25.5 Gauduchon metrics and positivity3C68E80772A5

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Gauduchon metrics replace the cohomological conservation of mass by a linear trace identity. In the form recalled by Tosatti and Weinkove immediately before [Lemma 2.3, TW10], Gauduchon's conformal theorem starts, in dimension $n\geq 2$, from a Hermitian form $\eta $ and constructs a smooth real function $f$ such that $\gamma =\mathrm e^ f\eta $ is Gauduchon; the multiplier in theorem 11.4.2 is then $\rho =\mathrm e^{(n-1)f}$. The resulting trace comparison, corollary 11.5.3, also underlies the first mixed-volume bounds of Guedj and Lu [Proposition 3.3, GL22].

For lemma 11.24.1, the constant-coefficient estimate is the order-zero multiplier theorem in Taylor [Chapter 13, Theorem 5.1 and its proof, pp. 16–21, Taylor13]; compactness is [Chapter 13, (6.9), p. 24, Taylor13]. Freezing the continuous coefficients gives the Sobolev estimate, and the proof writes out the classical Alexandrov–Bakelman–Pucci contact-set bound and tangent-ball maximum principle before applying the method of continuity and compact-perturbation index theory. It does not differentiate the coefficients.

The trace identity has a dual form: positivity can be tested against Gauduchon metrics. For an arbitrary smooth background, the generalized Lamari criterion theorem 11.6.1 is stated by Boucksom, Guedj and Lu [Lemma 1.15, BGL25]; its proof combines a separation argument with a right inverse of a scalar elliptic operator. In theorem 11.8.1, Lamari duality combines with normalized compactness and Siu analyticity to turn bounded transverse test pairings into a global hypersurface. The regularized logarithmic potentials in proposition 11.8.2 give the converse bound when a Cartier divisor agrees with the cutoff hypersurface throughout its chart. The same linear control gives the bound (11.5.6) for the normalization constant, which is used in the weak existence arguments of [Section 4, BGL25].

The deformation obstruction for convex norms in theorem 11.9.4 is due to Liu and Wan [v1, Theorem 3.2 and Sections 3.1–3.3, pp. 9–13, LiuWanKobayashi26], and the normalization of line metrics in lemma 11.9.1 is their Lemma 3.1, p. 8. The proof combines derivative estimates along the line axes with an average of the horizontal curvature and the formal adjoint on a fixed sphere bundle. The resulting $C^4$ neighborhood is uniform over all smooth complex norms, and the positive Gauduchon test forms may differ from one line summand to another.

The regularization of homogeneous functions in theorem 11.10.4 is the case of a compact base of a lemma of Kuang-Ru Wu [v1, Section 2.3, Lemma 13, pp. 10–13, WuFinslerIII23]. Its gluing step uses the closure under maxima, the stability under smooth perturbations and the semilocal approximation of Greene and Wu [Definitions 1.1, 1.3–1.4, pp. 48–49; Lemma 4.1 and Theorem 4.1, pp. 80–81, GreeneWu79], while the local positivity properties are proved directly from a holomorphic shear formula. The replacement in proposition 11.10.5 assumes that a strictly plurisubharmonic homogeneous function is given, and corollary 11.10.6 keeps the hypothesis of local strong convexity in the fibers.

The local example example 11.10.1 adapts the quadratic maximum of Fukuoka and Setti [v1, Example 3.18, p. 20, FukuokaSettiStrongConvex26] to a complex rank-two bundle over a disk. A geometric series of squared holomorphic moduli gives its total-space plurisubharmonicity. Its fiberwise quadratic margin tends to zero at nearby base points, showing why local strong convexity in theorem 11.10.4 requires a common margin on a total-space neighborhood. The example concerns this distinction between hypotheses; the general real-Finsler smoothing and connection-convergence theorems of that paper are not used.

The existence result of Liu and Wan [v1, Theorem 2.1, p. 7, LiuWanKobayashi26] that is needed here is proved directly by the tautological construction of lemma 11.11.1, proposition 11.11.2: a metric with positive curvature on the dual of the tautological line bundle gives a homogeneous squared norm that is strictly plurisubharmonic off the zero section. When the dual vector bundle is ample, such a line metric is a root of the Fubini–Study metric pulled back by a very ample embedding. The result is the strictly plurisubharmonic homogeneous function used in their regularization [v1, Lemma 3.4, pp. 13–14, LiuWanKobayashi26], and it gives corollary 11.11.3.

The construction of polar norms in 11.12 gives a proof of Liu and Wan's [v1, Proposition 3.5, pp. 14–16, LiuWanKobayashi26]. A Schur complement and a holomorphic correction of the second jet give local pluriharmonic supporting functions for the logarithm of the polar square. A uniform upper bound for the real Hessian of the primal squared metric gives strong convexity of the dual through its exact Legendre expression; neither real convexity of the primal metric nor uniqueness of a maximizer is assumed. The tautological construction for an ample dual and the homogeneous regularization then give the smooth replacement and the curvature obstruction of corollary 11.12.7, under the hypotheses stated there.

