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arxiv: 2606.01153 · v1 · pith:FUN6UDWNnew · submitted 2026-05-31 · 🧮 math.DG · math.CV

Scalar Curvature, Volumes and the Bergman Kernel

Bo-Yong Chen , Yuanpu Xiong , Liyou Zhang This is my paper

Pith reviewed 2026-06-28 16:37 UTC · model grok-4.3

classification 🧮 math.DG math.CV
keywords Kähler manifoldnef canonical bundleChern scalar curvaturecanonical volumeBergman kernelminimal volumeKähler-Einstein metricscalar curvature integral
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The pith

If the canonical bundle of a compact Kähler manifold is nef, then MinVol_C(M) equals I_C(M) equals I_C^-(M) equals (nπ)^n / n! times CanVol(M).

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

The paper proves that on any compact complex n-manifold, the volume of a Hermitian metric with Chern scalar curvature at least -1 and the integral of the negative part of that curvature are each bounded below by (nπ)^n / n! CanVol(M). When the manifold is Kähler and its canonical bundle is nef, these lower bounds are attained exactly, so the three infima coincide with the bound. The proof combines the asymptotic expansion of the Bergman kernel for high multiples of K_M, the Kähler-Ricci flow, singular Kähler-Einstein metrics, and a gluing construction that uses the Burns-Simanca metric. Equality holds in the volume bound only for the Kähler-Einstein metric of negative scalar curvature. The same conclusion applies when the manifold is obtained by blowing up finitely many points on a projective manifold whose canonical bundle is big and nef.

Core claim

For a compact Kähler manifold M with nef canonical bundle K_M, the quantities MinVol_C(M), I_C(M), and I_C^-(M) are all equal to (nπ)^n / n! CanVol(M). This follows from the general lower bounds that hold for arbitrary Hermitian metrics together with the fact that the bounds are achieved under the nef assumption through analytic constructions based on Bergman kernel asymptotics and gluing.

What carries the argument

The asymptotic behaviour of the Bergman kernel of mK_M as m tends to infinity, which supplies the relation between the canonical volume and the curvature-controlled volumes and integrals.

If this is right

  • If a Kähler metric with S_C >= -1 attains the volume bound, then it must be the Kähler-Einstein metric of negative scalar curvature.
  • The equalities hold when M is obtained by blowing up a finite number of points on a projective manifold with big and nef canonical bundle.
  • The lower bounds on volume and on the integral of |S_C^-|^n hold for every Hermitian metric on every compact complex manifold.
  • Under the nef assumption the three quantities MinVol_C(M), I_C(M), and I_C^-(M) coincide.

Where Pith is reading between the lines

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

  • The nef condition on K_M might turn out to be unnecessary, in which case the equalities would hold for all compact Kähler manifolds.
  • The Bergman-kernel and gluing techniques could be adapted to produce similar sharp bounds for other curvature functionals or for non-Kähler Hermitian metrics.
  • The canonical volume would then function as the sole invariant that determines the minimal volume attainable under a lower bound on Chern scalar curvature.

Load-bearing premise

The canonical bundle K_M must be nef.

What would settle it

A compact Kähler manifold with nef canonical bundle K_M on which there exists a Hermitian metric g with S_C(g) >= -1 and vol_g(M) strictly less than (nπ)^n / n! CanVol(M).

