24.8.3 Polar curvatureB245A62A943C

Let $Y$ be a compact connected Riemann surface with a smooth positive form $\theta $. For $u\in \mathrm{PSH}(Y,\theta )$, the polar curvature $\mu _ u=\mathbf1_{\{ u=-\infty \} }(\theta +\mathrm{dd}^{\mathrm{c}}u)$ determines the ordinary model of $u$, and its atomic part determines the ideal model. By theorem 13.28.4, $d_\theta (u,v)$ is the total variation of $\mu _ u-\mu _ v$, also at zero non-pluripolar mass. By corollary 13.28.7, all normalized ordinary models form a complete geodesic space, and the ideal models form the closed subspace obtained by completing the analytic models of positive mass. On ideal models of positive mass this agrees with the topology of the source metric in theorem 20.3.21, corollary 20.3.22, for which this space is not totally bounded. Its completeness therefore has a different content from the compact completion obtained from weak convergence of traces.

Keeping only the atomic part of the polar curvature is a $1$-Lipschitz retraction onto the ideal models and selects the unique nearest ideal model; by proposition 13.28.11, the distance to the analytic closure is exactly the mass of the diffuse part of the polar curvature. The model of zero mass in example 13.28.14 has trivial multiplier ideals but positive distance from that closure. Convergence in total variation implies convergence in $L^1$ and in capacity, even at zero mass (theorem 13.28.8), whereas the moving pole of example 13.28.10 shows that convergence in capacity does not imply convergence in total variation.

Degeneration of the background is also explicit on curves. Between smooth backgrounds of positive area, theorem 13.29.3 gives correspondences of the model spaces that rescale by the ratio of the areas and are compatible with ideal projection, also at models of zero mass. Under uniform convergence of the backgrounds, the corresponding normalized representatives vary uniformly in $L^1$ and in capacity (corollary 13.29.7). When the limiting area is zero, theorem 5.29.23 shows that all normalized potentials converge to the unique limiting potential, uniformly over the admissible families, and the unscaled diameters of the model spaces tend to zero.

The joint space of theorem 13.30.3 treats the smooth background as a further coordinate. It is complete for the combination of smooth convergence of backgrounds and total variation of polar curvature, its analytic closure is exactly the ideal locus, even at zero area, and on each fixed background it restricts to the curve topology described above. In higher dimension, the realization of compatible singularity data and a replacement for the comparison of finite upper volumes are unresolved.

Finite curve maps give another compatibility within the curve theory. For $r:C'\to C$ of degree $d$, a smooth base form $\beta $ of nonnegative area and $\sigma =r^*\beta $, 18.24.1 commutes both envelopes with pullback below every pulled-back smooth obstacle. The polar metric scales by $d$, and 18.24.2 gives a $1$-Lipschitz retraction with polar measure $d^{-1}r^*r_*\mu $, preserving polar mass and commuting with ideal projection. By 18.24.3, this describes the closed complete subspace of models that descend to the original base of a compact curve map, with its original-base normalization. The cubic example shows that this averaging need not select a nearest model. These conclusions do not supply a complete topology for general higher-rank backgrounds.

In dimension at least two the polar part of the first curvature current loses information. For $u\in \mathrm{PSH}(X,\omega )$, write $\operatorname {Pol}_1(u)=\mathbf1_{\{ u=-\infty \} }(\omega +\mathrm{dd}^{\mathrm{c}}u)$, and $\operatorname {Pol}_\pi (u)$ for the corresponding current of $u\circ \pi $ on a modification $\pi $ of $X$. By proposition 18.26.2, distinct analytic ideal models with isolated logarithmic poles, whose multiplier ideals differ at suitable weights, all have $\operatorname {Pol}_1=0$. The blowup at the pole detects the coefficient on the exceptional divisor (corollary 18.26.4), but every fixed countable family of modifications misses some such models (corollary 18.26.5).

Tracing polar curvature against Gauduchon forms gives a uniform normalization over modifications. Fix a family $\mathscr R$ of smooth modifications $\pi \colon Y_\pi \to X$ and Gauduchon forms $\eta _\pi $ on $Y_\pi $. The comparison $\delta ^{\mathrm{pol}}_\omega $ of (18.26.7) is the supremum over $\pi \in \mathscr R$ of the total variation of the polar curvature of $u\circ \pi $ traced against $\eta _\pi ^{n-1}$, normalized by the trace of $\pi ^*\omega $. By proposition 18.26.8, it separates analytic models and, on curves, recovers the complete curve topology up to a constant factor. Bounded equivalence preserves polar currents (proposition 5.46.3), so transport by a smooth gauge change, followed by ordinary projection, preserves the comparison (corollary 18.26.10). Since polar curvature is closed for every pole set (proposition 5.46.7), equality of the scalar traces recovers the full polar currents on each modification (corollary 18.26.11). Arbitrary independent choices of the Gauduchon forms can nevertheless fail to induce the positive-volume topology, even for analytic ideal models on $\mathbb P^2$ (proposition 18.26.13). The space of polar trace coordinates is complete (proposition 18.26.9), and for sequences that are Cauchy for the comparison, proposition 18.26.12 gives compatible weak limits of the polar currents, all carried by the pole set of a single $\omega $-plurisubharmonic function. Whether these limiting currents are realized by an ordinary model is a further problem.

