24.4.15 Null loci and the singularities of the Chern–Ricci flowF6F63D20E784

Return to the general manifold $X$ of dimension $n$. Let $\omega _0$ be a Hermitian metric, taken as the initial metric of the Chern–Ricci flow, and suppose that the flow has a finite maximal time $\tau $. With the Ricci convention of corollary 12.34.7, the limiting background $\theta _\tau =\omega _0-\tau \operatorname {Ric}(\omega _0)$ is nef. To identify the region where the flow converges smoothly as the complement of a null locus, one first needs bigness of $\theta _\tau $. The dual currents detect the failure of strict positivity, but the degeneration of their determinants does not by itself locate an analytic exceptional set.

For a big smooth background $\theta $, the non-Hermitian locus $E_{\mathrm{nH}}(\theta )$ is the intersection of the singular sets of the strict currents $\theta +\mathrm{dd}^{\mathrm{c}}u$, those bounded below by a positive multiple of a Hermitian form, whose potentials have almost analytic singularities; one such current attains the intersection. Here a potential has almost analytic singularities if it is locally a positive multiple of the logarithm of the norm of an ideal up to a bounded remainder, and this remainder becomes smooth after subtracting the divisorial logarithms on a suitable log resolution. This is the class of singularities used in the theorem stated in section 25.22.

For nef and big $\theta $, define

where bigness on a singular subvariety has its intrinsic analytic meaning. By the theorem recalled in section 25.22, the equality $E_{\mathrm{nH}}(\theta )=\operatorname {Null}(\theta )$ holds for nef and big smooth backgrounds on arbitrary compact complex manifolds. The numerical criterion recalled there has additional hypotheses: bounded mass on $X$ and positive lower volume on its positive-dimensional submanifolds.

The flow theorem recalled in section 25.22 identifies the finite-time singularities of the Chern–Ricci flow with the null locus under positive limiting lower volume, for arbitrary initial metrics in dimension two and for Guan–Li initial metrics in higher dimensions, that is, those with $\mathrm{dd}^{\mathrm{c}}\omega _0=0$ and $\mathrm{d}\omega _0\wedge \mathrm d^{\mathrm c}\omega _0=0$. Positivity of $\mathcal V_\mu $ cannot replace these hypotheses, as the example below shows.

The calculation with currents in corollary 12.34.7 applies to arbitrary initial metrics. When $\tau <\infty $, there are normalized Gauduchon currents $T$ annihilating $\theta _\tau $:

If $\mathcal V_\mu (\theta _\tau )>0$, the determinant-weighted measures of all such currents vanish, by proposition 12.34.2. This is a precise statement of degeneracy, but it does not imply that their traces concentrate on analytic sets, since a smooth positive semidefinite matrix can have deficient rank on a whole open set.

Theorem 12.34.10 shows that positivity of the variational volume does not imply bigness at the end of the flow: in every dimension at least three, there is a Chern–Ricci flow with finite maximal time whose limiting background is nef, not big, and has positive $\mathcal V_\mu $. Thus being the endpoint of a flow does not exclude the radial obstruction. In this model, proposition 12.34.12 shows that every normalized annihilating current has its trace supported on the degenerate fiber, with a vertical coefficient matrix of rank one. This support follows from the explicit semipositive representative, not from the vanishing of determinants alone.

If $\theta _\tau $ is big, whether by assumption or under additional hypotheses, the parabolic theorem recalled in section 25.22 gives convergence as currents and smooth local convergence off $E_{\mathrm{nH}}(\theta _\tau )$. Combined with nefness and the null-locus theorem, this identifies the analytic exceptional set. The example above does not contradict this conditional theorem, but any further numerical or geometric criterion must impose enough structure to exclude its nonbig endpoint. The local continuity of envelopes and the homogeneous replacement arguments of theorem 15.8.1 provide an elliptic foundation away from the poles of a strict barrier; the parabolic estimates are a separate input.

Completeness of the limiting metric requires an additional estimate. In theorem 10.24.1, the metric defined by the unique finite-energy solution of a divisorial exponential equation in a big closed class is incomplete on every nonempty connected open subset of its ample locus, because its minimal divisorial singularity cancels the prescribed pole. Nefness alone does not help: theorem 10.24.7 keeps a nonintegrable pole with smooth positive coefficient and a complete cusp, but has a disjoint null curve at finite distance; its solution has zero Lelong numbers and finite energy, but is not of minimal ordinary type. The construction of theorem 10.24.16 with compatible initial data also gives an incomplete auxiliary metric, with the exact identities for the resolution and the prescribed volume, but its class is not nef. Simultaneous nefness and compatibility remain a separate question, and none of these elliptic examples decides whether the limits of an evolving flow are complete; see section 25.19.

For integrable poles weakened by logarithmic factors, corollary 10.25.5 gives a sufficient condition for completeness in terms of the density, the potential and the tangential determinant; the proof controls normal lengths and applies in arbitrary Hermitian backgrounds. The torus models of example 10.25.11, example 10.25.12 attain its critical exponent and show that it is sharp, even for a Kähler background and full mass. Their pole coefficients need not extend smoothly and positively across the divisor, so the geometric smoothing question remains separate.

The regularity of the residual also matters. By theorem 5.39.1, a Hölder residual on a compact manifold that defines a smooth positive metric off its analytic singular set puts that set at finite distance. Continuity alone does not suffice: the prescribed radial densities of theorem 10.22.7 have bounded continuous minimal remainders and complete regular ends for the range $n<d\leq 2n$ of the exponent $d$ in that theorem. These examples concern the specified densities; they say nothing about compatibility with an evolving Chern–Ricci flow.