25.16 Section growth and partial kernelsCAFE5EFFA7C8

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The growth of sections links singular potentials to the algebraic dimension through a projective modification. The form of Moishezon's theorem used in section 19.9 is stated in [Introduction, p. 1332, Moi67]; it is used as a classical geometric input. A Kähler modification also gives the bounded mass conclusion of lemma 19.9.12, by Guedj and Lu's argument pulling back upper volumes [Lemma 3.6, GL22].

For section rings constructed through the algebraic reduction, see Xiaojun Wu's [Wu24]. The flag construction in section 19.9 uses a fixed compact flag on an auxiliary modification; it makes no assertion about the centers of arbitrary valuations of the field of meromorphic functions.

The maximum at maximal mapping rank in proposition 19.9.19 is Qin's [v1, Proposition B.1, pp. 35–36, QinCurvatureRank26]. The scalar and flag bounds in corollary 19.9.20 use the section-ratio field and period count already established in this book, with the curvature rank measured on its semipositive locus. The fixed-twist and endpoint deductions, Euler polynomial, surface estimate and sharp product follow [v1, Sections 10.2–10.3 and Example 11.1, pp. 32–34, QinCurvatureRank26]; they are independent of that paper's all-degree finite-jet estimate, Theorem A.

Principalization and its isomorphism off the base ideal are Włodarczyk's [Theorem 2.0.3, p. 35, and Lemma 4.0.3, p. 42, Wlo09]. The image dimension, proper mapping and Chow inputs are given with proofs in [Chapter II, Corollary 8.6, Theorem 8.8 and Theorem 8.10, pp. 118–121, DemCADG]. Serre duality uses the compact Hermitian setting of [Chapter VI, Theorem 7.3, p. 310, DemCADG]. The analytic Hirzebruch–Riemann–Roch formula used in proposition 19.9.23 is [Theorem (3.4), p. 163, Ati64]; it applies to compact complex manifolds and requires no Kähler hypothesis.

The formulas for derivatives in the weight and the Gevrey bounds in proposition 18.34.5, theorem 18.34.6, corollary 18.34.9 follow the mechanism for bounded weights of Marinescu and Xing [v1, Theorems 1.1–1.2, pp. 2–3, and Section 3, pp. 11–19, MarinescuXingParameter26], applied to the fixed space of selected sections on a compact manifold. The proof uses the inverse of a compressed Gram matrix and its finite jets, while the degree and the multiplier-ideal space stay fixed. The example example 18.34.12, concerning control at the boundary, is the explicit example on the disk on p. 3 of that preprint. Marinescu and Xing point to the earlier theory of bounded perturbations of Pasternak-Winiarski; its theorems involving the distance to the boundary require additional weighted openness estimates and are not used in these compact arguments.

The estimates for the probability measures in lemma 18.34.16, proposition 18.34.18, proposition 18.34.22 follow directly from the reproducing identities in finite dimension. Positivity of the symmetric coupling defined by the squared kernel gives the factor one half in the bound for infinitesimal variations, and integrating the logarithm of the odds of the probability of each Borel set gives a sharper modulus for the global oscillation; the smooth example of rank one, example 18.34.21, shows that this modulus is optimal as a supremum. Normalizing by the dimension gives a uniform radius of analyticity after rescaling. The calculation on a product in example 18.34.24 expresses the conditional fiber probabilities of section 18.40 as a perturbation of the norm weight of order inverse to the degree. Its nonzero limit of intrinsic measures and its unchanged limit of logarithmic potentials describe different parts of the kernel data.

In the fiber-integrability criterion of theorem 18.39.8, corollary 18.39.9, example 18.39.10, a radial calculation on a simplex detects the strict endpoint of integrability and the weighted gain $p\sum _ jb_ j^{-1}$, where $p$ and $b_ j$ are the exponents of the normal and the fiber variables in the anisotropic model of the selecting potential. A global lower bound and a single chart with an upper bound then give a bounded divisorial band, whose loss of sections is controlled by the dimension estimate lemma 18.39.1. This extends the isotropic theorem on sections vanishing along a divisor, without claiming that the condition is necessary for arbitrary selecting potentials.

Theorem 18.39.13 extends this argument to finite families of divisors with simple normal crossings. The calculation of the ideal on the base uses separated normal coordinates, and lemma 18.39.12 controls all the mixed twists arising in the restrictions. In the product model of example 18.39.14, integrability at points where several poles meet is established directly, by Fubini's theorem in distinct pairs of base and fiber coordinates.

The criterion on a modified base, theorem 18.39.16, combines the same argument with the transformation rule for adjoint ideals, lemma 5.43.22, and the pushforward identity for non-pluripolar products, proposition 18.35.1. The projective model with a point pole in example 18.39.17 makes explicit the Jacobian order $q-1$, where $q$ is the dimension of the base, and the positive residual on the blowup. The criterion assumes that the modification and its divisorial data are given; it does not assert that they exist for an arbitrary selecting potential.

