18.30 Selected spaces and partial Bergman kernelsF62F806EF43E

Fix a quasi-plurisubharmonic function $\varphi $ on $X$, the selecting potential, and a smooth positive probability measure $\nu $ on $X$. For each integer $m\geq 1$, the selected space of degree $m$ and its dimension are

Instead of $\varphi $, the Hilbert norm on $V_ m$ uses a real continuous weight $v\in C(X)$:

Since $v$ is bounded, this integral is finite for every global section. It vanishes only for $s=0$, because $\nu $ has a positive density and a holomorphic section vanishing almost everywhere vanishes identically. This separation of the selecting potential from the norm follows 18.28.37 Quantization of partial equilibrium measures.

Fix also an integer $r\geq 1$, which determines the normalization below; it will be chosen as the growth degree of $N_ m$, which may be smaller than $n$. Let $s_1,\ldots ,s_{N_ m}$ be an orthonormal basis of $V_ m$ for the norm $\| \cdot \| _{m,v,\nu }$. The partial Bergman kernel and the partial kernel measure are

By unitary invariance they do not depend on the orthonormal basis, and both are zero when $V_ m=\{ 0\} $. The evaluation argument of proposition 19.8.3 gives the extremal formula and the total mass:

The selecting potential enters only through $V_ m$. In particular, for fixed $h$, $v$ and $\nu $, if the multiplier ideals $\mathcal I(m\varphi )$ are contained in those of another selecting potential, then so are the selected spaces, and the extremal formula shows that the kernels are ordered in the same way.

Replacing $h$ by $h\mathrm{e}^{-a}$ and $v$ by $v-a$, for a smooth real function $a$, leaves $h^ m\mathrm{e}^{-mv}$, and hence the norm and the kernel measure, unchanged.