On the degenerate Arnold conjecture on T^{2m} x 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 Poincaré's last geometric theorem and represents one of the most famous problems in symplectic geometry — still open today in its full generality. 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 $\T^{2m}\times \mathbb C\mathbb P^n$, $m,n\geq 1$, which are $C^0$-close to the identity in the $\mathbb C \mathbb P^n$-direction, namely that any such $\phi$ has at least $\text{CL}\big(\T^{2m}\times \mathbb C\mathbb P^n\big)+1= 2m+n+1$ contractible fixed points.