read the original abstract

Motivated by the works of Gromov and LeBrun in Riemannian geometry, we study the analogous phenomena in complex geometry. We first show that both $\int_M |S_C^-(g)|^ndV_g$ and ${\rm vol}_g(M)$ (normalized by $S_C(g)\ge -1$) are bounded below by $\frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)$ for any Hermitian metric $g$ on a compact complex $n-$manifold $M$. Here $S_C$ denotes the Chern scalar curvature, $S_C^-=\max\{-S_C,0\}$ and ${\rm CanVol}(M)$ is the canonical volume of $M$, i.e., the volume of the canonical line bundle $K_M$. Moreover, if ${\rm vol}_g(M)=\frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)$ holds for some K\"ahler metric with $S_C\ge -1$, then it has to be the K\"ahler-Einstein metric of negative scalar curvature. The completely new phenomenon is that if $M$ is a compact K\"{a}hler manifold such that $K_M$ is nef, then ${\rm MinVol}_C(M)=\mathcal{I}_C(M)=\mathcal I_C^-(M)=\frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)$, where ${\rm MinVol}_C(M)$ is the infimum of ${\rm vol}_g(M)$ with $S_C(g)\ge -1$ and $\mathcal I_C^-(M)=\inf_g \int_M |S_C^-(g)|^ndV_g$, $\mathcal I_C(M)=\inf_g \int_M |S_C(g)|^ndV_g$. It remains unknown whether the nef condition is superfluous. The answer is positive when $M$ is obtained by blowing up a finite number of points from a projective manifold with big and nef canonical line bundle. The arguments are based on the asymptotic behaviour of the Bergman kernel of $mK_M$ as $m\rightarrow \infty$, the theory of K\"ahler-Ricci flow and singular K\"{a}hler-Einstein metric, as well as a very delicate gluing technique, using the Burns-Simanca metric.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper establishes lower bounds ∫_M |S_C^-(g)|^n dV_g ≥ (nπ)^n / n! CanVol(M) and vol_g(M) ≥ (nπ)^n / n! CanVol(M) for any Hermitian metric g with S_C(g) ≥ -1 on a compact complex n-manifold M. It further shows that if equality holds for a Kähler metric, then the metric is the Kähler-Einstein metric of negative scalar curvature. The central result is that when M is Kähler with K_M nef, the quantities MinVol_C(M), I_C(M) and I_C^-(M) all coincide with (nπ)^n / n! CanVol(M). The proofs rely on Bergman kernel asymptotics for mK_M as m o∞, the Kähler-Ricci flow, singular Kähler-Einstein metrics, and a gluing construction using the Burns-Simanca metric. The equality is verified explicitly for blow-ups of projective manifolds with big and nef canonical bundle; it is left open whether the nef hypothesis can be removed.

Significance. If the central equality holds, the work supplies a precise complex-geometric counterpart to the Gromov-LeBrun volume bounds in Riemannian geometry, expressing minimal volumes and L^n-norms of negative scalar curvature directly in terms of the canonical volume. The identification of equality cases with Kähler-Einstein metrics and the use of Bergman-kernel asymptotics together with Ricci-flow techniques constitute concrete analytic-algebraic links. The gluing construction with the Burns-Simanca metric is a technical strength that may be reusable in related problems.

major comments (2)
  1. [Abstract / main theorem] Abstract (and the statement of the main theorem): the equality MinVol_C(M)=I_C(M)=I_C^-(M)=(nπ)^n/n! CanVol(M) is asserted for every compact Kähler manifold with nef K_M. When K_M is nef but not big, CanVol(M)=0 and h^0(M,mK_M)=o(m^n), so the leading m^n term in the standard Bergman-kernel expansion is absent. The manuscript must supply an explicit argument showing that the lower-order asymptotics, the Kähler-Ricci flow, or the gluing construction still achieve the infimum; otherwise the claim should be restricted to the big-and-nef case already treated for blow-ups.
  2. [Abstract] Abstract: the lower bound ∫ |S_C^-|^n dV_g ≥ (nπ)^n / n! CanVol(M) is stated for arbitrary Hermitian metrics with S_C ≥ -1. The proof sketch invokes Kähler-Ricci flow and singular KE metrics, both of which presuppose a Kähler structure. The manuscript should clarify whether the Hermitian lower bound is obtained by a separate argument or by reduction to the Kähler case; if the latter, an approximation or density statement is required.
minor comments (2)
  1. Notation: the symbols MinVol_C, I_C and I_C^- are introduced without an explicit reference to the precise infima they denote (over which class of metrics, with which normalization). A short displayed definition would improve readability.
  2. [Abstract] The abstract mentions 'the theory of Kähler-Ricci flow and singular Kähler-Einstein metric' but does not indicate which convergence or regularity results are invoked; a brief pointer to the relevant theorems would help readers trace the argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on the abstract and main results. We address each major comment below and will revise the manuscript accordingly to clarify the statements and strengthen the arguments.

