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A Mirror Theorem for Non-split Toric Bundles: Appendix a and Referencesby@semaphores

A Mirror Theorem for Non-split Toric Bundles: Appendix a and References

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This research paper develops a new method (I-functions) for understanding mirror symmetry in complex spaces called non-split toric bundles.
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Author:

(1) Yuki Koto

Appendix A. Equivariant Fourier transformation




Note that this is a straightforward generalization of [20, Conjecture 1.7].


References

  1. Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398.


  2. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28.


  3. K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617.


  4. Nicole Berline and Mich`ele Vergne, Classes caract´eristiques ´equivariantes. Formule de localisation en cohomologie ´equivariante, C. R. Acad. Sci. Paris S´er. I Math. 295 (1982), no. 9, 539–541.


  5. Jeff Brown, Gromov-Witten invariants of toric fibrations, Int. Math. Res. Not. IMRN (2014), no. 19, 5437–5482.


  6. Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427.


  7. Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng, Computing genus-zero twisted GromovWitten invariants, Duke Math. J. 147 (2009), no. 3, 377–438.


  8. _________, A mirror theorem for toric stacks, Compos. Math. 151 (2015), no. 10, 1878–1912.


  9. Tom Coates and Alexander Givental, Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), no. 1, 15–53.


  10. Artur Elezi, A mirror conjecture for projective bundles, Int. Math. Res. Not. (2005), no. 55, 3445–3458.


  11. Honglu Fan and Yuan-Pin Lee, On Gromov-Witten theory of projective bundles, Michigan Math. J. 69 (2020), no. 1, 153–178.


  12. William Fulton, Intersection theory, second ed., 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. 2, Springer-Verlag, Berlin, 1998.


  13. Alexander Givental, A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996), Progr. Math., vol. 160, Birkh¨auser Boston, Boston, MA, 1998, pp. 141–175.


  14. Alexander B. Givental, Symplectic geometry of Frobenius structures, Frobenius manifolds, Aspects Math., vol. E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 91–112.


  15. T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518.


  16. Tam´as Hausel and Bernd Sturmfels, Toric hyperK¨ahler varieties, Doc. Math. 7 (2002), 495–534.


  17. Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079.


  18. _________, Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2909–2958.


  19. _________, Shift operators and toric mirror theorem, Geom. Topol. 21 (2017), no. 1, 315–343.


  20. Hiroshi Iritani, Quantum cohomology of blowups, 2023.


  21. Hiroshi Iritani and Yuki Koto, Quantum cohomology of projective bundles, 2023, arXiv:2307.03696 [math.AG].


  22. Hiroshi Iritani and Fumihiko Sanda, private communication.


  23. Yunfeng Jiang, Hsian-Hua Tseng, and Fenglong You, The quantum orbifold cohomology of toric stack bundles, Lett. Math. Phys. 107 (2017), no. 3, 439–465.


  24. Bumsig Kim, Andrew Kresch, and Tony Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), no. 1-2, 127–136.


  25. Chiu-Chu Melissa Liu, Localization in Gromov-Witten theory and orbifold Gromov-Witten theory, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 353–425.


  26. Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), no. 252, 1998, S´eminaire Bourbaki. Vol. 1997/98, pp. Exp. No. 848, 5, 307–340.


  27. Constantin Teleman, Gauge theory and mirror symmetry, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1309–1332.


  28. Valentin Tonita, Twisted orbifold Gromov-Witten invariants, Nagoya Math. J. 213 (2014), 141–187.


  29. Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670.


This paper is available on arxiv under CC 4.0 license.