Proposition 18.38.22FF3CE9DCBFE

Proof. Since $n=q+1$, proposition 18.37.3 identifies the selected space with the space of homogeneous polynomials of degree $m$ on $\mathbb C^{q+1}$ that vanish to order at least $r_ m$ at $b$. For $\ell =(\ell _0,\ldots ,\ell _ q)\in \mathbb Z_{\geq 0}^{q+1}$ write $|\ell |=\sum _ i\ell _ i$ and $Z^\ell =\prod _ iZ_ i^{\ell _ i}$. In the affine chart $Z_0=1$, the vanishing order of $Z^\ell $ at $b$ is $\ell _1+\cdots +\ell _ q$, so the selected space is spanned by the monomials $Z^\ell $ with $|\ell |=m$ and $\ell _1+\cdots +\ell _ q\geq r_ m$.

Distinct monomials are orthogonal, by invariance under the coordinate circle actions. Their squared norms with respect to $\sigma ^ q$ are

To verify this, integrate $|z^\ell |^2\mathrm{e}^{-|z|^2}$ over $\mathbb C^{q+1}$ in two ways. As a product of one-variable integrals it equals $\pi ^{q+1}\prod _ i\ell _ i!$, while in polar coordinates it is the average of $|z^\ell |^2$ over the unit sphere times a radial integral that contributes the factor $(m+q)!$. Dividing by the same computation for $\ell =0$ gives the right-hand side of (18.38.3) as the normalized sphere average, and the pushforward of the normalized sphere measure to $\mathbb P^ q$ is $\sigma ^ q$, since both are the probability measures invariant under the unitary group. This proves (18.38.3).

The reciprocal of the squared norm is $D_ mm!/(\ell _0!\cdots \ell _ q!)$. Summing first over the monomials with $\ell _1+\cdots +\ell _ q=j$, the multinomial formula gives the contribution

to the kernel on $\mathbb P^ q$. Since $\nu _ X$ is the lift of $\sigma ^ q$, the norms and kernels lift exactly to $X_ n$, which proves the kernel formula. There are $\binom {j+q-1}{q-1}$ monomials of affine degree $j$, and summing over the excluded degrees $0\leq j<r_ m$ gives $\binom {r_ m+q-1}{q}$. Since $0\leq r_ m\leq m$ and $r_ m/m\to a$, this proves the dimension formula, the nonvanishing and the limit. $\square$