Question

The "most symmetric" Mukai-Umemura 3-fold with automorphism group $PGL(2,\mathbb{C})$ admits a Kaehler-Einstein metric according to Donaldson's result.

On the contrary, there are some arbitrarily small complex deformations of the above $3$-fold which do not admit Kaehler-Einstein metrics, as shown by Tian. All examples considered by Tian seem to have no symmetries at all. Is it possible to find similarly arbitrarily small complex deformations with $\mathbb{C}^*$-action and which do not admit any Kaehler-Einstein metric ?

Answer 1

[Edited] Such a manifold cannot exist.

Indeed the small deformations of the "symmetric" Mukai-Umemura $3$-fold $X$ are described explicitly by Donaldson in this paper, pages 43-44. There he describes 5 classes of deformations. Classes 1,2,3 correspond to points in $H^1(X,TX)$ whose $PSL(2,C)$ orbits are closed, and the corresponding small deformations admit Kahler-Einstein metrics thanks to a theorem of Székelyhidi, see Propositions 7,8 and page 12 of this paper.

Cases 4,5 have orbits that are not closed, and do not admit Kahler-Einstein metrics thanks to Tian's argument (case 5 contains the manifold considered by Tian). These are the only possible cases where you could hope to find an example. Any such manifold has a nontrivial C* action precisely if the $PSL(2,C)$-stabilizer of the corresponding point in $H^1(X,TX)$ contains a C*. However one can check directly that in both cases the stabilizer cannot contain any C*