Symplectic manifolds with vanishing action–Maslov homomorphism

Abstract

The action-Maslov homomorphism I W 1 .Ham.X; !// ! R is an important tool for understanding the topology of the Hamiltonian group of monotone symplectic manifolds. We explore conditions for the vanishing of this homomorphism, and show that it is identically zero when the Seidel element has finite order and the homology satisfies property D (a generalization of having homology generated by divisor classes). We use these results to show that I D 0 for products of projective spaces and the Grassmannian of 2 planes in C 4 .53D45; 53D35, 53D40, 20F69

DESCENT AND C0-RIGIDITY OF SPECTRAL INVARIANTS ON MONOTONE SYMPLECTIC MANIFOLDS

“…McDuff's criteria essentially require that many of the genus zero Gromov-Witten invariants of M vanish. In [4], Branson expands on McDuff's work and proves that the asymptotic spectral invariants descend in many new cases such as monotone products of complex projective spaces and the Grassmanian G (2,4).…”

Calabi quasimorphisms for monotone coadjoint orbits

“…The following is an example due to Chaput, Manivel and Perrin [7] of a homogeneous space G C /P whose quantum cohomology ring QH * (G C /P ) is not semisimple. Example 6.3 (Quantum Euler class of IG (2,6)). Let us endow C 6 with a symplectic form Ω.…”

“…We denote by IG (2,6) the isotropic Grassmannian of isotropic 2-planes in C 6 . The dimension of IG (2,6) is equal to 7.…”

Quasi-morphisms and quasi-states in symplectic topology

“…Abusing the notation we will denote the resulting (partial) Calabi quasi-morphism on Ham (M ) also by µ a . The list of manifolds for which µ a is known to descend to Ham (M ) for all a includes symplectically aspherical manifolds [61], complex projective spaces [20] and their monotone products [20,13], a monotone blow-up of CP 2 at three points and the complex Grassmannian Gr(2, 4) [13]. The list of manifolds for which it is known that µ a does not descend to Ham (M ) at least for some a includes various symplectic toric manifolds and, in particular, the monotone blow-ups of CP 2 at one or two points [23,54].…”

“…Namely, to certain Lagrangian submanifolds L of M one can associate the quantum homology (or the Lagrangian Floer homology) QH(L) that comes with an open-closed map i L : QH(L) → QH(M ) [3,8,30,31]. 13 , and if i L (x) divides an idempotent a ∈ QH(M ), then L is a-superheavy -this is shown in [23] in the monotone case, cf. [8,30,31].…”