On Hirzebruch surfaces, the ample comparison quotient and the two prioritary points of proposition 11.13.2, lemma 11.13.3 are Liu and Wan's [v1, Proposition 4.2 and Lemma 4.3, pp. 17–20, LiuWanKobayashi26]. The facts about Hirzebruch surfaces that they need are proved directly in lemma 11.13.1, from sections along the ruling and a very ample embedding. The positive Gauduchon tests for the individual dual line bundles in proposition 11.13.4 are the divisor calculation in their proof of Theorem 1.1, p. 23. These constructions give the ample point and the test forms used in theorem 11.15.5, corollary 11.15.6.

The stability of extensions in proposition 11.14.1 is Liu and Wan's [v1, Proposition 4.4, pp. 20–21, LiuWanKobayashi26], including the equality case on $\mathbb F_0$. The proof of proposition 11.14.2 uses the exact sequences of $\operatorname {Hom}$ groups and applies to every extension, while restriction to a fiber of the ruling shows that none of them is ample. The explicit family of Dolbeault operators in proposition 11.14.3 converges to the split operator in every finite $C^ m$ norm. These stable, nonample approximations give the stable point used in the deformation argument of theorem 11.15.5.

The approximation theorem theorem 11.15.5 is Liu and Wan's [v1, Theorem 4.1, p. 17; proof, p. 22, LiuWanKobayashi26]. Its input on the stack of prioritary sheaves is Walter's [v1, Proposition 2, p. 2, WalterPrioritary93], which applies to all prioritary torsion-free sheaves of fixed rank at least two with fixed numerical Chern classes, without any semistability assumption. The required openness statements are proved directly by the family arguments in lemma 11.15.1, lemma 11.15.2, lemma 11.15.3; on a Hirzebruch surface, slope stability reduces to finitely many classes of line bundles. Parallel transport and compactness, in lemma 11.15.4, give convergence on the fixed smooth bundle and on its dual in every finite $C^ m$ norm. Combining this approximation with the Gauduchon-degree obstruction and the polar construction above gives corollary 11.15.6, which is Liu and Wan's [v1, Theorem 1.1, p. 2; proof, p. 23, LiuWanKobayashi26].

The fixed Stein hypersurface of theorem 11.16.3, corollary 11.16.4 corresponds to Liu and Wan's [v1, Proposition 5.1, pp. 24–27, LiuWanKobayashi26] for explicit algebraic bundles. A positive horizontal direction and the vertical critical hyperplane give the test on the Levi index, and a holomorphic quadratic jet makes the real Hessian nondegenerate without changing the Levi form. Separability of the relevant metric space gives countably many robust local tests, which lemma 11.16.1, lemma 11.16.2 place on a single global graph by Runge and Oka–Weil approximation. The graph is chosen before the metric, whereas the critical point that detects the failure may depend on the metric.

The dual norm of a concave neighborhood, theorem 11.17.2, is Liu and Wan's [v1, Lemma 5.2, pp. 27–32, LiuWanKobayashi26]. After a correction by a pure jet, a negative tangent plane of the boundary gives a supporting holomorphic section. Balls of uniform size tangent to the boundary from inside then express the squared dual norm as a supremum of functions with a common margin of strong real convexity, and ampleness together with homogeneous regularization gives a smooth replacement. Let $E$ be an ample bundle of rank $r$, and write $U_ E=\mathbb P(\mathcal O_ X\oplus E)\setminus \mathbb P(\mathcal O_ X)$ for the complement of the zero section in its projective completion. The argument at high levels of an exhaustion in corollary 11.17.4, which shows that $U_ E$ is not $r$-convex when $E^*$ admits no smooth complex norm with plurisubharmonic square, is their [v1, Corollary 5.3, p. 32, LiuWanKobayashi26]. Combined with the norm obstruction and the stable ample approximation, it gives corollary 11.17.5.

The case of positive mean curvature in theorem 11.18.2, corollary 11.18.4 is Liu and Wan's [v1, Proposition 5.4, pp. 32–33, LiuWanKobayashi26]. The calculation with the reciprocal of the norm gives the more general count, in which $s$ positive horizontal directions of the curvature give $(n+r-s)$-convexity of $U_ E$ for a bundle $E$ of rank $r$. For stable bundles of positive degree, the application uses the existence of Hermitian–Einstein metrics, proved by Uhlenbeck and Yau [UhlenbeckYau86] in the Kähler case and extended to Gauduchon metrics by Li and Yau [LiYau87]. The degree of definition 11.18.3 pairs the Bott–Chern first Chern class with the Gauduchon form; with the unscaled curvature $F_ h$ of a Hermitian metric $h$, the integrated trace carries the factor $2\pi n$. The resulting least convexity index three in corollary 11.18.5 is [v1, Corollary 5.5, p. 33, LiuWanKobayashi26]; the lower bound uses the dual-norm obstruction above.

