Proposition 18.43.3.BDAF8AE299D2 Let $\nu $ be a smooth positive probability volume form on $\mathbb P^2$, invariant under the rotations of $z_1$ and of $z_2$, and write
\[ \nu =\rho (z)\, \mathrm dA(z_1)\, \mathrm dA(z_2),\qquad \rho _0=\rho (0)>0 \]
in the chart above. Give $H^0(\mathbb P^2,L^ m)$ the norm $\int _{\mathbb P^2}|s|_{h^ m}^2\, \mathrm d\nu $, and let $B_ m$ be its kernel density. Then
\[ B_ m(p)=\frac{3}{\pi ^2\rho _0}\frac m{\log m} \left(1+O((\log m)^{-1})\right). \]
For the Fubini–Study probability volume, the leading constant is $3/2$.
Proof.
The homogeneous monomials form an orthogonal basis, because the metric and the volume are invariant under the two independent rotations and distinct monomials transform by distinct characters. At $p$, only $Z_0^ m$ has a nonzero value, equal to one in the frame $e^{\otimes m}$. Consequently $B_ m(p)=I_ m^{-1}$, where
\[ I_ m=\int _{\mathbb P^2}|Z_0^ m|_{h^ m}^2\, \mathrm d\nu =\int _{\mathbb C^2}(1+3W(|z_1|^2,|z_2|^2))^{-m/3} \rho (z)\, \mathrm dA(z_1)\, \mathrm dA(z_2). \]
Here the omitted projective line $\{ Z_0=0\} $ has zero volume. By rotational invariance, the first derivatives of $\rho $ vanish at zero, so $\rho (z)=\rho _0+O(|z_1|^2+|z_2|^2)$ near zero. In the variables $a=|z_1|^2$ and $b=|z_2|^2$, polar integration has Jacobian $\pi ^2\, \mathrm da\, \mathrm db$.
Choose a fixed small $\delta >0$. On $W\leq \delta $, set $\ell (W)=\tfrac 13\log (1+3W)$. For constants $c,C>0$ independent of $m$,
\[ cW\leq \ell (W)\leq W,\qquad 0\leq \mathrm e^{-m\ell (W)}-\mathrm e^{-mW} \leq CmW^2\mathrm e^{-cmW} \leq \frac Cm\mathrm e^{-cmW/2}. \]
The first estimate of lemma 18.43.1 bounds the integral of this error in the weight by $O(m^{-2}\log m)$, and the second bounds the error from replacing $\rho $ by $\rho _0$ by $O(m^{-4/3})$. On the complement $W>\delta $, the contribution to $I_ m$ is at most $(1+3\delta )^{-m/3}$, because $\nu $ has mass one, and the corresponding tail of the exponential model is at most $\mathrm e^{-m\delta /2}J(m/2)$. Therefore
\[ I_ m=\pi ^2\rho _0J(m)+O(m^{-4/3})+O(m^{-2}\log m) =\frac{\pi ^2\rho _0}{3m}\log m+O(m^{-1}). \]
Inverting proves the first assertion. The Fubini–Study probability volume has the local expression
\[ \nu =\frac2{\pi ^2}(1+|z_1|^2+|z_2|^2)^{-3} \, \mathrm dA(z_1)\, \mathrm dA(z_2). \]
It has mass one, since $2\int _0^\infty \int _0^\infty (1+a+b)^{-3}\, \mathrm da\, \mathrm db=1$, and its density at zero is $2/\pi ^2$. The leading constant is therefore $3/2$.
$\square$
Pome
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