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What is minimal $n$, such that there exists a smooth complex projective manifold $M^n$ satisfying the following two conditions:

1) The group of automorphisms of $M^n$ is $\mathbb C^*$.

2) All the fixed point of the $\mathbb C^*$-action are isolated and the set of fixed points is non-empty.

Is it possible to get some sort of classification of such manifolds in low dimensions?

Note that $n>2$...

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    $\begingroup$ "with isolated fixed points" should be interpreted as "such that there exists an isolated fixed point", or "such that all fixed points are isolated"? $\endgroup$
    – YCor
    Commented Dec 15, 2017 at 16:21
  • $\begingroup$ Sorry, I finally figured out all the conditions that I've meant and reformulated the question $\endgroup$
    – aglearner
    Commented Dec 15, 2017 at 16:47
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    $\begingroup$ I am probably missing something, but it seems to me that for the action of $\mathbb{C}^*$ on $\mathbb{P}^2$ given by $t\cdot (X,Y,Z)=(X,tY,t^2Z)$, the fixed points are $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$. $\endgroup$
    – abx
    Commented Dec 15, 2017 at 16:53
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    $\begingroup$ @abx there's an additional requirement that there are no further automorphisms. Otherwise the similar action on $\mathbb{P}^1$ already works. $\endgroup$
    – YCor
    Commented Dec 15, 2017 at 17:44
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    $\begingroup$ @YCor: The point has automorphism group a point, contradicting 1). $\endgroup$
    – Ben McKay
    Commented Dec 15, 2017 at 18:59

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