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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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]B A. Beauville, Sur la cohomologie de certains espaces de modules de fibrés vectoriels, Geometry and analysis, 37–40, Bombay, 1995.
[Be]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.
[Br]Br M. Brion, Equivariant cohomology and equivariant intersection theory, Representation theories and algebraic geometry (Montreal, 1997), 1–37, Kluwer Acad. Publ., Dordrecht, 1998.
[BDW]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.
[BF]BF G. Bini, C. Fontanari, On the cohomology of $\overline {M}_{0,n}(\mathbb P^1, d)$, Commun. Contemp. Math 4 (2002), 751–761.
[BH]BH K. Behrend, A. O’Halloran, On the cohomology of stable map spaces, Invent. Math. 154 (2003), no. 2, 385–450.
[C]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]CF I. Ciocan-Fontanine, The quantum cohomology ring of flag varieties, Transactions of the AMS, 351 (7) (1999), 2695–2729.
[D]D P. Deligne, Théorie de Hodge. II, III, Publ. Math. I.H.E.S, 40 (1972), 44(1974), 5–58 and 5–77. ;
[Dh]Dh A. Dhillon, On the cohomology of moduli of vector bundles, AG/0310299.
[EG]EG D. Edidin, W. Graham, Equivariant intersection theory, Invent. Math. 131 (1998), no. 3, 595–634.
[ES]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.
[FM]FM W. Fulton, R. MacPherson, A compactification of configuration spaces, Ann. of Math., 139 (1994), 183–225.
[FP]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.
[Ge]zero 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.
[Gi]Gi V. Ginzburg, Equivariant cohomology and Kähler geometry, Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 19–34, 96.
[GP]taut T. Graber, R. Pandharipande, Construction of nontautological classes on moduli spaces of curves, Michigan Math. J. 51 (2003), no. 1, 93–109.
[Gr]Gr A. Grothendieck, Quelques proprietes fondamentales en theorie des intersections, Seminaire Chevalley Anneaux de Chow et applications, 1959.
[Gr2]Gr2 A. Grothendieck, Le groupe de Brauer II. Dix exposes sur la cohomologie de schemas, North Holland, 1968.
[K]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.
[Kim]Kim B. Kim, Quot schemes for flags and Gromov invariants for flag varieties, preprint, AG/9512003.
[M]M Y. Manin, Frobenius manifolds, quantum cohomology and moduli spaces, A.M.S Colloquim Publications, vol. 47, 1999.
[MM]MM A. Mustata, A. Mustata, On the Chow ring of $\overline M_{0,m}(n, d)$, preprint, AG/0507464.
[O1]O1 D. Oprea, Tautological classes on the moduli spaces of stable maps to $\mathbb {P}^r$ via torus actions, to appear.
[O2]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).
[Pa1]chow R. Pandharipande, The Chow Ring of the nonlinear Grassmannian, J. Algebraic Geom. 7 (1998), no. 1, 123–140.
[Pa2]divisors 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.
[Pa3]three R. Pandharipande, Three questions in Gromov-Witten theory, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 503–512.
[S]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.
[St]quot 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.
[STi]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.
[Si]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.
[T]T B. Totaro, Chow groups, Chow cohomology, and linear varieties, Journal of Algebraic Geometry, to appear.
[V1]V1 A. Vistoli, Intersection theory on algebraic stacks and their moduli spaces, Invent. Math. 97 (1989), 613–670.
[V2]V2 A. Vistoli, Chow groups of quotient varieties, J. Algebra 107 (1987), 410–424.
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
MR Author ID:
734182
Email:
oprea@alum.mit.edu
Received by editor(s):
January 12, 2005
Received by editor(s) in revised form:
January 4, 2006
Published electronically:
June 20, 2006
