Proposition 19.8.134ABE18AC08D2

Proof. A section of $L_ k^ m$ lifts to a holomorphic function $f$ on $\mathbb C^ n\setminus \{ 0\} $ satisfying

Since $n\geq 2$, Hartogs' theorem extends $f$ holomorphically to $\mathbb C^ n$. In its Taylor series, a monomial of total degree $j$ is multiplied by $\mathrm{e}^{-j}$ under the contraction, so comparing coefficients shows that only monomials of degree $km$ occur. Conversely, every homogeneous polynomial of that degree has the required equivariance. Counting monomials gives the binomial coefficient; when $k=0$, only the constant polynomials remain.

The inner product defined by $h_ k^ m$ and $\mathrm{d}V$ is invariant under the unitary group $U(n)$, so the basis-independent function $K_{m,0}$ is $U(n)$-invariant. It is also invariant under positive radial dilations: every section is homogeneous of degree $km$, and the factor $|z|^{-2km}$ in its squared norm cancels the effect of a dilation. Unitary transformations and positive radial dilations act transitively on $\mathbb C^ n\setminus \{ 0\} $, so $K_{m,0}$ is constant. By proposition 19.8.3 its integral is $N_ m$, and $\mathrm{d}V$ has mass one. Thus $K_{m,0}=B_{m,0}=N_ m$.

The statements about the Bergman potential and current follow immediately. Since $\alpha ^ n=0$ and $\zeta ^2=0$,

The binomial coefficient grows at most like $m^{n-1}$, which proves the normalized limit, and $\alpha ^ n=0$ proves the vanishing of the curvature mass. $\square$