25.17 Gauduchon variational volume3747CBC0FB2D

3747CBC0FB2D

The Gauduchon variational volume extracts a homogeneous concave quantity from an elliptic normalization problem. The method goes back to Collins's construction [Theorem 2.13, Collins21]; section 12.1 develops the adjoint formula and its dependence on the chosen probability measure.

For a Kähler class represented by a closed form $\theta $, this construction meets the Hessian geometry of the Kähler cone: the metric of section 12.4 is $1/n$ times the Hessian of $-\log \int _ X\theta ^ n$ used by Magnússon [Section 1, Magnusson12]. Completeness and boundary points of positive volume at finite distance are studied in [Proposition 4.4 and Lemma 4.5, Magnusson12]; section 12.6 gives the argument by homogeneity in the present normalization.

The cones of invariant nonclosed forms in subsection 12.6.4 are computed directly. Stokes' theorem on the compact quotient shows that invariant $(n-1,n-1)$-forms are closed, and the arithmetic–geometric mean inequality for matrices then determines the quotient and its normalized minimizer. The resolvent argument of lemma 12.6.9 and the normalized chord in a prescribed linear subspace give the completeness statement. The Iwasawa example comes with an explicit compact quotient and a proof, by averaging coefficients, that this quotient is not Kähler. The initial construction uses the normalized invariant probability measure. In dimension three, proposition 12.6.22, corollary 12.6.25 follow from the formula for the variation of mass proved in the book and from the polynomial continuity of bounded products. On the Iwasawa quotient, theorem 12.6.26 extends the determinant formula to probability measures invariant along the central fibers, by means of a closed equation on the base for the Schur complement. The exact vertical Dirichlet defect characterizes all maximizing smooth densities, and the completion estimates for this larger family concern the means of the coefficients of the minimizing forms.

For the warped cones in subsection 12.6.31, a closed equation on the base and an explicit balanced cofactor give the full quotient. The pullback of the affine Hessian metric becomes flat, and the variance of the trace at the endpoints gives the three alternatives: a strip, a half-plane or the whole plane. Smooth existence for the equation on the closed base is theorem 14.2.1, and integration fixes its scalar; the warped background on the product may be nonclosed. That the finite faces are not big is tested against closed top-degree forms on the base, so the conclusion applies to singular current representatives as well. In the computations on projective space and on periodic tori, the density and the cone are given explicitly throughout.

The cones of currents in proposition 12.6.43 are determined by localized closed test forms and Stokes' theorem for distributions. At the finite faces of the warped cone, proposition 12.6.44 applies theorem 14.19.2, the existence theorem for continuous solutions with $L^ p$ densities, on the closed base; it fixes the scalar by invariance of the mass under bounded perturbations and expands the pulled-back products. The obstruction to global $C^{1,1}$ solutions is the general bound of currents by determinants in proposition 5.28.6, derived from the identity for coefficients in proposition 5.28.5.

The convergence of the whole sequence in proposition 12.6.47 combines bounds for the reciprocal densities with the compactness for a fixed background in theorem 14.19.21. Since the base is closed, the uniqueness theorem for invariant mass, 14.25.2, identifies every cluster point of the potentials. For any bounded competitor at the endpoint pulled back from the base, the density forced away from the extreme set already exhausts the total mass of the closed class, so positivity excludes residual mass on that set before uniqueness is applied.

The alternatives for the trace in proposition 12.6.49, proposition 12.6.51 follow from the explicit balanced cofactor and from integrals of reciprocal squares over tubes, while the mass on the closed base controls the remaining cofactor term. When the trace diverges, its normalization concentrates on the extreme components where the ratio is minimal; if there is a single extreme point, the whole sequence converges to a fiber current. Both the smooth numerator and the denominator then tend to zero, while their quotient remains equal to one.

In section 12.12, bigness of the closed nef core with positive top self-intersection follows from Demailly and Păun's criterion [Theorem 0.5, DP04], as recalled in [SGPT]; no such criterion for arbitrary nonclosed nef backgrounds is used.

