Star-shaped domain in $\mathbb{C}P^2$

Question

Consider $(\mathbb{C}P^2,\omega_{FS})$ where $\omega_{FS}$ is the standard Fubini-Study form. Let $L$ denote a sphere in $\mathbb{C}P^2$ in the class $\mathbb{C}P^1$. Further let $\int_{L} \omega_{FS} = \pi$.

Then the map

$$\begin{align} i : (B(1),\omega_0) &\to (\mathbb{C}P^2 \setminus L, \omega_{FS}) \\ (z,w) &\mapsto [\sqrt{1 - |z|^2 -|w|^2} : z : w] \end{align}$$

is a symplectomorphism. Where $(B(1),\omega_0)$ denotes the standard ball of radius 1 in $\mathbb{R}^4$ with the restriction of the symplectic standard form in $\mathbb{R}^4$.

Let $B_{\mathbb{C}P^2}(x_0, r)$ denote a metric ball in $\mathbb{C}P^2$ (for standard Kahler metric) of radius r and centred at $x_0 \in L$.

Then the paper I'm reading claims that $i^{-1} ( \mathbb{C}P^2 \setminus (L \cup B_{\mathbb{C}P^2}(x_0, r))$ is a star shaped neighbourhood in $B(1)$.

Could anyone help me prove the above statement?

Answer 1

One way to prove this is as follows.

First, from the assumption $B(1)\subset \mathbb C^2$ and the centre of $B(1)$ is $(0,0)$. Now we need the following two claims.

Claim 1. Any straight geodesic unit segment $I$ in $B(1)$ going through $(0,0)$ is sent by $i$ to a geodesic segment in $\mathbb CP^2$ through the point $i(0,0)$ in $\mathbb CP^2$.

Claim 2. Let us parametrise $I$ by its length for $t\in [0,1]$ let $p(t)$ be the point on $I$ on distance $t$ from $(0,0)\in I$. The the distance function $d_{\mathbb CP^2}(x_0, i(p(t)))$ is monotone decreasing for $t\in [0,1]$.

Clearly, it follows from Claims 1, 2, that the domain you consider is star-shaped with respect to $(0,0)$. If you want to prove Claim 2, one way to do this is to represent $\mathbb CP^2$ as the metric quotient $S^5/S^1$ and understands what Claim 1 means for this representation. Here $S^5$ is the unit sphere in $\mathbb C^3$ and $S^1$ is acting by multiplication by complex units. Claim 1 is rather simple. One can check it for the segment tangent to the vector $(1,0)\subset T_{0,0}(\mathbb C^2)$ and then use the fact that the map $i$ is $U(2)$ equivariant.