25.22 Geometric problemsBCAE26692B2F

BCAE26692B2F

A closed strictly positive current has geometric consequences that a positive nonclosed background need not have, and this distinction governs the passage from the normalization of equations to the geometric criteria below. The characterization of Fujiki's class by Kähler currents used in section 24.4 is Demailly and Păun's [Theorem 0.7, DP04], and the transcendental Grauert–Riemenschneider conjecture is [Conjecture 0.8, DP04]; in the arXiv version of that paper these are Theorem 0.6 and Conjecture 0.7.

For nef classes with positive top self-intersection, the equivalence between bigness and bounded mass cited in section 24.4 is Guedj and Lu's [Theorem 4.9, GL22]. The characterization of Fujiki's class by bounded mass and positive upper volume of the class is [Theorem C, BGL25]; that upper volume is different from the Gauduchon functional for a fixed probability measure.

The transcendental Morse inequality for closed classes on compact Kähler manifolds, and its consequences for differentiability of the volume and duality of cones, mentioned in section 24.4, are Tosatti's [Theorem 1.1 and Corollary 1.2, version 2, Tosatti26]; they are quoted from the literature, and their proofs are not reproduced. The qualitative criterion under bounded mass is [Theorem 4.12, GL22].

The non-Hermitian locus in section 24.4 is that of Dang [Definition 2.12 and Theorem 2.14, DangNull26], where almost analytic singularities have the precise meaning of [Definition 2.8, DangNull26], including a smooth remainder on a suitable log resolution. Dang's qualitative identification of this locus with the null locus, for nef and big backgrounds that may be nonclosed, is [Theorem 1.1, DangNull26]. Theorem A of the same paper gives a numerical criterion under different hypotheses: bounded mass on the ambient manifold and positive lower volume on its submanifolds of positive dimension.

Theorem C of [DangNull26] identifies the singularities of the Chern–Ricci flow at a finite time with the null locus when the limiting lower volume is positive, for arbitrary initial metrics in dimension two and for initial metrics of Guan–Li type in higher dimensions, that is, initial metrics $\omega _0$ with $\mathrm{dd}^{\mathrm{c}}\omega _0=0$ and $\mathrm{d}\omega _0\wedge \mathrm d^{\mathrm c}\omega _0=0$. Once the limiting background is big, [Theorem 4.3, DangNull26] gives convergence as currents and smooth local convergence away from its non-Hermitian locus. This conditional parabolic input should be distinguished from the implication from positive volume to bigness without further hypotheses, which fails by theorem 12.34.10.

The theorem on the maximal existence time, theorem 12.34.4, is Tosatti and Weinkove's [Theorem 1.2, (1.4), and Section 4, arXiv version, TW15]. The proof given here combines the positive affine path and time-weighted estimates for the potential with lemma 14.5.1 and classical parabolic regularity. In the conversion to the book's conventions, potentials are multiplied by $2\pi $ and time is unchanged. The conditions for geometric smoothing in [Section 3.1, DNGL26] include the further compatibility discussed in section 25.19.

Distance estimates for a twisted flow also depend on its parameters. In Section 5.2 of [v2, p. 18, Dang26] the twist is fixed to be $\operatorname {Ric}(\omega _ X)$, for the reference metric $\omega _ X$ of that paper, and Theorem 5.7, p. 21, there allows the constants to depend on the endpoint of a finite time interval. The flat flows of example 12.34.14 show why the formulation for arbitrary twists in Theorem C of the introduction of that paper needs further control: a fixed initial density and a fixed lifetime do not bound the diameter independently of the twist, and a single positive twist can give unbounded diameter over an infinite lifetime.

The conjecture on balanced and pluriclosed metrics referred to in section 24.4 is stated by Fino and Vezzoni [Conjecture 1.1, FV26]; there the balanced and the pluriclosed metric are allowed to be different. That a compact manifold in Fujiki's class admitting a pluriclosed metric is Kähler is Chiose's [Theorem 0.2 in the author manuscript, Chiose14]; the locator refers to Chiose's manuscript version of that paper.

The case of parallel torsion, theorem 24.4.20, is a theorem of Chen and Zheng [v1, Theorem 1.2, pp. 2 and 8, CZParallel26]. Their projection by integration of parallel forms and their description of the holonomy are [v1, Lemmas 3.1–3.2 and Propositions 3.3–3.5, pp. 3–6, CZParallel26]; lemma 24.4.17 proves both descriptions again in a form that keeps track of positivity and of commutation with the differential. The slice of fixed trace and the maximum of the volume are [v1, Propositions 4.2 and 4.5, Lemmas 4.3–4.4, pp. 7–8, CZParallel26]. The inverse of the positive cofactor map proves the step of taking roots directly, and lemma 24.4.19 proves, by linear algebra of primitive forms, that a Hermitian form which is both balanced and pluriclosed is closed. The additional obstruction of Zhao and Zheng for non-balanced metrics with Bismut-parallel torsion, used in their Corollary 1.3, is a separate ingredient.

The conjectured long-time existence and convergence of the pluriclosed flow on balanced manifolds with $c_1(X)=0$ is [Conjecture 1.2, FV26]. The discussion in section 24.4 distinguishes a conserved balanced trace from a pointwise bound for the metric.

The persistence of pluriclosed metrics under deformations, theorem 24.4.28, which assumes that the Aeppli number $h_ A^{1,1}$ is constant, is Cavalcanti's [v5, Theorem 8.11, Section 8.3, pp. 41–42, CavSKT13]. The fourth-order Aeppli Laplacian and the realization of Aeppli classes by harmonic forms used in lemma 24.4.24 are due to Schweitzer [v1, Section 2.c, pp. 9–10, SchAeppli07], who constructs the elliptic symbol in Section 2.b, pp. 7–8. The proof given here writes the projection onto $\partial \bar\partial $-closed forms explicitly and obtains its smooth dependence on the parameter from the resolvent and the constancy of the dimension of the kernel. It applies to any finite-dimensional smooth parameter space and keeps the prescribed initial metric. This is also the analytic input stated in [v1, Theorem 2.1, p. 7, BarDef26]; Barbaro's construction of deformations of Lie groups there requires a separate argument with transverse cohomology.