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Deformed Chern scalar curvature yields canonical metrics in Hermitian geometry

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load-bearing objection A lecture-note survey of the author's own Hermitian-curvature program: no new theorems, but the deformed-Yamabe proof is written out fully and the central computation verifies. the 2 major comments →

arxiv 2607.26673 v1 pith:TALCYHDL submitted 2026-07-29 math.DG math.CV

Almost-Hermitian and Hermitian metrics

Daniele Angella This is my paper
classification math.DG math.CV MSC 53C5553C2153D2058J05
keywords Chern scalar curvatureHermitian metricsYamabe problemmomentum maplocally conformally Kählerdeformed scalar curvaturevariational methodsSobolev inequality
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

These lecture notes argue that the scalar curvature of the Chern connection is not the right quantity to study on its own. The paper builds a one-parameter family of deformed scalar curvatures — Chern scalar curvature plus carefully weighted torsion terms — and shows that two natural problems become tractable for these deformations. First, a twisted momentum-map interpretation holds in locally conformally Kähler geometry, with the deformed curvature as the momentum map. Second, the corresponding deformed Yamabe equation is variational, and a sharp Sobolev bound plus a pointwise curvature condition guarantee existence of a smooth positive minimizer. The upshot is a concrete route to canonical Hermitian metrics in every conformal class, with the classical Riemannian Yamabe problem and the locally conformally Kähler momentum-map problem as endpoints t=2n−1 and t=2n.

Core claim

The central claim is Theorem 5.8: on a compact complex manifold of dimension n≥3 with Hermitian metric ω and t>0, the t-deformed Yamabe invariant satisfies λ^t_X({ω}) ≤ tΛ_n, where Λ_n is the sharp constant of the Euclidean Sobolev inequality; if the inequality is strict, the t-deformed Yamabe equation has a positive smooth minimizer; and the pointwise condition (t−2n+1)μ_0(ω)(p)>0 at some point p guarantees the strict inequality. Here μ_t(ω)=scal^Ch(ω)+((t−n)/(n−1))d*θ+((t−2n)/4)|T|², built so that its conformal transformation law has no gradient term, making the associated Einstein–Hilbert functional bounded below and variational. The paper also proves the momentum-map theorem 4.2: under t

What carries the argument

The carrying object is the t-deformed scalar curvature μ_t(ω)=scal^Ch(ω)+(t−n)/(n−1)d*θ+(t−2n)/4|T|². The coefficients are chosen so that under a conformal change ω→e^fω the term linear in ∇f cancels; this makes the deformed Yamabe equation the Euler–Lagrange equation of the deformed Einstein–Hilbert functional, whose energy is bounded below. The proof of existence combines the sharp Sobolev inequality with bubble test functions built from holomorphic normal coordinates; the Taylor expansion of the metric at a point is used to compute the leading correction, and the sign of (t−2n+1)μ_0 decides whether the energy drops below the universal constant tΛ_n.

Load-bearing premise

The momentum-map half of the paper assumes a compact connected Lie group of symmetries of a special Lee-type twisted-Hamiltonian kind; without such a group, the momentum-map picture for locally conformally Kähler manifolds is not established.

What would settle it

Compute the t-deformed Yamabe invariant on a compact Hermitian manifold of dimension n≥3 where the pointwise condition (t−2n+1)μ_0(p)>0 holds, and exhibit that it equals tΛ_n (so no minimizer exists). The paper leaves open whether equality can occur, so a single such example would refute the sufficiency claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For any compact Kähler manifold of dimension n≥3 and any t>2n−1, there exists a Hermitian metric conformal to the Kähler one with constant t-deformed scalar curvature (Corollary 5.13).
  • The deformed Yamabe invariant is always bounded above by tΛ_n, a universal constant depending only on dimension and t; this reduces existence questions to establishing a strict inequality.
  • The momentum-map picture gives a Futaki-type obstruction: if the invariant F(Y) is nonzero, no K-invariant locally conformally Kähler structure can have constant momentum-map curvature.
  • The constancy of the momentum map, rather than of Chern scalar curvature itself, is proposed as the natural canonical-metric equation in the locally conformally Kähler setting.
  • The family interpolates between the classical Riemannian Yamabe problem (t=2n−1) and the locally conformally Kähler momentum-map equation (t=2n), so results apply to both settings at once.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the momentum-map theorem extends beyond the assumed Lie-group symmetry, the Futaki-type obstruction would likely yield stability conditions for locally conformally Kähler manifolds analogous to K-stability — a direction the paper itself flags as open.
  • Because equality λ^t_X({ω})=tΛ_n seems unrealized for n≥3 (the paper notes only CP^1 realizes it), a plausible conjecture is that strict inequality always holds in complex dimension at least 3; testing this on explicit manifolds like Hopf or Calabi-Eckmann would be informative.
  • The paper's Hirzebruch-surface construction shows the existence result is not limited to locally homogeneous metrics; this suggests cohomogeneity-one ansätze could produce constant deformed scalar curvature metrics on many other ruled manifolds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper is a set of lecture notes surveying three problems in Hermitian geometry centered on the Chern scalar curvature: the Chern-Yamabe problem, the momentum-map interpretation of a deformed Chern scalar curvature in locally conformally Kähler geometry, and the variational theory of t-deformed Yamabe equations. It contains proofs of existence and uniqueness in the non-positive Gauduchon degree case, a conditional momentum-map theorem under a Lie-symmetry assumption, and an Aubin-type existence theorem for the deformed Yamabe problem. The exposition combines PDE, moment-map, and variational methods, with emphasis on the author's prior joint work.

