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The Arnold conjecture in $ \mathbb C\mathbb P^n $ and the Conley index

  • *Corresponding author: Luca Asselle

    *Corresponding author: Luca Asselle 

This research is supported by the DFG-project 380257369 "Morse theoretical methods in Hamiltonian dynamics". M.I. is supported by the Beethoven2-grant 2016/23/G/ST1/04081 of the National Science Centre, Poland. M.S. is supported by the DFG-grant 459826435 "The equivariant spectral flow and bifurcation for indefinite functionals with symmetries"

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  • In this paper we give an alternative, purely Conley index based proof of the Arnold conjecture in $ \mathbb C\mathbb P^n $ asserting that a Hamiltonian diffeomorphism of $ \mathbb C\mathbb P^n $ endowed with the Fubini-Study metric has at least $ n+1 $ fixed points.

    Mathematics Subject Classification: Primary: 37J45; Secondary: 58E05.


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  • [1] A. AbbondandoloMorse Theory for Hamiltonian Systems, CRC Press, 2001. 
    [2] L. Asselle and M. Starostka, Morse Homology for the Hamiltonian Action in Cotangent Bundles, preprint, 2022, arXiv: 2202.02324.
    [3] L. Asselle and M. Starostka, The Palais-Smale condition for the Hamiltonian action on a mixed regularity space of loops in cotangent bundles and applications, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 113, 28 pp. doi: 10.1007/s00526-020-01762-0.
    [4] C. Conley, Isolated Invariant Sets and the Morse Index, CMBS Regional Conf. Series 38, Amer. Math. Soc., 1978.
    [5] C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnold, Invent. Math., 73 (1983), 33-49.  doi: 10.1007/BF01393824.
    [6] Z. DzedzejK. Gȩba and W. Uss, The Conley index, cup-length and bifurcation, J. Fixed Point Theory Appl., 10 (2011), 233-252.  doi: 10.1007/s11784-011-0065-9.
    [7] Y. Eliashberg, Estimates on the number of fixed points of area preserving transformations, preprint. 1979, Syktyvkar Univerity.
    [8] A. Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Ergodic Theory Dynam. Sys., 7 (1987), 93-103.  doi: 10.1017/S0143385700003825.
    [9] A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom., 28 (1988), 513-547. 
    [10] A. Floer, Symplectic fixed points and holomorphic spheres, Commun. Math. Phys., 120 (1989), 575-611.  doi: 10.1007/BF01260388.
    [11] B. Fortune, A symplectic fixed point theorem for $\mathbb{C}\mathbb P^n$, Invent. Math., 81 (1985), 29-46.  doi: 10.1007/BF01388770.
    [12] K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology, 38 (1999), 933-1048.  doi: 10.1016/S0040-9383(98)00042-1.
    [13] K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant for general symplectic manifolds, The Arnoldfest. Proceedings of a conference in honour of V.I. Arnold for his 60th birthday, Toronto, Canada, June 15-21 (1997), (al., E. Bierstone (ed.) et, ed.), 24 (1999), 173-190.
    [14] K. GȩbaM. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications, Studia Math., 134 (1999), 217-233.  doi: 10.4064/sm-134-3-217-233.
    [15] A. B. Givental', A symplectic fixed point theorem for toric manifolìds, The Floer Memorial Volume, Birkhäuser (1995), 445-481.
    [16] R. Golovko, On variants of Arnold conjecture, Archivum Mathematikum (Brno), 56 (2020), 277-286.  doi: 10.5817/am2020-5-277.
    [17] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhauser Advanced Texts, 1994. doi: 10.1007/978-3-0348-8540-9.
    [18] M. Izydorek, A cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems, J. Diff. Equations, 170 (2001), 22-50.  doi: 10.1006/jdeq.2000.3818.
    [19] M. IzydorekT. O. RotM. StarostkaM. Styborski and R. C. A. M. Vandervorst, Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces, J. Diff. Equations, 263 (2017), 7162-7186.  doi: 10.1016/j.jde.2017.08.007.
    [20] T. Kragh, Fibrancy of symplectic Homology in cotangent bundles, Proc. Sympos. Pure Math., 85, Amer. Math. Soc., Providence, RI, (2012), 401-407. doi: 10.1090/pspum/085/1394.
    [21] T. Kragh, Parametrized ring-spectra and the nearby Lagrangian conjecture, Geom. Topol., 17 (2013), 639-731.  doi: 10.2140/gt.2013.17.639.
    [22] G. Liu and G. Tian, Floer homology and Arnold conjecture, J. Differential Geom., 49 (1998), 1-74. 
    [23] C. K. McCord, On the Hopf index and the Conley index, Trans. Amer. Math. Soc., 313 (1989), 853-860.  doi: 10.1090/S0002-9947-1989-0961594-0.
    [24] Y.-G. Oh, A symplectic fixed point theorem on $\mathbb T^{2n} \times \mathbb{C}\mathbb P^k$, Math. Z., 203 (1990), 535-552.  doi: 10.1007/BF02570755.
    [25] L. Polterovich, Hofer's diameter and Lagrangian intersections, Int. Math. Res. Not., 1998 (1998), 217-223.  doi: 10.1155/S1073792898000178.
    [26] Y. Ruan, Virtual neighborhoods and pseudo-holomorphic curves, Proceedings of 6th Gökova Geometry-Topology Conference, Turkish J. Math., 23 (1999), 161-231. 
    [27] Yu. B. Rudyak and J. Oprea, On the Lyustrnik-Schnirelmann category of symplectic manifolds and the Arnold conjecture, Math. Z., 230 (1999), 673-678.  doi: 10.1007/PL00004709.
    [28] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc., 291 (1985), 1-41.  doi: 10.1090/S0002-9947-1985-0797044-3.
    [29] M. Schwarz, A quantum cup-length estimate for symplectic fixed points, Invent. Math., 133 (1998), 353-397.  doi: 10.1007/s002220050248.
    [30] M. Starostka, Connected components of the space of proper gradient vector fields, J. Fixed Point Theory Appl., 24 (2022), Paper No. 2, 5 pp. doi: 10.1007/s11784-021-00900-1.
    [31] M. Starostka and N. Waterstraat, The $E$-cohomological conley index, cup-lengths and the Arnold conjecture on $T^{2n}$, Adv. Nonlinear Stud., 19 (2019), 519-528.  doi: 10.1515/ans-2019-2044.
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