Frontiers of Mathematics in China

, Volume 13, Issue 2, pp 483–508 | Cite as

Poincaré polynomials of moduli spaces of stable maps into flag manifolds

Research Article
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Abstract

By using the Bialynicki-Birula decomposition and holomorphic Lefschetz formula, we calculate the Poincaré polynomials of the moduli spaces in low degrees.

Keywords

Bialynicki-Birula decomposition Poincaré polynomial 

MSC

13D40 14M15 14D22 

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Notes

Acknowledgements

The author thanks Professor Jian Zhou for his comments and suggestions, he thanks Professor Dragos Oprea for telling him Edward’s work [13] and some reference typos, and he also wants to thank Xiaowen Hu for helpful discussion. This work was supported by the Science Foundation of Zhejiang Sci-Tech University (ZSTU) (Grant No. 17062079-Y).

References

  1. 1.
    Agrawal S. The Euler characteristic of the moduli space of stable maps into a Grassmannian. Undergraduate Thesis, Univ of California, San Diego, 2011Google Scholar
  2. 2.
    Atiyah M F, Singer I M. The index of elliptic operators: III. Ann of Math (2), 1968, 87(3): 546–604MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Behrend K. Cohomology of stacks. In: Intersection Theory and Moduli. ICTP Lect Notes, Vol 19. 2004, 249–294Google Scholar
  4. 4.
    Bertram A. Quantum schubert calculus. Adv Math, 1997, 128(2): 289–305MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bertram A. Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian. Internat J Math, 1994, 5(6): 811–825MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bertram A, Ciocan-Fontanine I, Kim B. Two proofs of a conjecture of Hori and Vafa. Duke Math J, 2005, 126(1): 101–136MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bialynicki-Birula A. Some theorems on actions of algebraic groups. Ann of Math, 1973, 98(3): 480–497MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bialynicki-Birula A. Some properties of the decompositions of algebraic varieties determined by actions of a torus. Bull Acad Polon Sci Sér Sci Math Astronom Phys, 1976, 24(9): 667–674MathSciNetMATHGoogle Scholar
  9. 9.
    Carrell J B. Torus Actions and Cohomology. In: Gamkrelidze R V, Popov V L, eds. Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action. Encyclopaedia Math Sci, Vol 131. Berlin: Springer, 2002Google Scholar
  10. 10.
    Chen L. Poincaré polynomials of hyperquot schemes. Math Ann, 2001, 321(2): 235–251MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ciocan-Fontanine I. On quantum cohomology rings of partialag varieties. Duke Math J, 1999, 98(3): 485–524MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ciocan-Fontanine I. The quantum cohomology ring of ag varieties. Trans Amer Math Soc, 1999, 351(7): 2695–2729MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Edwards G. On genus zero stable maps to the ag variety. Undergraduate Thesis, Univ of California, San Diego, 2013Google Scholar
  14. 14.
    Fulton W, PandharipandeR. Notes on stable maps and quantum cohomology. arXiv: alg-geom/9608011Google Scholar
  15. 15.
    Hori K. Mirror Symmetry, Vol 1. Providence: Amer Math Soc, 2003Google Scholar
  16. 16.
    Kac V, Cheung P. Quantum Calculus. Berlin: Springer, 2002CrossRefMATHGoogle Scholar
  17. 17.
    Kim B. Quot schemes forags and Gromov invariants for ag varieties. arXiv: alg-geom/9512003Google Scholar
  18. 18.
    Lian B H, Liu C -H, Liu K F, Yau S -T. The S1 xed points in Quot-schemes and mirror principle computations. In: Cutkosky S D, Edidin D, Qin Z B, Zhang Q, eds. Vector Bundles and Representation Theory. Contemp Math, Vol 322. Providence: Amer Math Soc, 2003, 165–194CrossRefMATHGoogle Scholar
  19. 19.
    Manin Yu I. Stable maps of genus zero to ag spaces. Topol Methods Nonlinear Anal, 1998, (2): 207–217CrossRefMATHGoogle Scholar
  20. 20.
    Martín A L. Poincaré polynomials of stable map spaces to Grassmannians. In: Rendiconti del Seminario Matematico della Università di Padova, Tome 131. 2014, 193–208Google Scholar
  21. 21.
    Oprea D. Tautological classes on the moduli spaces of stable maps to Pr via torus actions. Adv Math, 2006, 207(2): 661–690MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Skowera J. Bialynicki-Birula decomposition of Deligne-Mumford stacks. Proc Amer Math Soc, 2013, 141(6): 1933–1937MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Strømme S A On parametrized rational curves in Grassmann varieties. In: Ghione F, Peskine C, Sernesi E, eds. Space Curves. Lecture Notes in Math, Vol 1266. Berlin: Springer, 1987, 251–272MathSciNetCrossRefMATHGoogle Scholar

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© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ScienceZhejiang Sci-Tech UniversityHangzhouChina

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