24.10 Sections, kernels and the dimension of growth86B65C86A5FA

A Hermitian form is the curvature of a holomorphic line bundle only if it is closed, so the algebraic part of the theory must keep the closed curvature data separate from the Hermitian background. The weighted spaces of sections of section 19.8, their multiplicative variant and their intrinsic flag bodies do this, and their dimension of growth can be strictly smaller than the complex dimension of the manifold.

For the partial Bergman kernels of section 18.29, the upper bound by the envelope is proved in general; a matching lower bound is known in the explicit regimes below, and the general problem is open. Let $L\to X$ be a holomorphic line bundle with a smooth Hermitian metric of curvature $\theta $. If $X$ admits a modification from a compact Kähler manifold and $\varphi \in \mathrm{PSH}(X,\theta )$ has positive non-pluripolar mass, theorem 18.35.5 proves convergence of the kernel measures normalized in full dimension. The proof keeps track of the relative canonical bundle, whose Jacobian factor corrects both the integrability condition of the ideals and the Hilbert norm. This geometric assumption implies the bounded mass property, although the birational transformation identities themselves do not require it. On a holomorphic submersion with descended data, the equilibrium theorem on the base gives a different exact limit on the total space. On the Hopf $n$-fold, the dimension of the selected space of sections of the $m$th tensor power grows like $m^{n-1}$, so normalizing by $m^ n$ would erase the limit; adding the vertical Hermitian form recovers a nonzero measure, with the factor computed in (18.36.5).

Isolated point poles that do not descend are treated in theorem 18.37.4: their projected jet conditions differ from the multiplier ideals on the base by only finitely many orders. For a single point pole on the Hopf manifold, theorem 18.38.4 proves the lower asymptotics of the potentials as well as convergence of the kernel measures in total variation. Moving the pole along its fiber leaves every selected space unchanged, so these limits recover the projected equilibrium, while the multiplier-ideal type of the original Hermitian weight can be different.

Theorem 18.40.2 gives a further regime: a logarithmic pole along a section over a hyperplane, with arbitrary compact connected fibers of positive dimension. The growth degree is the dimension of the base. The partial potentials attain the domination envelope of the original singularity on the total space, and the kernel measures converge in total variation for every smooth positive volume used in the norm. The limit on the base is determined by the metric of the line bundle, and its lift keeps exactly the conditional probabilities on the fibers.

For an interior rational Newton polytope with integral affine offset, theorem 18.45.7, theorem 18.45.11 prove a lower bound by the envelope and a positive limit of the measures in the relative growth dimension. Here the smooth weight and the sampling probability are invariant under the torus, the logarithmic weight is strictly convex, and the target is the specified envelope of Newton growth. In lower dimension the limit lies on a real moment shell and differs from every mixed measure formed from the curvature of the envelope and a smooth Hermitian form on the ambient manifold. These results do not identify an arbitrary multiplier-ideal envelope.

The multiplier-ideal family of theorem 18.46.2 has positive growth exponent and potentials converging to its ideal envelope, yet its kernel measures have two different limits: rounding at two critical divisors makes the leading term of the section count depend on the parity of the degree. Convergence of the normalized dimensions is therefore an additional necessary condition for a finite limit of the measures. This selecting weight has zero top non-pluripolar mass, so it lies outside the positive-mass regime above.

For the two isolated base points of a pencil, theorem 18.47.3, theorem 18.47.7 identify the full ideal envelope and prove a global lower bound for the potentials and a positive limit of the measures. The selected spaces consist of three neighboring bands of monomials and have dimension exactly $3m+1$, although their class of competitors at all weights equals the upper ordinary type class. The specified compact logarithmic box and Gaussian norm give intrinsic mass three. Locally uniform convergence away from the poles, convergence in every finite $L^ p$ and uniform convergence after exponentiation coexist with an unbounded additive error at each fixed degree. These spaces are not multiplicative, and this selecting weight also has zero top non-pluripolar mass.

Beyond the regimes treated above (isolated points, horizontal projective poles, descended data, interior Newton polytopes and the pencil example), the problem involves two tasks: producing enough sections with the required ideal constraints, and controlling the resulting measures. A Bergman upper bound gives neither. On a manifold that is not Moishezon, normalization by the ambient degree may already force every kernel measure to vanish, for reasons independent of singularity theory.

Locality of Hölder regularity, and regularity of weighted extremal functions under a uniform density condition in capacity, give special cases of the regularity problem; see section 25.21. These hypotheses do not cover every singular envelope, and they do not settle the problem of lower asymptotics above.

On a fibration, the comparison of spaces of finite codimension in lemma 18.37.1 suggests a more specific question, which asks for an analytic description of the constraints on sections after descent.

For descended continuous weights and probability norms, such an approximation, combined with lemma 18.37.1 and the equilibrium theorem on the base, would give the limits of the kernel measures and of the determinant functionals, although not by itself convergence of the potentials. For finitely many isolated poles, theorem 18.37.4 gives the stronger error $O(m^{q-1})$ under its hypothesis on the projected poles. Theorem 18.39.3 gives the same error for a logarithmic singularity along a section over a smooth divisor of the base, under the curvature margin stated there; its base reference is explicitly divisorial, and example 18.39.4 shows that the order of the error is sharp. The problem does not presuppose that every selecting weight in $\mathrm{PSH}(X,\omega )$ admits such a base reference.

