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Yann
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The "most symmetric" Mukai-Umemura 3-fold with automorphism group $PGL(2,\mathbb{R})$$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 ?

The "most symmetric" Mukai-Umemura 3-fold with automorphism group $PGL(2,\mathbb{R})$ 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 ?

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 ?

Source Link
Yann
Yann
  • 289
  • 1
  • 7

Mukai-Umemura 3-fold and Kaehler-Einstein metrics

The "most symmetric" Mukai-Umemura 3-fold with automorphism group $PGL(2,\mathbb{R})$ 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 ?