read point-by-point responses

  1. Referee: [Abstract / main theorem] Abstract (and the statement of the main theorem): the equality MinVol_C(M)=I_C(M)=I_C^-(M)=(nπ)^n/n! CanVol(M) is asserted for every compact Kähler manifold with nef K_M. When K_M is nef but not big, CanVol(M)=0 and h^0(M,mK_M)=o(m^n), so the leading m^n term in the standard Bergman-kernel expansion is absent. The manuscript must supply an explicit argument showing that the lower-order asymptotics, the Kähler-Ricci flow, or the gluing construction still achieve the infimum; otherwise the claim should be restricted to the big-and-nef case already treated for blow-ups.

    Authors: We agree that the case of nef but not big K_M requires explicit treatment, since CanVol(M)=0 and the leading Bergman term vanishes. In this case the claimed equality reduces to showing that the infima MinVol_C(M), I_C(M) and I_C^-(M) are all zero. The current proofs via leading-term asymptotics apply directly only when K_M is big. We will add a dedicated paragraph (or subsection) in the revision that handles the non-big case separately: when CanVol(M)=0 we construct a sequence of Kähler metrics with S_C ≥ -1 whose volumes and L^n-norms of negative scalar curvature tend to zero, using a degeneration argument along the Kähler-Ricci flow combined with the gluing technique already developed for the big case. If this construction turns out to require substantial new work, we will instead restrict the main theorem to the big-and-nef setting (as already verified for the blow-up examples) and state the non-big case as a separate, open question. revision: yes

  2. Referee: [Abstract] Abstract: the lower bound ∫ |S_C^-|^n dV_g ≥ (nπ)^n / n! CanVol(M) is stated for arbitrary Hermitian metrics with S_C ≥ -1. The proof sketch invokes Kähler-Ricci flow and singular KE metrics, both of which presuppose a Kähler structure. The manuscript should clarify whether the Hermitian lower bound is obtained by a separate argument or by reduction to the Kähler case; if the latter, an approximation or density statement is required.

    Authors: The lower bound for arbitrary Hermitian metrics is obtained by reduction to the Kähler case. We first prove the inequality for Kähler metrics via the Kähler-Ricci flow and singular KE metrics, then extend it to Hermitian metrics by a standard approximation argument: any Hermitian metric with S_C ≥ -1 can be approximated in C^2 by Kähler metrics (in the same cohomology class when the manifold is Kähler, or locally) while preserving the lower bound on S_C up to a small error that can be absorbed into the constant -1. We will insert a short clarifying remark (and, if needed, a reference to the approximation lemma) in the revised manuscript to make this reduction explicit. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external analytic tools

full rationale

The paper establishes lower bounds on normalized volumes and integrals of |S_C| using Bergman kernel asymptotics of mK_M, Kähler-Ricci flow, singular KE metrics, and gluing via the Burns-Simanca metric. These are standard external results in complex geometry, not self-referential definitions or fitted inputs renamed as predictions. The central equality under the nef assumption on K_M is obtained by applying these tools to produce metrics achieving the infima; the paper explicitly notes the general nef case is open and only proves the result for blow-ups of projective manifolds with big and nef K_M. No load-bearing step reduces to a self-citation chain, ansatz smuggled via prior work, or renaming of a known empirical pattern. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard analytic results (asymptotics of Bergman kernel of mK_M, existence theory for singular Kähler-Einstein metrics, Kähler-Ricci flow convergence) whose details are not supplied in the abstract; no free parameters or invented entities are visible.

axioms (2)
  • domain assumption Asymptotic behaviour of the Bergman kernel of mK_M as m→∞ holds under the stated positivity or nef assumptions.
    Invoked in the abstract as one of the main arguments; this is a standard but non-trivial result in complex geometry.
  • domain assumption The gluing technique with the Burns-Simanca metric produces a metric satisfying S_C ≥ -1 with controlled volume.
    Cited as part of the proof strategy; requires verification that the glued metric meets the curvature lower bound.

pith-pipeline@v0.9.1-grok · 5972 in / 1658 out tokens · 25367 ms · 2026-06-28T16:37:49.256212+00:00 · methodology

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Reference graph

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