Pseudo-rotations and holomorphic curves
- PDF / 477,300 Bytes
- 31 Pages / 439.37 x 666.142 pts Page_size
- 52 Downloads / 165 Views
Selecta Mathematica New Series
Pseudo-rotations and holomorphic curves Erman Çineli1 · Viktor L. Ginzburg1 · Ba¸sak Z. Gürel2 Accepted: 26 October 2020 © Springer Nature Switzerland AG 2020
Abstract We prove a variant of the Chance–McDuff conjecture for pseudo-rotations: under certain additional conditions, a closed symplectic manifold which admits a Hamiltonian pseudo-rotation must have deformed quantum product and, in particular, some nonzero Gromov–Witten invariants. The only assumptions on the manifold are that it is weakly monotone and that its minimal Chern number is at least two. The conditions on the pseudo-rotation are expressed in terms of the linearized flow at one of the fixed points and are hypothetically satisfied for most (but not all) pseudo-rotations. Keywords Pseudo-rotations · Periodic orbits · Hamiltonian diffeomorphisms · Floer homology · Quantum product · Gromov–Witten invariants Mathematics Subject Classification 53D40 · 37J10 · 37J45
Contents 1 Introduction . . . . . . . . . . . . 2 Main results . . . . . . . . . . . . 2.1 Definitions . . . . . . . . . . . 2.1.1 Base group . . . . . . . 2.1.2 Loop contribution . . . . 2.2 Detecting the quantum product
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
The work is partially supported by NSF CAREER award DMS-1454342, NSF Grant DMS-1440140 through MSRI (BG) and by Simons Collaboration Grant 581382 (VG).
B
Viktor L. Ginzburg [email protected] Erman Çineli [email protected] Ba¸sak Z. Gürel [email protected]
1
Department of Mathematics, UC Santa Cruz, Sant Cruz, CA 95064, USA
2
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA 0123456789().: V,-vol
. . . . . .
. . . . . .
78
Page 2 of 31
2.3 Particular cases and refinements . . . . . . . . . . . . . . . . . 2.3.1 Toric pseudo-rotations . . . . . . . . . . . . . . . . . . . 2.3.2 Pseudo-rotations in dimension four . . . . . . . . . . . . 3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Conventions and notation . . . . . . . . . . . . . . . . . . . . . 3.2 Floer homology and the pair-of-pants product . . . . . . . . . . 3.2.1 Floer homology . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Pair-of-pants product . . . . . . . . . . . . . . . . . . . . 3.3 Regularity for zero-energy solutions . . . . . . . . . . . . . . . 4 From extremal partitions to the quantum product . . . . . . . . . . . 4.1 Extremal partitions: the first look . . . . . . . . . . . . . . . . . 4.1.1 Definitions and basic facts . . . . . . . . . . . . . . . . . 4.1.2 Combinatorial results: the existence of extremal partitions 4.2 Combinatorics of extremal partitions and the quantu
Data Loading...