On the warped paths of section 12.13, the same balanced cofactor as in subsection 12.6.31 computes the cost for every minimizing test. The finite improper identity follows from the first variation and the actual nef limit, while the five rates of proposition 12.13.4 come from the first and second inverse moments of the fiber coefficient. The exclusion of every closed nef positive-volume core in proposition 12.13.6 uses the smooth-test obstruction to bigness and Demailly–Păun's [arXiv v2, Theorem 2.12, pp. 12–13, DP04], together with the global potential consequence of the Kähler $\partial \bar\partial $-lemma. The constant closed case gives exactly the expression in Tosatti's Morse inequality [v2, Theorem 1.1, p. 1, Tosatti26]; this comparison does not assign a closed cohomology class to a nonconstant warped coefficient.

In the radial model on a projective manifold in section 12.14, a global bounded potential compensates the coefficient along the fibers and gives the lower bound against every smooth Gauduchon test, and explicit balanced metrics give matching values after perturbation. The test against closed top-degree forms on the base in theorem 12.14.6 excludes every quasi-plurisubharmonic potential whose curvature current is strictly positive. This disproves the conjecture that positive volume implies bigness without further restrictions; its versions for closed forms and for surfaces remain separate questions.

The limits of test forms in subsection 12.14.9 are computed from these explicit balanced metrics by Stokes' theorem for closed forms and radial integration. In dimension three the powers normalized by the trace concentrate on the degenerate fiber at a logarithmic rate, whereas in higher dimension their threshold quotient stays positive. Localized closed forms on the base, with a small Kähler correction, give a different balanced family approaching the fiber in every dimension of the model.

The initial metric of the Chern–Ricci flow in 12.34.8 is chosen so that, up to a smooth global change of gauge, the radial background is the limiting form at the maximal time, and the maximal-time theorem determines this lifetime. In the normalization $\operatorname {Ric}=-\mathrm{dd}^{\mathrm{c}}\log \omega ^ n$, time one corresponds to time $1/(2\pi )$ for the Chern–Ricci form $-2\pi \mathrm{dd}^{\mathrm{c}}\log \omega ^ n$ used in this book. A test against a closed form on the base excludes every positive representative at later times, and the support and the vertical rank of all annihilators at the endpoint follow directly from the semipositive representative and the formula for the trace matrix. The averaged trace is a special case of the general identity for the flow proved before.

In the criteria of 12.10.22, invariance of the mass of bounded potentials and the support of the measure on the contact set prevent the stretched obstacle envelopes from collapsing, and the sublinear condition provides competitors. The uniform bound for Lelong numbers in lemma 5.38.8 then gives vanishing Lelong numbers by scaling. The argument with smooth roots of rank two extends the scalar first-order equation across the analytic set, and the support theorem for currents carried by analytic sets, recalled in section 5.38, removes its residual current. The calculations on Hopf manifolds and quadrics give examples in which the dual problem is attained. The general implications involving moments and entropy remain separate questions.

In the example with a point pole, proposition 12.10.33, radial integration shows directly that the potential has full mass, and the equality case of the arithmetic–geometric mean inequality for matrices, combined with the local strong maximum principle, excludes every smooth semipositive maximizing test form. The argument does not need to identify the potential with the greatest threshold envelope. The thresholds for entropy and weighted integrability in corollary 12.10.35 follow directly from the computed radial density. They concern the density of this single equation, whereas the question of uniform moment or entropy conditions concerns the determinant-weighted probability measures of a maximizing sequence when the supremum is attained by a smooth semipositive test.

The bound for the determinant at the endpoint, theorem 12.9.2, is derived within the book from the criterion for mass invariance and the quotient by the lower mass; related criteria for the mass are [Propositions 3.2 and 3.3(iv), GL22]. The concentrating family in section 12.18 distinguishes the unique smooth maximizer for one density from maximizing sequences whose higher moments diverge.

The coordinate formula (12.18.7) identifies the intrinsic determinant measure on singular and degenerate positive currents. The proof of theorem 12.18.8 applies Tosatti and Weinkove's [arXiv v2, Corollary 1, p. 1, TW10] separately to each smooth positive root. The source uses $\mathrm i\partial \bar\partial \varphi $; the potential is multiplied by $2\pi $ here, and theorem 11.1.7 fixes the volume coefficient. The last paragraph of that source, p. 8, refers the existence argument to Cherrier and Demailly or the authors' earlier paper. No estimates uniform in the varying right-hand sides are used.