Significance. As a survey, the paper is useful for bringing together three techniques and for presenting the deformed Yamabe generalization with explicit conformal formulas and a pointwise curvature condition. The classical parts are mostly self-contained and the momentum-map computation is coherent under its stated assumptions. However, the sharp Sobolev constant is stated with a wrong exponent, creating an internal inconsistency that directly affects the main existence theorem. The final numerical values of the deformed Yamabe constant appear to be correct, and the overall framework is promising, but the manuscript needs a careful correction of the constant formulas before it can be accepted.

major comments (2)
  1. [Theorem 5.5 and Remark 5.6] The sharp Sobolev inequality in Theorem 5.5 is misstated: the optimal constant for ||u||^2_{L^{2*}} ≤ σ_m ||∇u||^2 is σ_m = 4/(m(m-2)) Vol(S^m)^{-2/m}, not Vol(S^m)^{2/m}. As written, for m=4 the claimed σ_m is 2.565, while the bubble (19) gives ratio ≈0.0975; the equality statement is then false. The same sign error appears in Remark 5.6 ('σ_{2n}=1/(n(n-1))Vol(S^{2n})^{1/n}'). The table values and the final Λ_n=2n Vol^{1/n} are consistent with the corrected negative exponent, so the rest of the proof can be repaired, but Theorem 5.5 and Remark 5.6 must be corrected and checked.
  2. [Theorem 5.7 proof] The displayed computation of ∂_i \tilde{h}_{j\bar{ℓ}} + ∂_j \tilde{h}_{i\bar{ℓ}} is not correct. The derivative of the anti-holomorphic factor \overline{∂z^ℓ/∂w^j} with respect to w_i vanishes, so the term with complex conjugates in the displayed formula should not be present. With C^k_{rs}=1/4(∂_s h_{\bar{k}r}+∂_r h_{\bar{k}s}), the correct second derivative is ∂^2 z^ℓ/∂w_i∂w_j(0)=-2C^ℓ_{ij}=-1/2(∂_i h_{j\bar{ℓ}}+∂_j h_{i\bar{ℓ}}), which cancels the first-order terms and yields (20). As written, the proof does not justify the conclusion; please correct the computation.
minor comments (5)
  1. [Theorem 5.8 statement] Theorem 5.8 is labeled '([Aub76, Heb03])' but the deformed scalar curvature and the torsion-dependent pointwise criterion are from [APS+26]; [Aub76, Heb03] support the method. Please adjust the attribution.
  2. [Section 4.3] The momentum-map theorem is conditional on an assumed compact Lie group K; the text mentions Vaisman metrics as a model case but does not prove existence of such K in general. Please state explicitly in the main text that the momentum-map picture is established only under this extra symmetry, and discuss whether it holds for all Vaisman metrics or only a subclass.
  3. [Theorem 5.8, Step 11] The passage from subcritical minimizers to the critical equation is compressed: the case λ^t_{2*}<0 and the left-continuity of λ^t_s are only implicit. A remark with the standard splitting would improve readability.
  4. [Section 5.1, Eq. (12)] The phrase 'the coefficient of the gradient term' for n-(n-1)a+4b should read 'the coefficient of the first-order drift term g(∇f,θ^♯)'.
  5. [Remark 5.6] The numerical values of Λ_n in the table are correct, but the displayed formula for σ_{2n} must be fixed as indicated in the major comment above.

