; Gromov-Witten; orbifold; cohomology; tautological ring},
language = {eng},
number = {4},
pages = {1469-1509},
publisher = {Association des Annales de l’institut Fourier},
title = {Chen–Ruan Cohomology of $\mathcal\{M\}_\{1,n\}$ and $\overline\{\mathcal\{M\}\}_\{1,n\}$},
url = {http://eudml.org/doc/275455},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Pagani, Nicola
TI - Chen–Ruan Cohomology of $\mathcal{M}_{1,n}$ and $\overline{\mathcal{M}}_{1,n}$
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 4
SP - 1469
EP - 1509
AB - In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable $n$-pointed curves of genus $1$. In the first part of the paper we study and describe stack theoretically the twisted sectors of $\mathcal{M}_{1,n}$ and $\overline{\mathcal{M}}_{1,n}$. In the second part, we study the orbifold intersection theory of $\overline{\mathcal{M}}_{1,n}$. We suggest a definition for an orbifold tautological ring in genus $1$, which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.
LA - eng
KW - moduli spaces; Gromov-Witten; orbifold; cohomology; tautological ring
UR - http://eudml.org/doc/275455
ER -

References

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  1. Dan Abramovich, Tom Graber, Angelo Vistoli, Algebraic orbifold quantum products, Orbifolds in mathematics and physics (Madison, WI, 2001) 310 (2002), 1-24, Amer. Math. Soc., Providence, RI Zbl1067.14055MR1950940
  2. Dan Abramovich, Tom Graber, Angelo Vistoli, Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math. 130 (2008), 1337-1398 Zbl1193.14070MR2450211
  3. Alejandro Adem, Johann Leida, Yongbin Ruan, Orbifolds and stringy topology, 171 (2007), Cambridge University Press, Cambridge Zbl1157.57001MR2359514
  4. Pavel Belorousski, Chow Rings of moduli spaces of pointed elliptic curves, (1998) MR2716762
  5. Lawrence Breen, On the classification of 2 -gerbes and 2 -stacks, (1994) Zbl0818.18005MR1301844
  6. Weimin Chen, Yongbin Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004), 1-31 Zbl1063.53091MR2104605
  7. Fred Diamond, Jerry Shurman, A first course in modular forms, 228 (2005), Springer-Verlag, New York Zbl1062.11022MR2112196
  8. Carel Faber, Rahul Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS) 7 (2005), 13-49 Zbl1084.14054MR2120989
  9. Barbara Fantechi, Lothar Göttsche, Orbifold cohomology for global quotients, Duke Math. J. 117 (2003), 197-227 Zbl1086.14046MR1971293
  10. William Fulton, Intersection Theory, (1984), Springer-Verlag, Berlin Zbl0885.14002MR732620
  11. Ezra Getzler, Operads and moduli of genus 0 Riemann surfaces, The moduli space of curves (Texel Island, 1994) 129 (1995), 199-230, Birkhäuser Boston, Boston, MA Zbl0851.18005MR1363058
  12. Ezra Getzler, Intersection theory on M ¯ 1 , 4 and elliptic Gromov-Witten invariants, J. Amer. Math. Soc. 10 (1997), 973-998 Zbl0909.14002MR1451505
  13. Ezra Getzler, The semi-classical approximation for modular operads, Comm. Math. Phys. 194 (1998), 481-492 Zbl0912.18007MR1627677
  14. Jean Giraud, Cohomologie non abélienne (French), 179 (1971), Springer-Verlag, Berlin-New York Zbl0226.14011MR344253
  15. Tom Graber, Rahul Pandharipande, Constructions of nontautological classes on moduli spaces of curves, Michigan Math. J. 51 (2003), 93-109 Zbl1079.14511MR1960923
  16. Tom Graber, Ravi Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, Duke Math. J. 130 (2005), 1-37 Zbl1088.14007MR2176546
  17. Sean Keel, Intersection theory of moduli space of stable n -pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545-574 Zbl0768.14002MR1034665
  18. Étienne Mann, Orbifold quantum cohomology of weighted projective spaces, J. Algebraic Geom. 17 (2008), 137-166 Zbl1146.14029MR2357682
  19. David Mumford, Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry Vol II 36 (1983), 271-328, Birkhäuser Boston, Boston, MA Zbl0554.14008MR717614
  20. Nicola Pagani, Chen–Ruan cohomology of moduli of curves, (2009) Zbl1236.14032
  21. Nicola Pagani, The Chen-Ruan cohomology of moduli of curves of genus 2 with marked points, Adv. Math. 229 (2012), 1643-1687 Zbl1236.14032MR2871153
  22. Nicola Pagani, The orbifold cohomology of moduli of hyperelliptic curves, Int. Math. Res. Not. IMRN (2012), 2163-2178 Zbl1259.14029MR2923163
  23. Nicola Pagani, Orsola Tommasi, The orbifold cohomology of moduli of genus 3 curves Zbl1314.14051
  24. Dan Petersen, The structure of the tautological ring in genus 1  Zbl1291.14045
  25. Joseph H. Silverman, The arithmetic of elliptic curves, 106 (1992), Springer-Verlag, New York Zbl0585.14026MR1329092
  26. James E. Spencer, The stringy Chow ring of the moduli stack of genus-two curves and its Deligne-Mumford compactification, (2004) MR2705012
  27. James E. Spencer, The orbifold cohomology of the moduli of genus-two curves, Gromov-Witten theory of spin curves and orbifolds 403 (2006), 167-184, Amer. Math. Soc., Providence, RI Zbl1115.14018MR2234890
  28. Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), 613-670 Zbl0694.14001MR1005008

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