The closed and nonclosed torus families in theorem 12.18.13, theorem 12.18.15 preserve the complete determinant data and positive mass extrema while ellipticity degenerates. For the nonclosed family, the lower-power identities are verified on the two-coordinate block, retaining its fixed nonzero torsion. In theorem 12.18.17, the Poisson kernel gives uniformly convergent continuous potentials and exact maximizers for varying roots. The limiting current has rank-deficient absolutely continuous and singular coefficient parts, so its determinant measure vanishes although the bounded potential has positive Monge–Ampère mass. These calculations concern scalar perturbations and current limits; they supply no flow existence or convergence theorem.

The identity (12.17.3) of lemma 12.17.2, for bounded potentials in a background $g$ with $\mathrm{dd}^{\mathrm{c}}(g^2)=0$, uses the calculus of bounded products and of weighted decreasing limits in proposition 5.28.2, lemma 5.27.2, whose classical local foundations are [Section 2, Theorems 2.1 and 2.4, pp. 4–9, BT82]. That the sign $\mathrm{dd}^{\mathrm{c}}g\leq 0$ gives positive lower mass is part of the criteria in terms of the sign of the torsion of Angella, Guedj and Lu [Theorem 3.2, pp. 4641–4642, AGL23]. The curvature calculation in theorem 12.17.7 identifies this sign on principal elliptic threefolds, and the two quotient constructions give explicit mass constants in the present convention for the operator. The conclusion for the determinant of the full current then follows from the endpoint theorem proved in the book and a change of background.

The estimate for local families in theorem 12.9.7 controls the solution form and its Gauduchon factor by two contractions. The elliptic input is the parametrix and Fredholm argument on compact manifolds in Taylor [Chapter 7, Sections 4 and 10, especially (10.4)–(10.6), Taylor7], together with the mapping properties on Hölder spaces in [Chapter 13, (8.7), (8.17) and (8.22), Taylor13]. Invariance of the index is [Section 7, Proposition 7.4, TaylorFA], and smooth regularity for nonlinear elliptic equations is also formulated in [Proposition, TaylorReg]. The estimates for the equation and for the normalized factor give explicit constants for the densities without assuming positive lower mass; instead, the data are restricted to smooth functions in a small $C^{2,\alpha }$ neighborhood.

The factor bound in proposition 12.9.15 starts from the conformal theorem of Gauduchon [Théorème 2 and its proof, pp. 132–133, GauduchonStandard85]. Its general statement has no restriction on the first Betti number; the even-Betti assumption belongs to the surface theorem at the beginning of that article. The smooth maximum principle is recalled and applied in [pp. 131–132, GauduchonStandard85]. The quantitative estimate here writes the cofactor equation in real divergence form and proves its Harnack bound by power iteration and a dyadic mean-oscillation argument. Since $d(\omega +\mathrm{dd}^{\mathrm{c}}v)=d\omega $, the vector coefficient is controlled by the fixed torsion. This supplies the exact-adjoint bounds of theorem 12.9.18 without derivative bounds on the varying solution form. Gauduchon uses $d^ c=\mathrm i(\bar\partial -\partial )$ and $L=-\operatorname {tr}(dd^ c)$; the positive factor equation is unchanged by this scaling and sign conversion.

The classical analytic input in section 12.22 is Hironaka's elimination of indeterminacies: every modification is dominated by a composition of blowups with smooth centers. The argument by principalization and domination is explained by Abramovich, Karu, Matsuki and Włodarczyk [Sections 1.2.3–1.2.4, pp. 538–539, and Section 5.4, p. 564, AKMW02], where the analytic case is treated explicitly. The curvature of the exceptional divisors then gives the uniform bound for the pairings used here; no factorization into a sequence of blowdowns is needed for this estimate.

For the fixed projective cones in section 12.25, every Gauduchon test is compared with a closed form of constant rank through an explicit logarithm of an exceptional divisor, which gives the exact loss of rank when the prescribed effective divisors are removed. For a general modification whose source is Kähler, proposition 12.25.11 obtains in addition two-sided bounds by powers from the intersection polynomial for closed backgrounds in section 12.20. Fixed coefficient functions that vanish, cones of residuals without restrictions, and sources that are not Kähler lie outside this construction.

