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Bott-Chern complexity of K\"ahler pairs

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Pith's one-line read This paper introduces the Bott-Chern complexity of compact Kähler pairs and proves that values below 3 force projectivity, while the non-projective equality case is a toric fibration over a singular K3 surface.

desk verdict A new, clean invariant for Kähler pairs with a sharp projectivity cutoff, but the equality case leans on preprints I can't fully check. read the letter →

arxiv 2505.04303 v1 pith:YNMN5AID submitted 2025-05-07 math.AG math.CV

Christopher Hacon , Joaquín Moraga , José Ignacio Yáñez This is my paper · ORCID
classification math.AGmath.CV MSC 32J2714E3014M2514J28
keywords Bott-CherncohomologyKählervarietiesCalabi-YaupairscomplexityprojectivetoricK3surfacesminimalmodelprogramprojectivitycriterion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces an invariant for compact Kähler pairs, the Bott-Chern complexity, which combines the dimension of the variety, the size of its first Bott-Chern cohomology group, and the total coefficient of a boundary divisor. The central threshold result says that under a standard mild-singularity assumption, if the anticanonical divisor is nef and the Bott-Chern complexity is below 3, the variety must be projective. If the complexity equals 3 and the variety is not projective, then up to bimeromorphic modification the variety is a fibration whose general fibers are projective toric varieties over a singular non-projective K3 surface with Picard rank zero and $h^{1,1}_{BC}=1$. The value 3 is optimal, realized by non-projective singular K3 surfaces with empty boundary. A sympathetic reader should care because this gives a numerical, cohomological criterion for projectivity in the Kähler setting, where such criteria are scarce.

What carries the argument

The carrying object is the Bott-Chern cohomology $H^{1,1}_{BC}(X)$, the space of real $(1,1)$-forms with local potentials modulo $d d^c$-exact forms, which plays the role of the Néron–Severi space in the Kähler setting; the Bott-Chern complexity is $\dim X + h^{1,1}_{BC}(X) - |B|$ minimized over decompositions of $B$ into $\mathbb{Q}$-Cartier divisors. The proof also uses the fine complexity, where the span of the components of $B$ in numerical $\mathbb{R}$-Weil divisors replaces $H^{1,1}_{BC}$, along with two structural inputs: the dimension of the base of the maximally rationally connected fibration is bounded by the fine complexity, and the canonical bundle formula for generalized Kähler sub-pairs with possibly negative boundary coefficients governs the extremal case. The examples rely on the classification of extremal elliptic K3 surfaces to produce singular K3 surfaces with Picard rank zero and $h^{1,1}_{BC}=1$.

What would settle it

Search for a strongly $\mathbb{Q}$-factorial compact Kähler log canonical pair $(X,B)$ with $-(K_X+B)$ nef and Bott-Chern complexity strictly below 3 for which $X$ is not projective; the theorem asserts that no such pair exists, so any such pair would settle the claim false.

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Extended reading notes

Core claim

The paper's central claim is that the Bott-Chern complexity $c_{BC}(X,B)$, defined as the minimum over decompositions of $B$ into $\mathbb{Q}$-Cartier components of $\dim X + h^{1,1}_{BC}(X) - |B|$, controls projectivity for strongly $\mathbb{Q}$-factorial compact Kähler log canonical pairs with $-(K_X+B)$ nef. Theorem 4.1 states that $c_{BC}(X,B)<3$ forces $X$ to be projective; if $c_{BC}(X,B)=3$ and $X$ is not projective, then there is a dlt modification $(X',B')\to (X,B)$, a small modification $Y'\to X'$, and an MRC fibration $Y'\to W$ whose general fiber is a projective toric variety and whose base $W$ is a singular non-projective K3 surface of Picard rank zero and $h^{1,1}_{BC}(W)=1$. Consequently, Calabi-Yau compact Kähler pairs have nonnegative Bott-Chern complexity, and non-projective ones have complexity at least 3; the examples constructed by contracting configurations of curves in degenerations of extremal elliptic K3 surfaces show the bound is attained.

Load-bearing premise

The equality-case argument leans on a technical adjunction formula and on termination of a minimal-model program for generalized Kähler pairs whose boundary may have negative coefficients, both cited to recent preprints; if these apply only in narrower settings, the description of the base as a K3 surface could fail.

