# zbMATH — the first resource for mathematics

The quantum cohomology ring of flag varieties. (English) Zbl 0920.14027
Summary: We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work [I. Ciocan-Fontanine, Int. Math. Res. Not. 1995, No. 6, 263-277 (1995; Zbl 0847.14011)]. We also give a geometric proof of the quantum Monk formula.

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 14F99 (Co)homology theory in algebraic geometry 81T10 Model quantum field theories
Full Text:
##### References:
 [1] V.V. Batyrev, Quantum cohomology rings of toric varieties, Astérisque 218 (1991), 9-34. · Zbl 0806.14041 [2] K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127(3) (1997), 601-617. CMP 97:07 · Zbl 0909.14007 [3] K. Behrend and Y. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. Journal 85 (1996), 1-60. CMP 97:02 · Zbl 0872.14019 [4] I. N. Bernšteĭn, I. M. Gel$$^{\prime}$$fand, and S. I. Gel$$^{\prime}$$fand, Schubert cells, and the cohomology of the spaces \?/\?, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3 – 26 (Russian). [5] Aaron Bertram, Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian, Internat. J. Math. 5 (1994), no. 6, 811 – 825. · Zbl 0823.14038 [6] -, Quantum Schubert calculus, Adv. Math. (to appear). CMP 97:14 · Zbl 0945.14031 [7] Aaron Bertram, Georgios Daskalopoulos, and Richard Wentworth, Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians, J. Amer. Math. Soc. 9 (1996), no. 2, 529 – 571. · Zbl 0865.14017 [8] A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des groupes de Lie compacte, Ann. of Math. (2) 57 (1953), 115-207. · Zbl 0052.40001 [9] Ionuţ Ciocan-Fontanine, Quantum cohomology of flag varieties, Internat. Math. Res. Notices 6 (1995), 263 – 277. · Zbl 0847.14011 [10] -, The quantum cohomology ring of flag varieties, University of Utah Ph.D.Thesis (1996). CMP 98:06 [11] Bruce Crauder and Rick Miranda, Quantum cohomology of rational surfaces, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 33 – 80. · Zbl 0843.14014 [12] Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53 – 88 (French). Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I. · Zbl 0312.14009 [13] C. Ehresmann, Sur la topologie des certaines espaces homogènes, Ann. of Math. 35 (1934), 396-443. · JFM 60.1223.05 [14] Sergey Fomin, Sergei Gelfand, and Alexander Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), no. 3, 565 – 596. · Zbl 0912.14018 [15] William Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), no. 3, 381 – 420. · Zbl 0788.14044 [16] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in Proceedings of the 1995 AMS Summer Institute in Santa Cruz (to appear). CMP 98:07 · Zbl 0898.14018 [17] Alexander B. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 13 (1996), 613 – 663. · Zbl 0881.55006 [18] Alexander Givental and Bumsig Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), no. 3, 609 – 641. · Zbl 0828.55004 [19] A. Grothendieck, Techniques de construction et théorèmes d’existence en géometrie algébrique IV: Les schémas de Hilbert, Séminaire Bourbaki 221 (1960/61). CMP 98:09 [20] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001 [21] B. Kim, Quot schemes for flags and Gromov invariants for flag varieties, preprint (1995). [22] -, On equivariant quantum cohomology, Internat. Math. Res. Notices, no. 17 (1996), 841-851. CMP 97:04 [23] -, Quantum cohomology of flag manifolds $$G/B$$ and quantum Toda lattices, preprint (1996). [24] Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287 – 297. · Zbl 0288.14014 [25] János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. · Zbl 0877.14012 [26] M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525 – 562. · Zbl 0853.14020 [27] Alexander Kuznetsov, Laumon’s resolution of Drinfel$$^{\prime}$$d’s compactification is small, Math. Res. Lett. 4 (1997), no. 2-3, 349 – 364. · Zbl 0910.14026 [28] Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447 – 450 (French, with English summary). · Zbl 0495.14031 [29] Alain Lascoux and Marcel-Paul Schützenberger, Symmetry and flag manifolds, Invariant theory (Montecatini, 1982) Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 118 – 144. · Zbl 0542.14031 [30] Gérard Laumon, Un analogue global du cône nilpotent, Duke Math. J. 57 (1988), no. 2, 647 – 671 (French). · Zbl 0688.14023 [31] G. Laumon, Faisceaux automorphes liés aux séries d’Eisenstein, Automorphic forms, Shimura varieties, and \?-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 227 – 281 (French). · Zbl 0773.11032 [32] J.Li and G. Tian, Quantum cohomology of homogeneous varieties, preprint (1995). [33] -, Virtual moduli cycles and Gromov-Witten invariants, preprint (1996). [34] Benjamin M. Mann and R. James Milgram, On the moduli space of \?\?(\?) monopoles and holomorphic maps to flag manifolds, J. Differential Geom. 38 (1993), no. 1, 39 – 103. · Zbl 0790.53053 [35] D. Monk, The geometry of flag manifolds, Proc. London Math. Soc. (3) 9 (1959), 253 – 286. · Zbl 0096.36201 [36] David Mumford, Lectures on curves on an algebraic surface, With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. · Zbl 0187.42701 [37] Z. Qin and Y. Ruan, Quantum cohomology of projective bundles over $$\mathbb{P}^{n}$$, Trans. Amer. Math. Soc. (to appear). CMP 97:05 · Zbl 0932.14030 [38] Yongbin Ruan and Gang Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), no. 2, 259 – 367. · Zbl 0860.58005 [39] Edward Witten, Topological sigma models, Comm. Math. Phys. 118 (1988), no. 3, 411 – 449. · Zbl 0674.58047 [40] E. Witten, Topological sigma model, Commun. Math. Phys. 118 (1988), 411-449. · Zbl 0674.58047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.