The theorem that the line bundle of a smooth anticanonical curve of self-intersection zero is unitary flat near the curve, theorem 12.27.3, is due to Chen, Filip, Sun, Tosatti and Zhang [Theorem 1.1, with the standing degree-zero hypothesis in Section 1.1, CFSTZ26]. Its proof uses their identification of the holomorphic fibers in Proposition 4.1, the elliptic inverse independent of the holonomy in Lemma 5.2, and the weighted horizontal inverse and contraction of Propositions 5.4–5.5. The cutoff in proposition 12.27.5 is the construction at the end of their proof in Section 6. The variational bounds in rank one follow from proposition 12.26.1, and the examples obtained by blowing up nine points on a plane cubic, together with their torsion criterion, follow by restriction to the cubic. Only this local theorem and its direct consequences are used, not the global classification of surfaces in that paper.

The pencil in the case of finite holonomy and the rigidity in the non-torsion case, proposition 12.28.2, theorem 12.28.3, give the case of a smooth reduced divisor of Koike's [Lemma 3.5, Koike26]. The Levi-flat level sets of the flat norm have a smooth suspension model, so the density of their leaves and the maximum modulus argument are proved directly. The hypothesis is that the line bundle of the divisor is unitary flat near the smooth compact divisor; the global criteria for semipositivity and the dichotomy between fibrations and the Hartogs phenomenon in Koike's paper are not used. Together with the theorem on anticanonical neighborhoods, these local results explain the non-Stein and Hartogs properties of the nine-point examples with non-torsion normal bundle.

The quotient construction in proposition 12.29.3 follows the mechanism of invariant fibers of Denisi and Höring [v1, Lemma 2.10, p. 6, Proposition 4.2, p. 12 and pp. 14–18, and Theorem 5.1, p. 19, DH26]. Their action has $\kappa =2\pi \mathrm{i}\rho $, where $\kappa $ is the parameter of proposition 12.29.3 and $\rho $ is theirs, and their function $\mu _ Q$ equals $1+2\sigma $, with $\sigma $ the function defined before that proposition. The Laurent argument in lemma 12.29.1 proves directly the obstruction for punctured line bundles used in their Example 2.1, while example 12.29.2 computes the degree of an explicit factor of automorphy by its invariant metric. The statement about the quotient includes the two exceptional parameters, which are treated with compact elliptic curves. The computation of classes in theorem 12.30.4 gives their Proposition 4.2 and Lemmas 4.5 and 4.7–4.8, with explicit generators and conventions for projectivization. The flat unitary character in lemma 12.30.2 treats line bundles of every positive degree $d$ directly; the coefficient of the quotient is then $\kappa =2\pi \mathrm{i}d\rho $. This gives their non-Stein conclusion for split elliptic ruled surfaces without the reduction to a finite cover in their Lemma 5.2.

For the split elliptic ruled surface $M$, the algebra $\bigoplus _{r\geq 0}H^0(M,\operatorname {Sym}^ rT_ M)$ in theorem 12.31.4 has the bigness and nonfinite generation stated in [v1, Theorem 1.4, p. 2, with proof on pp. 9–10, DH26]. The decomposition by polynomial weights in lemma 12.31.1 gives the exact dimensions and the tautological volume in theorem 12.31.3, and its map to leading coefficients gives a direct proof of nonfinite generation through a quotient by an ideal of square zero. For canonical extensions, [v1, Remark 3.8, pp. 10–11, DH26] also states bigness and nonfinite generation, leaving the proof to the reader. The homogeneous quadric in lemma 12.32.1 and the decomposition of coefficients under the unipotent action give the description as a polynomial algebra in theorem 12.32.2. The exact dimensions, the tautological volume and nonfinite generation in corollary 12.32.3 follow for every extension class that is nonzero on the fibers of the ruling, including all Kähler canonical extensions on the split surface. The case of an extension class vanishing on the fibers of the ruling, proposition 12.33.2, follows directly from the ordinary cotangent quadric and the same decomposition of coefficients. Its section algebra is the same as in the case of nonzero restriction, while proposition 12.33.3 identifies the universal cover of the corresponding torsor, which is not Stein, as the product of $\mathbb C^2$ with the total space of $\mathcal O_{\mathbb P^1}(-2)$.

The question whether a weak Monge–Ampère power admits a scalar conformal factor is raised by Lu [Chapter 3, Section 5, LuHDR22]. The representation by finite currents in theorem 12.8.2 gives a supporting probability measure without defining that weak power or its scalar factor.