Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

The tautological rings of the moduli spaces of stable maps to flag varieties


Author: Dragos Oprea
Journal: J. Algebraic Geom. 15 (2006), 623-655
DOI: https://doi.org/10.1090/S1056-3911-06-00452-8
Published electronically: June 20, 2006
MathSciNet review: 2237264
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Abstract | References | Additional Information

Abstract: We show that the rational cohomology classes on the moduli spaces of genus zero stable maps to $ SL$ flag varieties are tautological.


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  • [B] A. Beauville, Sur la cohomologie de certains espaces de modules de fibrés vectoriels, Geometry and analysis, 37-40, Bombay, 1995. MR 1351502 (96f:14011)
  • [Be] K. Behrend, Cohomology of stacks, Intersection theory and moduli, 249-294 (electronic) ICTP Lect. Notes, XIX, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004. MR 2172499
  • [Br] M. Brion, Equivariant cohomology and equivariant intersection theory, Representation theories and algebraic geometry (Montreal, 1997), 1-37, Kluwer Acad. Publ., Dordrecht, 1998. MR 1649623 (99m:14005)
  • [BDW] A. Bertram, G. Daskalopoulos, R. Wentworth, Gromov Invariants for Holomorphic Maps from Riemann Surfaces to Grassmannians, J. Amer. Math. Soc. 9 (1996), no. 2, 529-571. MR 1320154 (96f:14066)
  • [BF] G. Bini, C. Fontanari, On the cohomology of $ \overline{M}_{0,n}(\mathbb{P}^1, d)$, Commun. Contemp. Math 4 (2002), 751-761. MR 1938492 (2003h:14044)
  • [BH] K. Behrend, A. O'Halloran, On the cohomology of stable map spaces, Invent. Math. 154 (2003), no. 2, 385-450. MR 2013785 (2004k:14002)
  • [C] J. Cox, A presentation for the Chow ring $ A^*(\bar{M}_{0,2}(\mathbb{P}^1,2))$, Ph.D. Thesis, 2005, math.AG/0505112.
  • [CF] I. Ciocan-Fontanine, The quantum cohomology ring of flag varieties, Transactions of the AMS, 351 (7) (1999), 2695-2729. MR 1487610 (2000b:14072)
  • [D] P. Deligne, Théorie de Hodge. II, III, Publ. Math. I.H.E.S, 40 (1972), 44(1974), 5-58 and 5-77. MR 0498551 (58:16653a); MR 0498552 (58:16653b)
  • [Dh] A. Dhillon, On the cohomology of moduli of vector bundles, AG/0310299.
  • [EG] D. Edidin, W. Graham, Equivariant intersection theory, Invent. Math. 131 (1998), no. 3, 595-634. MR 1614555 (99j:14003a)
  • [ES] G. Ellingsrud, S. A. Stromme, Towards the Chow ring of the Hilbert scheme of $ \mathbb{P}^2$, J. Reine Angew. Math. 441 (1993), 33-44. MR 1228610 (94i:14004)
  • [FM] W. Fulton, R. MacPherson, A compactification of configuration spaces, Ann. of Math., 139 (1994), 183-225. MR 1259368 (95j:14002)
  • [FP] W. Fulton, R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry--Santa Cruz 1995, 45-96, Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, 1997. MR 1492534 (98m:14025)
  • [Ge] E. Getzler, Operads and moduli spaces of genus 0 Riemann surfaces, The moduli space of curves, 199-230, Progr. Math., 129, Birkhäuser Boston, Boston, MA, 1995. MR 1363058 (96k:18008)
  • [Gi] V. Ginzburg, Equivariant cohomology and Kähler geometry, Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 19-34, 96. MR 0925070 (89b:58013)
  • [GP] T. Graber, R. Pandharipande, Construction of nontautological classes on moduli spaces of curves, Michigan Math. J. 51 (2003), no. 1, 93-109. MR 1960923 (2004e:14043)
  • [Gr] A. Grothendieck, Quelques proprietes fondamentales en theorie des intersections, Seminaire Chevalley Anneaux de Chow et applications, 1959.
  • [Gr2] A. Grothendieck, Le groupe de Brauer II. Dix exposes sur la cohomologie de schemas, North Holland, 1968. MR 0244270 (39:5586b)
  • [K] S. Keel, Intersection theory on the moduli space of stable $ n$-pointed curves of genus zero, Trans. Amer. Math. Soc, 330 (1992), no. 2, 545-574. MR 1034665 (92f:14003)
  • [Kim] B. Kim, Quot schemes for flags and Gromov invariants for flag varieties, preprint, AG/9512003.
  • [M] Y. Manin, Frobenius manifolds, quantum cohomology and moduli spaces, A.M.S Colloquim Publications, vol. 47, 1999. MR 1702284 (2001g:53156)
  • [MM] A. Mustata, A. Mustata, On the Chow ring of $ \overline M_{0,m}(n, d)$, preprint, AG/0507464.
  • [O1] D. Oprea, Tautological classes on the moduli spaces of stable maps to $ \mathbb{P}^r$ via torus actions, to appear.
  • [O2] D. Oprea, Divisors on the moduli spaces of stable maps to flag varieties and reconstruction, Journal für die reine und angewandte Mathematik, 586 (2005). MR 2180604
  • [Pa1] R. Pandharipande, The Chow Ring of the nonlinear Grassmannian, J. Algebraic Geom. 7 (1998), no. 1, 123-140. MR 1620694 (99f:14005)
  • [Pa2] R. Pandharipande, Intersection of Q-divisors on Kontsevich's Moduli Space $ {\overline M}_{0,n}(\mathbb{P}^r,d)$ and enumerative geometry, Trans. Amer. Math. Soc. 351 (1999), no. 4, 1481-1505. MR 1407707 (99f:14068)
  • [Pa3] R. Pandharipande, Three questions in Gromov-Witten theory, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 503-512. MR 1957060 (2003k:14069)
  • [S] J. Steenbrink, Mixed Hodge Structure on the Vanishing Cohomology, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Olso, 1976), 525-563. MR 0485870 (58:5670)
  • [St] S. Stromme, On parametrized rational curves in Grassmann varieties, Space curves (Rocca di Papa, 1985), 251-272, Lecture Notes in Math, 1266, Berlin-New York, 1987. MR 0908717 (88i:14020)
  • [STi] B. Siebert, G. Tian, On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997), no. 4, 679-695. MR 1621570 (99d:14060)
  • [Si] B. Siebert, An update on the (small) quantum cohomology, Proceedings of the conference on Geometry and Physics (D.H. Phong, L. Vinet, S.T. Yau eds.), Montreal 1995, International Press 1998. MR 1673112 (2000d:14060)
  • [T] B. Totaro, Chow groups, Chow cohomology, and linear varieties, Journal of Algebraic Geometry, to appear.
  • [V1] A. Vistoli, Intersection theory on algebraic stacks and their moduli spaces, Invent. Math. 97 (1989), 613-670. MR 1005008 (90k:14004)
  • [V2] A. Vistoli, Chow groups of quotient varieties, J. Algebra 107 (1987), 410-424. MR 0885804 (89b:14012)


Additional Information

Dragos Oprea
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, Stanford University, 450 Sera Mall, Stanford, California 94305
Email: oprea@alum.mit.edu

DOI: https://doi.org/10.1090/S1056-3911-06-00452-8
Received by editor(s): January 12, 2005
Received by editor(s) in revised form: January 4, 2006
Published electronically: June 20, 2006

American Mathematical Society