Acta Mathematica

The quantum orbifold cohomology of weighted projective spaces

Tom Coates, Alessio Corti, Yuan-Pin Lee, and Hsian-Hua Tseng

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Abstract

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S1-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.

Article information

Source
Acta Math., Volume 202, Number 2 (2009), 139-193.

Dates
Received: 26 October 2006
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892396

Digital Object Identifier
doi:10.1007/s11511-009-0035-x

Mathematical Reviews number (MathSciNet)
MR2506749

Zentralblatt MATH identifier
1213.53106

Rights
2009 © Institut Mittag-Leffler

Citation

Coates, Tom; Corti, Alessio; Lee, Yuan-Pin; Tseng, Hsian-Hua. The quantum orbifold cohomology of weighted projective spaces. Acta Math. 202 (2009), no. 2, 139--193. doi:10.1007/s11511-009-0035-x. https://projecteuclid.org/euclid.acta/1485892396


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References

  • A bramovich, D., C orti, A. & V istoli, A., Twisted bundles and admissible covers. Comm. Algebra, 31:8 (2003), 3547–3618.
  • A bramovich, D., G raber, T., O lsson, M. & T seng, H.-H., On the global quotient structure of the space of twisted stable maps to a quotient stack. J. Algebraic Geom., 16 (2007), 731–751.
  • A bramovich, D., G raber, T. & V istoli, A., Algebraic orbifold quantum products, in Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemp. Math., 310, pp. 1–24. Amer. Math. Soc., Providence, RI, 2002.
  • — Gromov–Witten theory of Deligne–Mumford stacks. Amer. J. Math., 130:5 (2008), 1337–1398.
  • A rkhipov, S. & K apranov, M., Toric arc schemes and quantum cohomology of toric varieties. Math. Ann., 335 (2006), 953–964.
  • A tiyah, M. F. & B ott, R., The moment map and equivariant cohomology. Topology, 23 (1984), 1–28.
  • A ustin, D.M. & B raam, P. J., Morse–Bott theory and equivariant cohomology, in The Floer Memorial Volume, Progr. Math., 133, pp. 123–183. Birkhäuser, Basel, 1995.
  • B ehrend, K. & F antechi, B., The intrinsic normal cone. Invent. Math., 128 (1997), 45–88.
  • B ertram, A., Another way to enumerate rational curves with torus actions. Invent. Math., 142 (2000), 487–512.
  • B oissiere, S., M ann, É. & P erroni, F., Crepant resolutions of weighted projective spaces and quantum deformations. Preprint, 2007.
  • B orisov, L.A., C hen, L. & S mith, G.G., The orbifold Chow ring of toric Deligne–Mumford stacks. J. Amer. Math. Soc., 18 (2005), 193–215.
  • C hen, B. & H u, S., A deRham model for Chen–Ruan cohomology ring of abelian orbifolds. Math. Ann., 336 (2006), 51–71.
  • C hen, W. & R uan, Y., Orbifold Gromov–Witten theory, in Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemp. Math., 310, pp. 25–85. Amer. Math. Soc., Providence, RI, 2002.
  • — A new cohomology theory of orbifold. Comm. Math. Phys., 248 (2004), 1–31.
  • C oates, T., C orti, A., I ritani, H. & T seng, H.-H., Computing genus-zero twisted Gromov–Witten invariants. Duke Math. J., 147 (2009), 377–438.
  • C oates, T., I ritani, H. & T seng, H.-H., Wall-crossings in toric Gromov–Witten theory I: Crepant examples. Preprint, 2008.
  • C ohen, R. L., J ones, J. D. S. & S egal, G. B., Floer’s infinite-dimensional Morse theory and homotopy theory, in The Floer Memorial Volume, Progr. Math., 133, pp. 297–325. Birkhäuser, Basel, 1995.
  • C orti, A. & G olyshev, V., Hypergeometric equations and weighted projective spaces. Preprint, 2006.
  • G ivental, A.B., Homological geometry. I. Projective hypersurfaces. Selecta Math., 1 (1995), 325–345.
