# on the density of hypersurfaces in complex projective spaces

Good morning,

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

1. 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 $$d(x,y)^2 = \frac{\sum_{i<j} |x_i \bar{y_j}-x_j\bar{y_i}|^2}{\sum |x_i|^2 \cdot \sum |y_i|^2}?$$
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

• This quantity is of course bounded from above by the diameter of $CP^n$. – Alexandre Eremenko Jan 10 '13 at 23:03
• Indeed, that was my first thought. I think by "the expected quantity must go to zero" the OP wanted an upper bound decreasing in the degree (which there cannot be unless you impose further conditions like epsilon-transversality). – Jonny Evans Jan 11 '13 at 0:39
• Where does the explicit formula for squared distance come from? – Georges Elencwajg Mar 22 '15 at 14:03
• I think it comes from the usual formula in projective geometry. I'm sorry for the ambiguous reply but I asked this question for a long time. – Đức Anh Mar 23 '15 at 4:01
• Thank you for answering, dear Duc [sorry I can't write your name correctly :-)] – Georges Elencwajg Mar 23 '15 at 21:48