On the degenerate Arnold conjecture on $\mathbb T^{2m}\times \mathbb C\mathbb P^n$

TL;DR

This paper proves a version of the Arnold conjecture for Hamiltonian diffeomorphisms on the product of a torus and complex projective space, showing they have at least a certain number of fixed points under specific conditions.

Contribution

The authors extend previous results to show the degenerate Arnold conjecture holds for certain Hamiltonian diffeomorphisms on $ ext{T}^{2m} imes ext{CP}^n$, especially those close to the identity in the complex projective component.

Findings

At least $2m + n + 1$ contractible fixed points for the considered diffeomorphisms.

The result applies to $C^0$-close Hamiltonian diffeomorphisms in the $ ext{CP}^n$-direction.

The approach builds on recent methods for the Arnold conjecture on $ ext{CP}^n$.

Abstract

In the 1960s Arnold conjectured that a Hamiltonian diffeomorphism of a closed connected symplectic manifold $(M,\omega)$ should have at least as many contractible fixed points as a smooth function on $M$ has critical points. Such a conjecture can be seen as a natural generalization of Poincar\'e's last geometric theorem and is one of the most famous (and still nowadays open in its full generality) problems in symplectic geometry. In this paper, we build on a recent approach of the authors and Izydorek to the Arnold conjecture on $\mathbb C\mathbb P^n$ to show that the (degenerate) Arnold conjecture holds for Hamiltonian diffeomorphisms $\phi$ of $\mathbb T^{2m}\times \mathbb C\mathbb P^n$, $m,n\geq 1$, which are $C^0$-close to the identity in the $\mathbb C \mathbb P^n$-direction, namely that any such $\phi$ has at least $\text{CL}(\mathbb T^{2m}\times \mathbb C\mathbb P^n)+1= 2m+n+1$ contractible fixed points.