Symplectic and Kähler structures on biquotients

On the signature of biquotients

“…, i(E 55 − E 66 )} of t max to be positively oriented. Analogously we obtain an orientation on su(2), su (5), and then also on…”

“…Biquotients were originally considered by Eschenburg [4] in the context of Riemannian geometry, but also appear naturally in other geometries, such as symplectic [5] or Sasakian geometry [2]. In all these considerations, symmetries play an essential role.…”

“…Moreover, everything from the su (5) factor is mapped to its negative. From this, one computes the map (4.1):…”

Double sided torus actions and complex geometry on $SU(3)$

“…On the other hand, it has been shown that there exists the 2 dimensional torus S 12 in SU (3)× SU (3) acting freely on SU (3) not contained in 1 × SU (3) such that the quotient space SU (3)/S 12 admits a structure of Kähler "manifold" as well as the flag manifold, in [3] and [4] implicitly and [5] explicitly. However, it is also shown in [5,Theorem 3.3] that there is no Kähler structure on SU (3)/S 12 which is invariant under the action of 2 dimensional torus T × T /S 12 , not like the flag manifold. Thus, such example seem to be essentially different from our objects we obtain by Theorem 1.1.…”

Symplectic and Kähler structures on $$\mathbb CP^1$$-bundles over $$\mathbb CP^2$$

“…In [31] Tolman constructed a remarkable family of symplectic forms ω λ 1 ,λ 2 on a compact 6-manifold M T , that have a Hamiltonian T 2 -action, and yet don't admit a T 2invariant compatible Kähler metric. 1 Recently, in [16,17] Goertsches, Konstantis, and Zoller proved that Tolman's manifold M T is diffeomorphic to the projectivisation P(E) of a complex rank two bundle E over CP 2 , and hence it admits some Kähler structure.…”