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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

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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
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up vote 4 down vote accepted

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" (the relevant part for integrable complex structures is Section 6).

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thank you very much. I have seen how naive my question is. Thank you also for recommanding the paper. –  Đức Anh Jan 11 '13 at 2:07
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I wouldn't say it's naive, it seems like a reasonable idea that high degree hypersurfaces get complicated and fill up space. But as with all good reasonable ideas, it takes some effort to make sense of it. –  Jonny Evans Jan 11 '13 at 6:40
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Thank you. I thought that since I've received a downvote. But I will continue to ask some more questions. –  Đức Anh Jan 11 '13 at 17:42
    
The world is full of people who will think things are naive, but don't let that put you off: you ask questions for your sake, not for their sake. And often they're wrong anyway. –  Jonny Evans Jan 11 '13 at 22:48
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