Editorial extensions

If this is right

  • If the main theorem is correct, any Calabi-Yau compact Kähler pair has nonnegative Bott-Chern complexity, and a non-projective one has complexity at least 3.
  • For strongly $\mathbb{Q}$-factorial log canonical pairs with nef anticanonical divisor, Bott-Chern complexity below 3 is a projectivity criterion: no non-projective example can exist below that threshold.
  • In the minimal non-projective case, a bimeromorphic model is a fibration with projective toric general fibers over a singular non-projective K3 surface with Picard rank zero and $h^{1,1}_{BC}=1$.
  • The fine-complexity analogues hold: fine complexity below 1 forces a projective toric variety, and fine complexity below 2 forces projectivity together with $H^{1,1}_{BC}(X)=\mathrm{Pic}(X)_{\mathbb{R}}$.
  • The bound 3 is sharp, since singular non-projective K3 surfaces with empty boundary realize exactly $c_{BC}=3$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One testable extension is a converse classification: check whether every Kähler fibration with projective toric general fibers over a singular K3 surface of Picard rank zero and $h^{1,1}_{BC}=1$ attains Bott-Chern complexity exactly 3.
  • Because the main hypothesis is only that $-(K_X+B)$ is nef, the threshold may extend beyond Calabi-Yau pairs to log Fano type Kähler pairs; a natural test is to allow $B$ to have negative coefficients and track how the canonical bundle formula absorbs them.
  • The examples are produced from two extremal elliptic K3 fibrations; surveying the full classification of such fibrations could reveal whether the equality value 3 is isolated or occurs in a family of singular K3 models with $h^{1,1}_{BC}=1$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper introduces the Bott-Chern complexity c_BC(X,B) = dim X + h^{1,1}_{BC}(X) - |B| for compact Kähler pairs, paralleling the projective notion of complexity with the Bott-Chern cohomology in place of the Néron–Severi rank. The main results are: (1) if (X,B) is log canonical and -(K_X+B) is nef, then c_BC(X,B) >= 0; (2) if c_BC(X,B) < 3, then X is projective; (3) if c_BC(X,B) = 3 and X is non-projective, then a bimeromorphic model of X carries an MRC fibration whose general fiber is a projective toric variety and whose base is a singular non-projective K3 surface with Picard rank zero and h^{1,1}_{BC}=1. These results are proved via a fine-complexity invariant, bounds on the dimension of the MRC base, and the canonical bundle formula for generalized Kähler sub-pairs. Two examples of non-projective singular K3 surfaces with c_BC=3 are constructed using deformation theory and classification of extremal elliptic K3 surfaces.

Significance. If the main theorem is correct, this is a valuable and novel connection between the complexity program, Kähler geometry, and projectivity criteria. The invariant is natural and likely to find further applications, and the extremal non-projective examples are concrete and convincing. The paper is well-structured and builds on recent substantial advances in the Kähler MMP. However, the proof of the equality case (Theorem 4.1(2)) depends on two recent preprints, [16] and [20], whose exact hypotheses are not stated or verified in the manuscript; this is a load-bearing point for the production of the K3 base. The paper would be significantly strengthened by making those dependencies explicit and justifying their applicability.

major comments (2)
  1. [Theorem 4.1(2), proof in §4] The proof applies the canonical bundle formula for generalized Kähler sub-pairs to (X'',B''+M'') over the compact Kähler surface Z, and later terminates the MMP over Z using [20, Lemma 2.6] and [22, Lemma 2.9]. These are load-bearing applications: they produce the surface (Z,B_Z+M_Z) and the model Y' after contraction. The manuscript does not state the precise hypotheses of [16, Theorem 0.3], and it is not clear that it covers sub-pairs with vertical negative coefficients over a non-projective Kähler base. Moreover, [20] is announced for Kähler varieties with projective Albanese map, while [22] is an algebraic statement; neither obviously applies to an arbitrary compact Kähler surface base. The authors should state the exact theorems they are invoking and prove that the hypotheses are satisfied in this situation, or give alternative arguments.
  2. [Theorem 4.1(2), proof in §4, final paragraph] The assertion "KY' + GY' + MY' is Q-linearly equivalent to an effective divisor which is exceptional over X'. Thus, we conclude that KY' + GY' + MY' is Q-linearly trivial" is too quick. While it is true that a semiample divisor cannot be Q-linearly equivalent to a nonzero effective exceptional divisor, this deserves a one-sentence justification, especially because the argument also relies on the preceding semiampleness of the same divisor. Adding this justification would make the proof more transparent.
minor comments (5)
  1. [Abstract and Introduction] The notation "dim H^{1,1}_{BC}(X)" is used where "h^{1,1}_{BC}(X)" is standard; please unify with the body of the paper.
  2. [Definitions 2.7 and 2.8] The notation c(X,B) is used both for the fine complexity and for the projective complexity from the literature; consider a distinct notation, for instance c_f(X,B), to avoid ambiguity.
  3. [Proof of Theorem 1.4] The proof says "if c(X,B)<2, then X is a projective variety and so [1, Theorem 1.2] applies to prove (1) and (2)", but parts (1) and (2) concern c(X,B)>=0 and c(X,B)<1. Please spell out that the non-negativity and toric conclusions follow from the projective result after the projectivity is established.
  4. [Section 4, after diagram] The notation B^{=1}_Z and B''^{=1} is used without definition; please define it as the sum of the components with coefficient exactly one.
  5. [Examples 5.1 and 5.2] The deformation argument is sketched; it would be helpful to state explicitly that the general fiber X_c has Picard lattice equal to L (by a standard semi-continuity argument), which is what makes the contraction possible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Bott-Chern complexity is defined independently, and the theorems are proved from external results and algebraic identities, not from the conclusions.

