From pseudo-rotations to holomorphic curves via quantum Steenrod squares

Abstract

In the context of symplectic dynamics, pseudo-rotations are Hamiltonian diffeomorphisms with finite and minimal possible number of periodic orbits. These maps are of interest in both dynamics and symplectic topology. We show that a closed, monotone symplectic manifold, which admits a nondegenerate pseudo-rotation, must have a deformed quantum Steenrod square of the top degree element, and hence non-trivial holomorphic spheres. This result (partially) generalizes a recent work by Shelukhin and complements the results by the authors on non-vanishing Gromov-Witten invariants of manifolds admitting pseudo-rotations.

Pseudo-rotations and Steenrod squares revisited

“…Theorem A was proven by the author in [13] under the additional assumption that (M, ω) satisfies the Poincaré duality property for Hamiltonian spectral invariants (which holds in particular whenever the minimal Chern number of (M, ω) satisfies N > n). Under the assumption that φ is strongly non-degenerate, Theorem A was also proved by C ¸ineli, Ginzburg, and Gürel in [2], using different additional arguments extending [13]. A related, but quite different, result relating the existence of pseudo-rotations to pseudo-holomorphic spheres was shown in [3].…”

Covariant constancy of quantum Steenrod operations

“…Quantum Steenrod operations, originally introduced by Fukaya [7], have recently appeared in a variety of contexts: their properties have been explored in [18] (which also contains the first nontrivial computations); they can be used to study arithmetic aspects of mirror symmetry [14]; and in Hamiltonian dynamics, they are relevant for the existence of pseudo-rotations [16,2,17]. Nevertheless, computing quantum Steenrod operations remains a challenging problem in all but the simplest cases.…”

“…This is interesting because of its implications for Hamiltonian dymanics: by the criterion from [2,17], it means that M cannot admit a quasi-dilation. We refer to Section 6a for further discussion.…”

Rational Quantum Cohomology of Steenrod Uniruled Manifolds

“…In particular, [She20] and subsequently [She19], [CGG19a] establish that if M is a monotone symplectic manifold admitting an F 2 -pseudorotation, then M is F 2 -uniruled, a notion introduced in [She20]. We review this notion in Section 2; being F-uniruled implies that for any compatible almost complex structure on M , there is a pseudoholomorphic sphere passing through every point of M .…”