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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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  • $\begingroup$ This quantity is of course bounded from above by the diameter of $CP^n$. $\endgroup$
    – Alexandre Eremenko
    Commented Jan 10, 2013 at 23:03
  • $\begingroup$ 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). $\endgroup$
    – Jonny Evans
    Commented Jan 11, 2013 at 0:39
  • $\begingroup$ Where does the explicit formula for squared distance come from? $\endgroup$
    – Georges Elencwajg
    Commented Mar 22, 2015 at 14:03
  • $\begingroup$ 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. $\endgroup$
    – Đức Anh
    Commented Mar 23, 2015 at 4:01
  • $\begingroup$ Thank you for answering, dear Duc [sorry I can't write your name correctly :-)] $\endgroup$
    – Georges Elencwajg
    Commented Mar 23, 2015 at 21:48

1 Answer 1

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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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  • $\begingroup$ thank you very much. I have seen how naive my question is. Thank you also for recommanding the paper. $\endgroup$
    – Đức Anh
    Commented Jan 11, 2013 at 2:07
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    $\begingroup$ 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. $\endgroup$
    – Jonny Evans
    Commented Jan 11, 2013 at 6:40
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    $\begingroup$ Thank you. I thought that since I've received a downvote. But I will continue to ask some more questions. $\endgroup$
    – Đức Anh
    Commented Jan 11, 2013 at 17:42
  • $\begingroup$ 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. $\endgroup$
    – Jonny Evans
    Commented Jan 11, 2013 at 22:48

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