  • — Homological geometry and mirror symmetry, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pp. 472–480. Birkhäuser, Basel, 1995.
  • — Equivariant Gromov–Witten invariants. Int. Math. Res. Notices, 13 (1996), 613–663.
  • G oldin, R., H olm, T. S. & K nutson, A., Orbifold cohomology of torus quotients. Duke Math. J., 139 (2007), 89–139.
  • G raber, T. & P andharipande, R., Localization of virtual classes. Invent. Math., 135 (1999), 487–518.
  • G rothendieck, A., Catégories cofibrées additives et complexe cotangent relatif. Lecture Notes in Mathematics, 79. Springer, Berlin–Heidelberg, 1968.
  • H ofer, H. & S alamon, D. A., Floer homology and Novikov rings, in The Floer Memorial Volume, Progr. Math., 133, pp. 483–524. Birkhäuser, Basel, 1995.
  • I ano-F letcher, A. R., Working with weighted complete intersections, in Explicit Birational Geometry of 3-folds, London Math. Soc. Lecture Note Ser., 281, pp. 101–173. Cambridge Univ. Press, Cambridge, 2000.
  • I llusie, L., Complexe cotangent et déformations. I. Lecture Notes in Mathematics, 239. Springer, Berlin–Heidelberg, 1971.
  • Complexe cotangent et déformations. II. Lecture Notes in Mathematics, 283. Springer, Berlin–Heidelberg, 1972.
  • I ritani, H., Quantum D-modules and equivariant Floer theory for free loop spaces. Math. Z., 252 (2006), 577–622.
  • J iang, Y., The Chen–Ruan cohomology of weighted projective spaces. Canad. J. Math., 59 (2007), 981–1007.
  • K apranov, M. & V asserot, E., Formal loops. III. Additive functions and the Radon transform. Adv. Math., 219:6 (2008), 1852–1871.
  • K im, B., K resch, A. & P antev, T., Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee. J. Pure Appl. Algebra, 179 (2003), 127–136.
  • K ontsevich, M., Enumeration of rational curves via torus actions, in The Moduli Space of Curves (Texel Island, 1994), Progr. Math., 129, pp. 335–368. Birkhäuser, Boston, MA, 1995.
  • L i, J. & T ian, G., Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties. J. Amer. Math. Soc., 11 (1998), 119–174.
  • L ian, B. H., L iu, K. & Y au, S.-T., Mirror principle. I. Asian J. Math., 1 (1997), 729–763.
  • L upercio, E. & U ribe, B., Loop groupoids, gerbes, and twisted sectors on orbifolds, in Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemp. Math., 310, pp. 163–184. Amer. Math. Soc., Providence, RI, 2002.
  • M anin, Y. I., Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces. American Mathematical Society Colloquium Publications, 47. Amer. Math. Soc., Providence, RI, 1999.
  • M ann, É., Cohomologie quantique orbifolde des espaces projectifs à poids. Ph.D. Thesis, IRMA, Strasbourg, 2005.
  • M oerdijk, I., Orbifolds as groupoids: an introduction, in Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemp. Math., 310, pp. 205–222. Amer. Math. Soc., Providence, RI, 2002.
  • O kounkov, A. & P andharipande, R., Gromov–Witten theory, Hurwitz numbers, and matrix models, I. Preprint, 2001.
  • O lsson, M. C., (Log) twisted curves. Compos. Math., 143 (2007), 476–494.
  • P andharipande, R., Rational curves on hypersurfaces (after A. Givental), in Séminaire Bourbaki, Vol. 1997/98, Exp. No. 848. Astérisque, 252 (1998), 5, 307–340.
  • R eid, M., Young person’s guide to canonical singularities, in Algebraic Geometry (Bowdoin College, Brunswick, ME, 1985), Proc. Sympos. Pure Math., 46, pp. 345–414. Amer. Math. Soc., Providence, RI, 1987.
  • R omagny, M., Group actions on stacks and applications. Michigan Math. J., 53 (2005), 209–236.
  • T seng, H.-H., Orbifold quantum Riemann–Roch, Lefschetz and Serre. Preprint, 2005.
  • V lassopoulos, Y., Quantum cohomology and Morse theory on the loop space of toric varieties. Preprint, 2002.