Even all modifications together do not make the first polar comparison separate models or give the right analytic closure. On every compact Kähler manifold of dimension at least two, proposition 18.26.17 constructs a nonzero ordinary model whose polar currents vanish on every modification, whose non-pluripolar volume is positive but strictly smaller than that of the zero model, and whose multiplier ideals are all trivial. More precisely, corollary 18.26.18 gives a complete ordered interval of such models with volumes $A-Mt^ n$ and mutual distances $M|s^ n-t^ n|$, where $A=\int _ X\omega ^ n$ and $M>0$ is a constant of the construction; for every $0<\delta <A$, the interval can be chosen inside $\mathcal M_\delta (\omega )$. Every first polar coordinate vanishes on this interval, and only its zero endpoint is ideal. Consequently every choice of Gauduchon forms identifies distinct models, and the closure of the analytic models for the polar comparison contains nonideal models.

The same diffuse cutoff can be placed in a Hermitian background. Proposition 18.26.19 separates the loss $Mt^ n$ in the mass of the potential from lower-degree terms coming from the background. If $\mathrm{dd}^{\mathrm{c}}(\omega ^ q)=0$ for $1\leq q<n$, the mass is the same for all bounded-equivalent representatives, and corollary 18.26.20 extends the whole interval of models to this Hermitian background; in particular the obstruction occurs on every compact complex surface with a Gauduchon form. For an arbitrary Hermitian background, proposition 18.26.21 computes the mass of each representative from finitely many torsion coefficients and the same loss $Mt^ n$. On a surface, corollary 18.26.22 shows that the mass is constant over the bounded-equivalence class exactly when $\omega $ is Gauduchon; otherwise even admissible smooth bounded perturbations change it, and the exact values would require optimizing these terms, which depend on the representative.

Under bounded mass and positive lower mass, proposition 18.26.23 still gives positive lower volume and a strict gap in upper volume for the diffuse ordinary models, while their first polar currents vanish on every modification. By the same proposition, this obstruction holds for every Hermitian metric on every compact complex surface. The theorem on preservation of residuals, theorem 13.19.6, distinguishes the diffuse parameters after ordinary projection. Consequently, in every background with both mass properties, corollary 18.26.24 gives a complete ordered interval of models throughout each range $0\leq t\leq t_0<1/C$, with $C$ the constant of proposition 18.26.19. Its upper-volume coordinate is strictly decreasing and Lipschitz, although the torsion terms prevent the explicit power formula from being inferred, and every first polar coordinate still vanishes. The same theorem preserves the entire residual measure, so its location matters: in proposition 18.26.25, two models carry residual mass $r$ in disjoint regions, have zero first polar data, and are at distance at least $2r$; when $\mathrm{dd}^{\mathrm{c}}(\omega ^ q)=0$ for $1\leq q<n$, their two singularity volumes also coincide. A complete extension must therefore distinguish this spatial residual information.

Even the full residual measure on the base is not enough. When $\mathrm{dd}^{\mathrm{c}}(\omega ^ q)=0$ for $1\leq q<n$, proposition 18.26.26 uses anisotropic maxima on one diffuse pole set to build a compact metric family of ordinary models indexed by positive weight vectors. Fixing the product of the weights gives complete subfamilies of real dimension $n-1$ whose members have the same residual measure, the same two singularity volumes, trivial multiplier ideals and zero first polar currents on every modification. The comparison distance nevertheless separates them, through the symmetric difference of the associated coordinate rectangles. A complete extension must therefore also detect singularity information lost by the full base residual. The proposition does not assert that the full residual products agree on every modification.

Return to a curve $Y$ with a smooth positive form $\theta $. On ideal models, the ordinary-type topology $\tau _{\theta ,0}$ is the topology of total variation of the atomic data, and its canonical metric includes the boundary of zero mass (theorem 13.28.17). On general ordinary models it is weaker than polar total variation: diffuse polar curvature can concentrate into an atom in the limit of the envelopes, which produces the Cauchy sequence without an ordinary-type limit in proposition 13.28.19. Characterizing the analytic closure on the whole ordinary space and extending the model topology to a complete one are therefore separate requirements.

Any such extension must keep the singularity information that weak limits lose. Potentials of infinite mass are already covered by $\tau _{\omega ,0}$: for instance, the potential of proposition 18.5.5 has infinite mass, while its canonical envelopes, the ordinary and the ideal one, both equal zero. On curves, zero volume does not prevent completeness, but in higher dimension the stretching argument still needs a positive lower bound for the volume. There is nevertheless a restricted locus where a complete metric is known at zero volume. On the divisorial product locus of proposition 18.15.8, polar curvature on the curve factor gives a complete metric whose analytic closure is exactly the set of ideal models; the top-degree volume comparison vanishes identically on this locus and therefore cannot produce this metric. The same restricted metric and analytic closure exist in a positive nonclosed background on a projective manifold (corollary 18.15.9), and the locus, the metric and the ideal projection persist when the second curve coefficient vanishes and the limiting background is not big (corollary 18.15.10). Proposition 18.16.5 gives a complete sum metric, with the same properties of ideal projection and analytic closure, on the locus induced by several independent curve models, and corollary 18.16.6 preserves this locus and metric through a nonclosed auxiliary coefficient and its nonbig limit at zero. When the curve coefficients vary, proposition 18.17.3 gives compatible weighted metrics, discarding vanishing coordinates, with uniform $L^1$ collapse when all curve areas vanish, and proposition 18.17.5 compares these metrics quantitatively across different sets of active coordinates, with the exact distance for ordered coefficients. Proposition 18.18.3, Corollary 18.18.5 extend the construction to arbitrary smooth admissible curve backgrounds and identify the limiting singleton, and proposition 18.20.2 preserves the restricted complete metric through arbitrary compact complex auxiliary factors with semipositive vertical terms, independently of bounded mass on the ambient manifold.