Good morning,

Let $H$ be a hypersurface in a complex projective space $\mathbb{CP}^N.$ Let $d$ be the distance de Fubini-Study on $\mathbb{CP}^N.$

- Let $x = [x_0: \ldots :x_N]$ and $y=[y_0:\ldots:y_N]$ two points in $\mathbb{CP}^N.$ Is the following formula true $$d(x,y)^2 = \frac{\sum_{i<j} |x_i \bar{y_j}-x_j\bar{y_i}|^2}{\sum |x_i|^2 \cdot \sum |y_i|^2}?$$
- (principal question) Is the following quantity $$\max_{z\in \mathbb{CP}^N} d(z,H)$$ bounded from above by a quantity which depends only on the degree of $H?$ The expected quantity (which depends on the degree of $H$) must converge to $0$ as the degree of $H$ increases to $\infty.$

Any help is appreciated. Thanks in advances.

Duc Anh