The band criterion of theorem 18.39.8, which places $V_ m$ between two spaces of sections vanishing along a divisor, requires a global logarithmic lower bound for $\varphi $, a positive residual on the base after a smooth gauge change, and a single chart over a smooth connected divisor in which $\varphi $ has an anisotropic upper bound. It constructs a base reference of positive mass and gives the error $O(m^{q-1})$ for every tensor power. Corollary 18.39.9 computes the selected series exactly, including pole sets of positive dimension inside the fibers, and example 18.39.10 attains the order of the error. These are sufficient conditions; general selecting weights and necessary conditions remain to be understood.

For finitely many divisors with simple normal crossings, theorem 18.39.13 gives the same error from a global lower bound by a sum of logarithms, one anisotropic witness for each divisor and a positive residual after subtracting all the divisorial terms. Example 18.39.14 computes the exact series for simultaneous poles in the fibers and attains the order of the error. The normal-crossing hypothesis provides a separated integrability test on the base; general ideal constraints on the base remain outside these results.

Theorem 18.39.16 allows the divisorial data to live on a specified smooth modification of the base, provided the residual there is positive and the pulled-back selecting weight has the required lower bound and witnesses; the transformation rule for adjoint multiplier ideals keeps the Jacobian in the coefficient of the sections. Example 18.39.17 computes the resulting orders of point ideals on projective space and gives a sharp error for a base pole of positive mass in higher codimension. Which general selecting weights admit such modification data is open.

In all three band criteria, corollary 18.39.19 allows a smooth semipositive residual with positive top integral. Example 18.39.20 lies outside the regime of strictly positive residuals: contracted exceptional lines obstruct every strictly positive smooth gauge, although the base reference has positive mass. There the exceptional fixed divisor makes the selected and reference spaces identical for every tensor power, which separates the local gap between orders from the global defect of sections; the positive volume of the reference remains part of the sufficient hypotheses.

Proposition 18.39.22 computes this mechanism on the same family of blowups for every positive selecting coefficient, every integral fixed multiplicity of the exceptional divisor and any two anisotropic powers. A coefficient at most the fixed multiplicity leaves the whole space of sections. Above it, the selected spaces impose point jets of linearly growing order. As long as the coefficient stays below the fixed multiplicity plus the projective degree, the explicit reference has positive mass and the defect of sections is $O(m^{q-1})$. At that threshold the reference has zero mass and the sections grow with degree $q-1$; beyond it the series eventually vanishes. These exact model computations leave the general geometric criterion open.

Example 18.39.25 shows that a nonzero sharp defect also occurs outside the regime of strictly positive residuals. On the blowup of projective space at two points, the residual is the sum of the pullbacks of the Fubini–Study forms under the two resolved linear projections; it has degree zero on the line through the two centers but positive top mass $2^ q-2$. Simultaneous interpolation of point jets gives the defect $2\binom {m+q-2}{q-1}$ for every tensor power. Thus a semipositive reference of positive volume does not force the local gap between orders to be absorbed by fixed divisors. Whether such conditions are necessary in general is open.

24.10.3 Growing vanishing orders on curves5FCA86C5A588

Fix an integer $r\geq 1$ and, for each positive integer $m$, a nonnegative integer $q_ m$. The model of section 18.41 on $\mathbb C$ has the smooth weight $|z|^{2r}$ and the selecting ideals $(z^{q_ m})$. When $q_ m\to \infty $, the kernel vanishes in the natural rescaled limit, but it has a Gaussian transition at the larger radius $(q_ m/(rm))^{1/(2r)}$. For $q_ m=\lfloor m\lambda \rfloor $ with a real $\lambda >0$, the limiting kernel measure misses the atom at the pole of the limiting envelope current. On a compact curve with the curvature and local expansion of theorem 18.42.1, that theorem transfers the Gaussian transition to global sections when $q_ m=o(m^{1/(2r+1)})$, and the construction with curvature boundaries in theorem 18.42.15 extends the corrected transition to every sublinear order. We record the original question; it is answered by theorem 18.42.27 together with the corrections from the weight in subsection 18.42.30.

For the data of this problem, theorem 18.42.27 shows that the law holds in the optimal universal range $q_ m=o(m^{1/(r+1)})$, uniformly in the direction of observation; outside that range, a compact example obtained by a holomorphic change of coordinate violates the uncorrected law along a subsequence. For a fixed weight, the exact criterion is (18.42.29).

In the whole range $q_ m\to \infty $, $q_ m=o(m)$, theorem 18.42.15 centers the transition on the boundary of the curvature quadrature domain and uses the normal length scale $1/(2\sqrt{m\rho })$, where $\rho $ is the density of the curvature. The first effective Taylor coefficient and the ranges of the corrections of finite order are computed in subsection 18.42.30; a flat perturbation there shows that no fixed finite Taylor jet of the weight determines the center to within the width of the transition for every sublinear sequence. This resolves the problem on compact curves as stated.

In higher dimension, for ordinary point jets of growing total order on the explicit anisotropic product of section 18.44, theorem 18.44.2 proves a Gaussian transition throughout the sublinear range of orders. The correct radial center includes the nonzero remainder terms of the weight, whereas proposition 18.44.8 gives the sharp, smaller range in which the homogeneous center can be used in this class. Theorem 18.44.11 transfers these laws through a submersion with compact connected fibers, with a pulled-back metric and a matched smooth volume; there the isolated selecting weight varies with the tensor power. Mixed or angular remainders, observation near the coordinate faces, other bases and general transfers through fiber selectors remain open directions.