18.29 Partial kernels and quotient equilibriumB52A879692D6

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Let $X$ be a connected compact complex manifold of dimension $n\geq 1$. Throughout this chapter, $L$ is a holomorphic line bundle on $X$ with a smooth Hermitian metric $h$. In a local holomorphic frame $e$ write $|e|_ h^2=\mathrm{e}^{-\chi }$; the curvature form $c_1(L,h)=\mathrm{dd}^{\mathrm{c}}\chi $ is a smooth closed real $(1,1)$-form, independent of the frame, as in section 19.8.

A quasi-plurisubharmonic function $\varphi $ on $X$, the selecting potential, selects in each tensor power $L^ m$ the holomorphic sections $s$ for which $|s|_{h^ m}^2\mathrm{e}^{-m\varphi }$ is locally integrable. A partial Bergman kernel is the Bergman kernel of this space of sections for a Hilbert norm defined by a continuous weight: the singular function $\varphi $ decides which sections occur, but it does not enter their norms. On a compact Kähler manifold, if $\varphi $ is $c_1(L,h)$-plurisubharmonic with positive non-pluripolar Monge–Ampère mass, the normalized kernel measures converge to the Monge–Ampère measure of a partial envelope. The competitors defining this envelope have multiplier ideals contained in those of $\varphi $ at every positive real weight, and lie below the continuous weight on the set where the norm is computed. The envelope is the largest of these $c_1(L,h)$-plurisubharmonic functions. This chapter studies how much of this equilibrium theory survives when $X$ is not Kähler, and when the selecting potential is plurisubharmonic only with respect to a Hermitian form unrelated to the curvature of $L$.

One direction of the Kähler argument is local and works on every compact complex manifold. A Bernstein–Markov estimate, which follows from the submean inequality, keeps the partial Fubini–Study potentials below the weight, and Lelong-number estimates on local modifications keep their limits within the multiplier-ideal class of $\varphi $. Together these bound every limit of the potentials by the partial envelope. A matching lower bound requires global sections approximating the envelope, and this is where the Kähler hypothesis enters. If $X$ admits a modification from a compact Kähler manifold, the relative canonical bundle with a Jacobian metric makes the selected spaces upstairs and downstairs isometric, and the Kähler theorem gives convergence of the kernel measures on $X$ itself.

Fibrations show what can happen without such a modification. On a holomorphic submersion with compact connected fibers, every section of a pulled-back line bundle is constant along the fibers and descends to the base. Probability measures on the fibers make this descent an isometry, so the kernel measures on the total space are lifts of kernel measures on the base. Their limit, the quotient equilibrium measure, is the lift of the equilibrium measure of the base, and it appears at the growth rate $m^{\dim B}$ of the section spaces rather than at $m^{\dim X}$. On the Hopf manifold it is a Hermitian Monge–Ampère measure; in dimension three this gives a nonzero limit on a manifold without the bounded mass property.

A selecting potential that is plurisubharmonic only with respect to a Hermitian form need not descend, but the sections it selects still do, so its singularities act only through a vanishing condition on the base. For isolated logarithmic poles, for poles along a section over a base divisor, and for anisotropic singularities controlled by integrability along the fibers, this condition differs from a multiplier-ideal condition on the base by a bounded number of vanishing orders. Such shifts cost only $O(m^{\dim B-1})$ sections, and for point poles and for poles along a section over a divisor the kernel measures still converge to a quotient equilibrium measure. The price is a loss of information: moving a pole along its fiber changes the multiplier ideals on $X$ but leaves every kernel unchanged. Explicit point and divisor models on projective bases compute the limits through binomial tails, and a change of the integration volume separates the base equilibrium measure from the conditional fiber probabilities carried by its lift.

Two mechanisms control the explicit models that follow: the local metric weight determines the scale of evaluation, and the number of selected sections determines the mass of the kernel measure. When vanishing orders grow sublinearly in $m$, an entire-function model gives a Gaussian transition on a shrinking scale. Transferring it to a compact curve requires a uniform local comparison of norms and a global $L^2$ correction preserving the vanishing order; the remainder of the smooth weight moves the center of the transition. In two dimensions, competing monomials at a fixed curvature zero can already introduce a logarithmic factor into the scale of the full kernel.

The section count also distinguishes selections with similar potential limits. A Newton polytope gives a graded monomial family whose growth constraint determines an envelope and whose lattice points determine the limiting mass. For a lower-dimensional polytope, concentration of the monomial norms gives a positive limit carried by real tori, although the higher curvature powers of the envelope vanish. Multiplier-ideal spaces need not form a graded family: rounding at critical divisors can make the leading count oscillate between even and odd degrees, while at the base points of a pencil three neighboring monomial bands contribute to a single positive limit. Convergence of the potentials therefore needs to be separated from convergence of the normalized measures.

We use the following notation from earlier chapters. For quasi-plurisubharmonic functions $u,v$, we write $u\preceq v$ if $u\leq v+C$ for a constant $C$, and $u\sim v$ if $u\preceq v\preceq u$ (definition 16.7.1); we write $u\preceq _{\mathcal I}v$ if $\mathcal I(ku)\subseteq \mathcal I(kv)$ for every real $k>0$, and $u\sim _{\mathcal I}v$ if these ideals are equal for every $k>0$ (section 5.33). For a smooth real $(1,1)$-form $\theta $, $P_\theta [u]_{\mathcal I}$ is the multiplier-ideal envelope of definition 18.4.1, and $V_\theta $ is the upper envelope of all nonpositive $\theta $-plurisubharmonic functions, which has minimal singularities (section 5.40). The non-pluripolar Monge–Ampère measure of $u\in \mathrm{PSH}(X,\theta )$ is written $\mathrm{MA}_\theta (u)$, or $\mathrm{MA}^{\mathrm{ext}}_\theta (u)$ when its mass may be infinite (theorem 13.14.3).