When can compact group actions be complexified?
Suppose a compact (real) Lie group $K$ acts holomorphically on a complex manifold $M$. Let $G$ be the complexification of $K$. Is there a natural way to obtain an action of $G$ on $M$ extending the action of $K$?
The example of $S^1$ acting on $Delta subset mathbb{C}$ would show that this is not always possible. But this seems pathalogical in the sense that the action does complexify to an action of $mathbb{C}^*$ on $mathbb{C}$. Is this sort of behaviour the only obstruction? For example if $M$ is $mathbb{C}^n$, does the action always complexify?
differential-geometry lie-groups complex-geometry
asked Nov 29 at 16:27
Curious Geometer
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Suppose a compact (real) Lie group $K$ acts holomorphically on a complex manifold $M$. Let $G$ be the complexification of $K$. Is there a natural way to obtain an action of $G$ on $M$ extending the action of $K$?
The example of $S^1$ acting on $Delta subset mathbb{C}$ would show that this is not always possible. But this seems pathalogical in the sense that the action does complexify to an action of $mathbb{C}^*$ on $mathbb{C}$. Is this sort of behaviour the only obstruction? For example if $M$ is $mathbb{C}^n$, does the action always complexify?
differential-geometry lie-groups complex-geometry
asked Nov 29 at 16:27
Curious Geometer
162
add a comment |
Suppose a compact (real) Lie group $K$ acts holomorphically on a complex manifold $M$. Let $G$ be the complexification of $K$. Is there a natural way to obtain an action of $G$ on $M$ extending the action of $K$?
The example of $S^1$ acting on $Delta subset mathbb{C}$ would show that this is not always possible. But this seems pathalogical in the sense that the action does complexify to an action of $mathbb{C}^*$ on $mathbb{C}$. Is this sort of behaviour the only obstruction? For example if $M$ is $mathbb{C}^n$, does the action always complexify?
differential-geometry lie-groups complex-geometry
asked Nov 29 at 16:27
Curious Geometer
162
Suppose a compact (real) Lie group $K$ acts holomorphically on a complex manifold $M$. Let $G$ be the complexification of $K$. Is there a natural way to obtain an action of $G$ on $M$ extending the action of $K$?
The example of $S^1$ acting on $Delta subset mathbb{C}$ would show that this is not always possible. But this seems pathalogical in the sense that the action does complexify to an action of $mathbb{C}^*$ on $mathbb{C}$. Is this sort of behaviour the only obstruction? For example if $M$ is $mathbb{C}^n$, does the action always complexify?
differential-geometry lie-groups complex-geometry
differential-geometry lie-groups complex-geometry
asked Nov 29 at 16:27
Curious Geometer
162
asked Nov 29 at 16:27
Curious Geometer
162
asked Nov 29 at 16:27
Curious Geometer
162
asked Nov 29 at 16:27
Curious Geometer
162
asked Nov 29 at 16:27
Curious Geometer
162
162
add a comment |
add a comment |
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