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



on the density of hypersurfaces in complex projective spaces
Concerning part 2), you could take a hyperplane with multiplicity k and then find a nearby hypersurface, so you could make the quantity you want arbitrarily close to the distance from a point to a hyperplane (hence independent of the degree). If you're interested in hypersurfaces which fill out projective space, Donaldson has a construction of sections of high degree line bundles which vanish "\epsilon-transversely" (i.e. pass steeply through the zero-section), and he proves that these converge as currents to the Kaehler form (in particular fill out the ambient space). See his famous 1996 JDG paper ["Symplectic submanifolds and almost complex geometry](http://www.intlpress.com/JDG/archive/1996/44-4-666.pdf)" (the relevant part for integrable complex structures is Section 6).