The negative-bundle weight in proposition 11.19.1 is the construction of Kusakabe and Matsumura [v1, Lemma 3.2(a,b,d), pp. 20–21, KusakabeMatsumuraOka26]. The complete metric of corollary 11.19.3 uses the two exhaustion barriers in the first paragraph of their proof of [v1, Lemma 3.3, p. 21, KusakabeMatsumuraOka26]. Their jet-vanishing condition [v1, Lemma 3.4, p. 21, KusakabeMatsumuraOka26] follows here from the smooth-center multiplier-ideal formula, which gives the exact ideal and the sharp threshold for every real weight in corollary 11.19.2. In their argument the logarithmic weight supports weighted $\bar\partial $ estimates and the gluing of local sprays; here its direct Levi calculation relates negative bundle curvature to the singularity along the zero section.

Suppose that, for a holomorphic bundle $F$ of rank $r$, the complement $F^\times $ of the zero section carries a smooth positive plurisubharmonic function that is homogeneous of degree two. Taking maxima over the level spheres, proposition 11.21.1 then shows that the least convexity index of $U_ F$ is $n+r$: a negative radial derivative makes the Levi form restricted to the level set nonpositive on a complex hyperplane. Unitary-flat metrics give corollary 11.21.3. For the line bundles $E_ a$ on a Hopf manifold in example 11.21.4, restriction to an elliptic fiber and positive curvature give the least index two when $|a|<1$, whereas a homogeneous plurisubharmonic metric gives the least index $n+1$ when $|a|\geq 1$; the Gauduchon degree distinguishes the three signs, although the de Rham first Chern class vanishes. A metric with Griffiths-nonpositive curvature gives the same homogeneous function in corollary 11.21.2. For split bundles on Hopf manifolds, proposition 11.21.5 combines the weighted curvature with the exact loss of dimension under restriction to closed submanifolds in lemma 11.17.3. It gives the least indices $r+1$ and $n+r$ in the two pure regimes; in the mixed regime, only separate upper and lower bounds are obtained.

For external tensor products, theorem 11.20.2 applies the same exhaustion with one positive curvature direction from each factor. The positive mean curvatures lifted from the factors control every vector of the tensor product, including the indecomposable ones, and give corollary 11.20.3. For the tangent bundle of a surface with a nonflat Ricci-flat Kähler metric, corollary 11.27.5 combines the exhaustion instead with the directional positivity of theorem 11.27.4: this tangent bundle admits no metric whose mean curvature is positive definite, but its RC-positive metric still gives the required exhaustion.

For the logarithmic parabolic Monge–Ampère flow on Kähler manifolds, the calculation of $L^ p$ norms of the volume ratio is Jiyuan Han's [v1, Lemma 3.2, pp. 13–14, HanParabolic26]. The chain rule and Stokes computation in proposition 11.23.8 keep the additional term coming from the derivative of the Hermitian background. When the forms along the flow remain Gauduchon, convex entropies decrease (corollary 11.23.11), and varying the target probability measure gives the exact criterion at the initial time in proposition 11.23.12. These statements apply on any interval on which a smooth solution exists. Han's uniform estimates for degenerating families of Calabi–Yau manifolds, and the Green-function estimates on which they depend, are not part of this calculation.

The calculation of the total variation in proposition 11.23.16 is a direct variation of determinants on a flat torus, whose coefficient is given by the two Haar averages of lemma 11.23.14. Corollary 11.23.17 compares this growth of the distance between two solutions with the initial derivatives of the individual entropies.

The support lemma for currents carried by a smooth hypersurface, lemma 11.22.1, is the local statement used by Dinh and Sibony in [proof of Theorem 4.1, pp. 965–966, DinhSibonyRegularization04], where it is attributed to Bassanelli. The proof given here tests against a quadratic function of the normal coordinate and polarizes horizontal forms; it applies whenever $\mathrm{dd}^{\mathrm{c}}R$ has order zero, $R$ being the positive current supported on the hypersurface. The estimate for pullbacks in lemma 11.22.2 follows directly from the classical $\mathrm{dd}^{\mathrm{c}}$-lemma for currents on compact Kähler manifolds, applied to a fixed closed pushforward; the current that is paired with the distributional primitive is smooth.

