Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier transformation Note that this is a straightforward generalization of [20, Conjecture 1.7]. References Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398.


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


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


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.


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


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


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.


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


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


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


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


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.


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.


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


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


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


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


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


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


Hiroshi Iritani, Quantum cohomology of blowups, 2023.


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


Hiroshi Iritani and Fumihiko Sanda, private communication.


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.


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.


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.


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.


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.


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


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. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier transformation Note that this is a straightforward generalization of [20, Conjecture 1.7]. References Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617. 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. Jeff Brown, Gromov-Witten invariants of toric fibrations, Int. Math. Res. Not. IMRN (2014), no. 19, 5437–5482. Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427. 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. _________, A mirror theorem for toric stacks, Compos. Math. 151 (2015), no. 10, 1878–1912. Tom Coates and Alexander Givental, Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), no. 1, 15–53. Artur Elezi, A mirror conjecture for projective bundles, Int. Math. Res. Not. (2005), no. 55, 3445–3458. Honglu Fan and Yuan-Pin Lee, On Gromov-Witten theory of projective bundles, Michigan Math. J. 69 (2020), no. 1, 153–178. 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. 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. Alexander B. Givental, Symplectic geometry of Frobenius structures, Frobenius manifolds, Aspects Math., vol. E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 91–112. T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. Tam´as Hausel and Bernd Sturmfels, Toric hyperK¨ahler varieties, Doc. Math. 7 (2002), 495–534. Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079. _________, Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2909–2958. _________, Shift operators and toric mirror theorem, Geom. Topol. 21 (2017), no. 1, 315–343. Hiroshi Iritani, Quantum cohomology of blowups, 2023. Hiroshi Iritani and Yuki Koto, Quantum cohomology of projective bundles, 2023, arXiv:2307.03696 [math.AG]. Hiroshi Iritani and Fumihiko Sanda, private communication. 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. 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. 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. 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. 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. Valentin Tonita, Twisted orbifold Gromov-Witten invariants, Nagoya Math. J. 213 (2014), 141–187. Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617. K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617. 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. 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. Jeff Brown, Gromov-Witten invariants of toric fibrations, Int. Math. Res. Not. IMRN (2014), no. 19, 5437–5482. Jeff Brown, Gromov-Witten invariants of toric fibrations, Int. Math. Res. Not. IMRN (2014), no. 19, 5437–5482. Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427. Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427. 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. 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. _________, A mirror theorem for toric stacks, Compos. Math. 151 (2015), no. 10, 1878–1912. _________ , A mirror theorem for toric stacks, Compos. Math. 151 (2015), no. 10, 1878–1912. _________ Tom Coates and Alexander Givental, Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), no. 1, 15–53. Tom Coates and Alexander Givental, Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), no. 1, 15–53. Artur Elezi, A mirror conjecture for projective bundles, Int. Math. Res. Not. (2005), no. 55, 3445–3458. Artur Elezi, A mirror conjecture for projective bundles, Int. Math. Res. Not. (2005), no. 55, 3445–3458. Honglu Fan and Yuan-Pin Lee, On Gromov-Witten theory of projective bundles, Michigan Math. J. 69 (2020), no. 1, 153–178. Honglu Fan and Yuan-Pin Lee, On Gromov-Witten theory of projective bundles, Michigan Math. J. 69 (2020), no. 1, 153–178. 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. 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. 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. 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. Alexander B. Givental, Symplectic geometry of Frobenius structures, Frobenius manifolds, Aspects Math., vol. E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 91–112. Alexander B. Givental, Symplectic geometry of Frobenius structures, Frobenius manifolds, Aspects Math., vol. E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 91–112. T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. Tam´as Hausel and Bernd Sturmfels, Toric hyperK¨ahler varieties, Doc. Math. 7 (2002), 495–534. Tam´as Hausel and Bernd Sturmfels, Toric hyperK¨ahler varieties, Doc. Math. 7 (2002), 495–534. Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079. Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079. _________, Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2909–2958. _________ , Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2909–2958. _________ _________, Shift operators and toric mirror theorem, Geom. Topol. 21 (2017), no. 1, 315–343. _________ , Shift operators and toric mirror theorem, Geom. Topol. 21 (2017), no. 1, 315–343. _________ Hiroshi Iritani, Quantum cohomology of blowups, 2023. Hiroshi Iritani, Quantum cohomology of blowups, 2023. Hiroshi Iritani and Yuki Koto, Quantum cohomology of projective bundles, 2023, arXiv:2307.03696 [math.AG]. Hiroshi Iritani and Yuki Koto, Quantum cohomology of projective bundles, 2023, arXiv:2307.03696 [ math.AG ]. math.AG Hiroshi Iritani and Fumihiko Sanda, private communication. Hiroshi Iritani and Fumihiko Sanda, private communication. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Valentin Tonita, Twisted orbifold Gromov-Witten invariants, Nagoya Math. J. 213 (2014), 141–187. Valentin Tonita, Twisted orbifold Gromov-Witten invariants, Nagoya Math. J. 213 (2014), 141–187. Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. 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. This paper is available on arxiv under CC 4.0 license. available on arxiv

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