24.6 Degeneration orders and finite-distance boundaries7B777ACFF4E8

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Approximating a nef background by positive ones yields two kinds of boundary data: the rate at which the variational volume vanishes, and the distance traveled in the Hessian metric of section 12.6. These need not detect the same boundary points. Throughout this section $\mu $ is a smooth strictly positive probability form on $X$.

For a nef background $\theta $ and a Hermitian form $h$, the lower and upper degeneration orders $\underline a(\theta )\leq \overline a(\theta )$ of proposition 12.20.1 are the lower and upper limits of $\log \mathcal V_\mu (\theta +\epsilon h)/\log \epsilon $ as $\epsilon \downarrow 0$. They do not depend on $\mu $ or on the perturbation direction $h$, but they need not coincide. On a fixed product of two elliptic curves, proposition 12.21.5 realizes every pair $(1+\alpha ,1+\beta )$ with $0<\alpha <\beta <1$, as well as irrational common orders, and proposition 12.21.4 gives zero boundary volume with decay slower than every positive power of $\epsilon $. Equality of the two orders and their integrality therefore require additional geometry, as in the closed Kähler case, where the order is governed by the numerical dimension.

Pulling back by a modification can only increase both orders (proposition 12.22.5), and a single blowup can increase them strictly, or remove positive limiting volume while both orders remain zero. Subtracting a positive current gives the same one-sided comparison when the residual stays nef. The perturbative minimal multiplicities of a nef background vanish, whereas in corollary 12.24.5 the boundary class contains a unique positive current, the current of integration along a divisor; it has minimal singularities, and removing it changes the order. The remaining questions, collected in 12.33.4, concern equality of the orders and geometric restrictions on these behaviors.

The Hessian metric gives boundary information of a different kind. Let $V$ be a finite-dimensional space of smooth real $(1,1)$-forms containing no nonzero form $\mathrm{dd}^{\mathrm{c}}f$ with $f$ smooth, and let $\mathscr C$ be the open cone of elements of $V$ that admit a smooth positive representative. By proposition 12.6.1, the metric $-D^2\log \lambda _\mu $ on $\mathscr C$ splits isometrically as the product of the logarithmic volume direction with the unit-volume hypersurface $\mathscr S=\{ \lambda _\mu =1\} $; consequently every path along which the volume tends to zero or to infinity has infinite length. Nevertheless, the blowup example of example 12.6.3 has boundary points at finite distance, represented by nef classes of positive volume. Completing the metric cone therefore cannot mean simply adjoining all nef classes, nor declaring every nef boundary point infinitely far away.

Problem 24.6.1.B93ADA1B6064 Let $\mu $ be a smooth strictly positive probability form on $X$, and let $\mathscr C$ be a cone of backgrounds in a specified finite-dimensional space $V$ as in section 12.6, with the Hessian metric $-D^2\log \lambda _\mu $. Describe the metric completion of its unit-volume hypersurface $\mathscr S=\{ \theta \in \mathscr C:\lambda _\mu (\theta )=1\} $. Which boundary points admit a current representative, and which estimates for the minimizing Gauduchon metrics distinguish boundary points at finite distance from those at infinite distance?

On a compact quotient of a complex Lie group, with $\mu $ the normalized invariant probability, theorem 12.6.12 answers the question for every fixed linear space of invariant backgrounds containing a positive form: the unit-volume hypersurface is complete, and its Cauchy sequences are exactly those whose coefficient matrices converge to a positive matrix. After normalization, nonzero singular limits of the coefficients lie at infinite distance. These assertions depend on the invariant probability and do not classify arbitrary cones of backgrounds or reference densities.

On the Iwasawa manifold, theorem 12.6.26 extends the same coefficient metric and completion to every smooth reference density that is invariant along the central elliptic fibers; these are exactly the probabilities that maximize the variational volume of each invariant positive background. The minimizing forms need not be invariant, but their coefficient means still satisfy the spectral distance bounds. A reference density that varies along the fibers gives smaller volume, and the completion of its Hessian metric is a separate question.

For the warped product cones of subsection 12.6.31, on the product of a compact Kähler manifold with a curve, theorem 12.6.41 shows that the completion of the unit-volume hypersurface is a closed strip, a closed half-plane or the whole plane. Which ends lie at finite distance is decided by two ratios of real codimension to vanishing order, attached to the minimum set and the maximum set of the warping coefficient. Every end at finite distance is represented by a smooth semipositive, nef, nonbig form of volume one. The length estimates of lemma 12.6.39 are bounds for the variance of the traces of the balanced minimizers; they use the product probability and extreme sets consisting of finitely many disjoint strata with power-type vanishing. Other densities and other geometries of the extreme sets remain to be studied.

The cone must be specified in 24.6.1, since allowing additional directions can shorten distances along paths. The finite-dimensional calculations above do not define a distance on the whole infinite-dimensional quotient of backgrounds modulo $\mathrm{dd}^{\mathrm{c}}$ of smooth functions.