The smoothing of positive pluriharmonic forms with continuous coefficients in lemma 11.22.3 uses the heat equation of the Hodge Laplacian, the Kähler identities and a positive correction tending to zero. For continuous potentials and closed backgrounds on a compact Kähler manifold, Dinh and Sibony construct the product of the curvature current with a positive pluriharmonic current through an energy [Theorem 5.1, Proposition 5.2 and identity (12), pp. 967–969, DinhSibonyRegularization04]. The proofs of lemma 11.31.10, theorem 11.31.11 integrate by parts against the current of bidegree $(n-1,n-1)$ itself, so that no auxiliary power of a Kähler form is needed. They give a positive measure as product, depending jointly continuously on the factors, on any compact Hermitian manifold, for every smooth real background and continuous potential.

The descent of horizontal currents in lemma 11.22.4 follows directly from the positivity of the coefficient matrix together with the Liouville theorem for harmonic functions on the complex line, or the maximum principle on a compact connected curve. It therefore applies both to bundles with fiber $\mathbb C$ and to bundles of compact curves. On the elliptic threefold, the exact vertical trace form gives lemma 10.19.1 for positive $\mathrm{dd}^{\mathrm{c}}$-closed currents, and its closed case gives the calculation of the Bott–Chern cone.

The mean-curvature criterion for RC-positivity and the conformal construction in lemma 11.26.2, theorem 11.26.3 correspond to Chou and Wu's [v1, Definition 5 and Lemmas 7–8, pp. 5–6, ChouWuRC26]. The scalar equation in the construction is the Gauduchon compatibility condition of proposition 11.23.2, and taking the curvature margin over a closed ball makes it uniform in both the base point and the unit fiber vector. Example 11.26.6 shows directly that the order of these two quantifiers matters, and proposition 11.26.7 gives a global obstruction to interchanging them when the determinant bundle admits a flat Hermitian metric.

The RC-positive tangent metrics on Ricci-flat surfaces in theorem 11.27.4 are Chou and Wu's [v1, Theorem 1 and Lemmas 9–15, pp. 7–12, ChouWuRC26]. The pointwise criterion is the invertibility of the anti-self-dual curvature operator. The identities behind (11.27.3) are Derdziński's [equations (33)–(35), p. 414, Derdzinski83], with the nonnegative Laplacian on functions used there replaced by $\operatorname {div}\nabla $ and with the same norm on endomorphisms; summing the three equations for the eigenvalues gives the coefficient $36$. The commuting curvature factors in [Lemma 2, p. 412, Derdzinski83] also show directly that a nonzero parallel operator with three distinct eigenvalues cannot occur here. This gives the required nonempty open set and a conformal tangent metric whose associated form is not closed.

The symmetrized sectional-curvature average and the integrated trace identity in lemma 11.28.2, proposition 11.28.3 are Xueyuan Wan's [v1, Lemma 2.1 and Proposition 3.3 with Lemmas 3.1–3.2, pp. 5–8, WanHermitianSectional26]. The average is proved here by complex Gaussian moments, and the connection calculation retains the exact term until Stokes' theorem removes it. The auxiliary metric $g$ is Kähler, while the tangent metric $h$ can be nonclosed. Their identity gives corollary 11.28.5 directly from the signs of its three terms. This use of the identity does not require the auxiliary Hessian equation or the canonical-bundle positivity argument of that paper. The torus and finite torus-cover statements of proposition 11.28.7, corollary 11.28.8 follow from this identity and periodic integration by parts: the symmetric connection difference vanishes, forcing the Hermitian coefficients to be constant. In dimension one, the same conclusion is the maximum principle for the logarithm of the metric coefficient.

The first-jet characterization of Griffiths positivity on projective manifolds, theorem 11.29.9, is due to Du and Xie [v1, Theorem 1.1 and Proposition 2.1, pp. 1–4, DuXieFirstJet26]; the identity in normal frames is written in the curvature convention used above. The density with two derivatives that the argument requires is proved directly in lemma 11.29.6, lemma 11.29.7, through normalized projective coordinates and Bernstein polynomials. The immersion test without twist, corollary 11.29.10, is the calculation in [v1, Section 3.1, pp. 4–5, DuXieFirstJet26]. The positive measure in the dual alternative lives on the sphere bundle of the dual bundle, so the condition that it annihilates all metrics is different from Gauduchon duality on the base.

The double cover in section 11.30, branched over $2q$ points, gives an explicit model with a computable space of sections in which a strictly positive line bundle fails the test without twist. The decomposition of sections into even and odd parts and the projective embedding are proved explicitly. A root of a very ample Fubini–Study metric gives a separately defined weighted family of tests, and proposition 11.30.3 identifies its first successful level $j=q-1$, together with obstructions by one or two atoms at the lower levels. The original criterion with a fixed very ample twist $m$ follows the subsequence $j=qm$, so this calculation does not determine its first successful twist. The obstructions at a single level should also be distinguished from an adjoint measure annihilating every smooth metric.