Corollary 18.39.19 separates the curvature requirements in these proofs: a smooth semipositive residual gives a positive reference current, and a positive top-degree integral gives it the required mass, while the estimates for integrability and restriction need no strict positivity of the curvature. In example 18.39.20 the original base is a blowup. Its contracted lines prevent any choice of gauge from making the residual strictly positive, while Hartogs extension shows that, because of the fixed exceptional divisor, the selected space and the reference space coincide.

The calculation of coefficients in proposition 18.39.22 combines this Hartogs identification with the anisotropic local threshold; the remaining conditions are ordinary jet conditions at points of projective space. Exact counts of homogeneous monomials then identify the range of the references in which the mass is positive, the drop in degree at the boundary of this range, and the tensor power at which a more singular selecting potential removes every section.

The model with two points in example 18.39.25 keeps positive top-degree volume while the residual has degree zero on the curve joining the two centers. Its nonzero defect, of degree $q-1$, is computed by lemma 18.39.24, which constructs simultaneous jets from affine linear factors and finite Taylor division and requires the sum of the jet orders to be at most the projective degree plus one. Thus a sharp loss of sections and the obstruction to a smooth strictly positive residual occur in the same explicit family.

The metric on projective space in section 18.43 is the one in Chiang's [v1, Example I.15, pp. 9–10, ChiangKernel26]. The proof of proposition 18.43.3 computes the global norms of monomials directly, in the convention $|e|_ h^2=\mathrm e^{-\phi }$ for a local frame $e$ with weight $\phi $, and for a given integration volume; it uses neither the general spectral localization theorem nor the chain of expansions by Newton polyhedra of that preprint. The transition for growing jets on a compact curve, theorem 18.42.1, is the result obtained by comparison with a disk in the preceding section on curves, and its full extension to all sublinear orders, at the boundary of the region where the curvature vanishes, is theorem 18.42.15.

The correction on shrinking scales in lemma 18.42.13 uses the $L^2$ estimate with pointwise inverse curvature of [Theorem 5.2 and its singular-metric extension, pp. 34–35, DemAM11], whose Hilbert-space estimate comes from the preceding Theorem 5.1 and its proof on pp. 33–34. Applying the estimate to $L^ m\otimes K_ C^{-1}$, where $L$ is the line bundle on the compact curve $C$ and $K_ C$ is the canonical bundle of $C$, takes the Ricci curvature into account, and a bounded function varying on the scale $R$ of the flat region restores strictly positive curvature inside it. The particular free boundary in lemma 18.42.10 is constructed by solving an equation for Fourier coefficients in fixed Hölder spaces of finite order and applying the implicit function theorem in Banach spaces. The argument with Gaussian annuli then gives theorem 18.42.15, theorem 18.42.27 directly, including slowly growing orders, uniformity in the angle, sharpness on compact curves and the limitation of finite Taylor corrections.

On the explicit compact product in section 18.44, global orthogonal monomials turn ordinary jet conditions of total order into a selection by a half-space. The transition for one curve factor gives the marginal Gaussian laws used in theorem 18.44.2, and the calculation of moments at a fixed positive radius, together with the exact trace, gives convergence in total variation. The sharp radial correction in proposition 18.44.8 follows from the positive first remainder assumed there. Theorem 18.44.11 then uses the integrability calculation at an isolated pole and constancy along compact connected fibers to transfer norms and kernels exactly; the normalization on the base and the matching of volumes are part of its statement.

The selection by an interior Newton polytope in section 18.45 is set up with an integral affine offset, a polytope in the interior, a strictly convex invariant logarithmic weight and a positive invariant sampling probability on the whole compact manifold. Angular orthogonality gives exact norms, lattice approximation gives the lower bound for the envelope, and uniform concentration gives the shell measure in the relative dimension. The local argument with Hessian minors distinguishes this measure from mixed non-pluripolar curvature in lower dimension. The construction does not identify an arbitrary multiplier-ideal envelope.

The critical product of multiplier ideals in section 18.46 is computed directly from the integrability formula for smooth centers and the identity for an atom in the plane. Exhausting the curvature mass fiber by fiber identifies its multiplier-ideal envelope. Exact norms given by beta integrals, together with concentration, give two kernel limits depending on parity, whereas dividing by the exact dimension gives a single limit of probability measures. The calculation shows that a fixed growth exponent does not imply convergence of the leading coefficient in the count of sections, even when the growth dimension is positive.

For the selection by multiplier ideals at the base points of a pencil in section 18.47, the lower bound for Lelong numbers implied by inclusions of multiplier ideals at all weights, together with a radial convex maximum, identifies the full classes of competitors for both the multiplier-ideal and the ordinary envelopes, and restriction to the curves of degree two in the pencil controls the competitors that are not invariant. Exact norms of truncated Gaussians keep track of the three bands of finite degree, whose count gives the limiting intrinsic mass three. The bounds at the poles also give convergence in every finite $L^ p$ and uniform convergence of the exponentials, while excluding a uniform additive error on the whole manifold. The compact sampling box and the obstacle are part of the statement.