Circularity Check

0 steps flagged

No significant circularity: Theorem 5.8 and the momentum-map picture are internally derived; self-citations are survey-level, not load-bearing reductions.

full rationale

I examined the central existence theorem (Theorem 5.8). The deformed scalar curvature μ_t is defined in (11) with coefficients chosen so the conformal law (14) has no gradient term; this is a definition, not a fitted prediction. The Euler-Lagrange equivalence (Theorem 5.1) and the lower bound/invariant (17) follow directly from (14)-(16) and Hölder. The bubble computation in Steps 1-10 uses only the sharp Sobolev inequality (Theorem 5.5), holomorphic normal coordinates (Theorem 5.7), and the algebraic identities in Step 8 and Step 10; the strict-inequality criterion is derived from the expansion, not assumed. Step 11 is a standard Aubin-type subcritical argument with explicit estimates. The theorem is credited to [Aub76, Heb03], and the proof is largely self-contained, so [APS+26] is a description source, not a reduction. Section 4's momentum-map theorem is explicitly conditional on the existence of a compact Lee-type twisted-Hamiltonian group K (Section 4.3); this is an honest scope limitation rather than circularity. The proof uses linearization formulas and Weyl-connection computations, with technical lemmas cited from the authors' published [ACPS25]; those citations are not equivalent to assuming the target identity. The Chern-Yamabe results of Section 3 are proven by standard continuity/bifurcation methods and do not reduce to a fit. Overall the lecture notes are a self-survey, but no equation-level prediction is an input by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 2 invented entities

The leading assumptions are the standard compact smooth setting, the sharp Sobolev machinery, the extra compact Lie symmetry in the momentum-map part, and the LCK torsion identity. The free parameter t and the coefficient convention shape the deformed problems but are not fitted to data. No physical entities are hypothesized; the invented quantities are mathematical definitions.

free parameters (2)
  • deformation parameter t
    The family μₜ in (11) depends on arbitrary t>0; special values t=2n−1 and t=2n recover the Riemannian scalar curvature and the LCK momentum-map quantity. Coefficients are fixed by killing the g(df,θ) term, not fitted to data.
  • deformation coefficients a,b = a=(t−n)/(n−1), b=(t−2n)/4
    Chosen in Section 5.1 so that the gradient term in the conformal transformation rule vanishes; a convention rather than a datum.
axioms (5)
  • domain assumption Compact connected complex manifold, smooth tensors, non-Kähler in general
    Section 2.1 states the standing setting; excludes singular and non-compact cases, which the author notes are also interesting.
  • standard math Sharp Sobolev inequality with optimal constant and bubble functions (Aubin–Talenti)
    Theorem 5.5 is the basis for the universal upper bound λ≤tΛₙ and the test-function expansion in Theorem 5.8.
  • standard math Elliptic regularity tools: Fredholm alternative, maximum principle, Calderón–Zygmund and Schauder estimates
    Used in the continuity method (Theorem 3.4) and the subcritical approximation (Theorem 5.8).
  • ad hoc to paper Existence of a compact Lie group K of Lee-type twisted-Hamiltonian symmetries with central V
    Section 4.3; without it the momentum-map theorem (Theorem 4.2) is not available for general LCK manifolds. This is the key extra symmetry assumption.
  • domain assumption For LCK metrics, |T^Ch|² = 2/(n−1)|θ|²
    Used in the proof of Theorem 4.2 to rewrite μ as a function of scalar curvature and Lee-form terms; a standard identity for locally conformally Kähler metrics.
invented entities (2)
  • t-deformed scalar curvature μₜ no independent evidence
    purpose: Defines a family of curvatures interpolating between the Riemannian scalar curvature and the momentum-map quantity; studied via the t-deformed Yamabe equation.
    Equation (11); a mathematical definition introduced in Section 5.1 (and [APS+26]) with no independent external handle.
  • t-deformed Yamabe invariant λᵗ_X({ω}) no independent evidence
    purpose: Infimum of the deformed Einstein-Hilbert functional; its position relative to tΛₙ controls existence.
    Equation (17); a new invariant of the theory, not a physical entity.

pith-pipeline@v1.3.0-daily-deepseek · 44691 in / 18733 out tokens · 174379 ms · 2026-08-01T11:01:49.074680+00:00 · methodology

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read the original abstract

We describe three problems in Hermitian geometry related to the scalar curvature of the Chern connection, aimed at identifying and constructing canonical metrics. They are chosen as toy examples where different techniques naturally interact: analytic methods for partial differential equations, the notion of momentum map, and variational methods.

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