On the degenerate Arnold conjecture on T^2m x CP^n - Asselle, Luca; Starostka, Maciej (Topological Methods in Nonlinear Analysis, 2025)
In the 1960s Arnold conjectured that a Hamiltonian diffeomorphism of a closed connected symplectic manifold (M,ω) 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 CP^n to show that the (degenerate) Arnold conjecture holds for Hamiltonian diffeomorphisms φ of ^2m× CP^n, m,n≥ 1, which are C^0-close to the identity in the C P^n-direction, namely that any such φ has at least CL(^2m× CP^n)+1= 2m+n+1 contractible fixed points.