Summary
Two symplectic diffeomorphisms,φ 0,φ 1 of a symplectic manifold (X, ω) are said to be homologous if there exists a smooth homotopyφ 1,t∋[0, 1] of symplectic diffeomorphisms between them such that the timedependent vector fieldξ t defined byd/dt(φ t -ξ t ºφ t is a globally hamiltonian vector field for allt, i.e. there exists a smooth real-valued timedependent hamiltonian functionh(x, t) onX x [0, 1] such thatξ t ⌋ω=dh t , whereh t=h(x,t).
V.I. Arnold [Ar] conjectured that any symplectic diffeomorphism ø of a compact symplectic manifoldX, homologous to the identity, has as many fixed-points as a function onX has critical points.
We prove Arnold's conjecture for complex projective spaces, with their standard symplectic structures, i.e. we prove that any symplectic diffeomorphism of ℂℙn homologous to the identity has at leastn+1 fixed-points.
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Research partially supported by a CSIR grant.
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Fortune, B. A symplectic fixed point theorem for ℂℙn . Invent Math 81, 29–46 (1985). https://doi.org/10.1007/BF01388770
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DOI: https://doi.org/10.1007/BF01388770