Communications in Number Theory and Physics

Volume 9 (2015)

Number 4

Fermat and the number of fixed points of periodic flows

Pages: 643 – 687

DOI: http://dx.doi.org/10.4310/CNTP.2015.v9.n4.a1

Authors

Leonor Godinho (Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Portugal)

Álvaro Pelayo (Department of Mathematics, University of California at San Diego)

Silvia Sabatini (Mathematisches Institut, Universität zu Köln, Germany)

Abstract

We compute explicit lower bounds for the number of fixed points of a circle action on a compact almost complex manifold $M^{2n}$ with nonempty fixed point set, provided the Chern number $c_1 c_{n-1} [M]$ vanishes. The proofs combine techniques originating in equivariant $K$-theory with celebrated number theory results on polygonal numbers, stated by Fermat. These lower bounds depend only on $n$ and, in some cases, are better than existing bounds. If the fixed point set is discrete, we also prove divisibility properties for the number of fixed points, improving similar statements obtained by Hirzebruch in 1999. Our results apply, for example, to a class of manifolds which do not support any Hamiltonian circle action, namely those for which the first Chern class is torsion. This includes, for instance, all symplectic Calabi–Yau manifolds.

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