full rationale

The central invariant, Bott-Chern complexity, is introduced in Definition 2.8 as the minimum over Q-Cartier decompositions of dim(X) + h^{1,1}_{BC}(X) minus the sum of coefficients, with the fine complexity defined independently in Definition 2.7. The main theorems are then proved by comparing these two invariants and applying established results: the MRC fibration for Kähler varieties [3], the relative Moishezon property [4], non-pseudoeffectivity of canonical divisors [29], and the projective complexity inequalities of Brown–McKernan–Svaldi–Zong [1]. In the proof of Theorem 4.1, the step from c_{BC}(X,B;Σ)=3 to c(X,B;Σ)=2 is an equality forced by the definitions: c_{BC}-c_fine = h^{1,1}_{BC}(X)-dim_R<Σ>, an integer, so the dichotomy is a direct arithmetic comparison, not a restatement of the theorem. The extremal case invokes the canonical bundle formula for generalized Kähler sub-pairs from [16] and termination lemmas from [20]; these are external supporting results, and the authors do not fit parameters or define the invariant in terms of the K3 base. The examples are constructed from the Shimada–Zhang classification of extremal elliptic K3 surfaces [30], an external classification, and the claimed values h^{1,1}_{BC}=1 and ρ=0 are computed from the construction. Even if the hypotheses of [16, Theorem 0.3] or [20, Lemma 2.6] were questioned, that would be a correctness risk about cited machinery, not a circular reduction of the paper's claims to its own inputs. No prediction is obtained by fitting, no uniqueness theorem by the same authors is used to forbid alternatives, and no known result is renamed as a new invariant. Accordingly, the paper is self-contained against external benchmarks and no circular step is identified.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The new complexity invariants are definitions, not entities requiring independent evidence; no new particles, forces, or geometric objects are postulated. All load-bearing assumptions are external theorems from the cited literature.

assumptions (8)
  • domain assumption Nonnegativity and toric characterization of projective log canonical pairs with anticanonical class nef ([1, Theorem 1.2] and related results)
    Used to bound c(F,B_F) >= 0 and to identify dimension-zero fibers as toric in Sections 3 and 4.
  • domain assumption Existence and almost-holomorphic property of the MRC fibration for compact Kähler manifolds, with non-uniruled base ([3])
    Defines the base Z and the general fibers used in Theorem 1.3 and Theorem 4.1.
  • domain assumption If the base Z of the MRC fibration is not uniruled, then K_Z is pseudo-effective ([29])
    Used in Section 3 to conclude that K_Z is pseudo-effective.
  • domain assumption Projectivity criterion for Kähler morphisms and Moishezon Kähler spaces with rational singularities ([4], [28])
    Used to show that Y is projective when dim Z = 1 and that the general fiber F is projective.
  • domain assumption Canonical bundle formula for generalized Kähler sub-pairs with possibly negative boundary coefficients ([16, Theorem 0.3])
    Central engine of the non-projective case in Section 4; it produces the sub-log Calabi-Yau surface (Z,B_Z+M_Z).
  • domain assumption Termination of the MMP for Kähler varieties over a projective morphism to a Kähler base, including contraction of vertical exceptional divisors ([15,20,22])
    Used in Section 4 to contract gamma_Y and pass to the model Y' over Z'.
  • domain assumption Shimada-Zhang classification of extremal elliptic K3 surfaces, including the two specific entries used in Examples 5.1 and 5.2 ([30])
    Provides the starting projective K3 surfaces with the prescribed reducible fibers.
  • domain assumption Kuranishi deformations of K3 surfaces can preserve a negative-definite sublattice of Picard rank 19 and allow contraction of the corresponding (-2)-curves
    Used in Examples 5.1 and 5.2 to obtain non-projective singular K3 surfaces with h11_BC = 1 and rho = 0.

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Pith. "Pith review of Bott-Chern complexity of K\"ahler pairs." pith.science (2026). https://pith.science/paper/YNMN5AID

@misc{pith2026250504303,
  author       = {Pith},
  title        = {Pith review of: Bott-Chern complexity of K\"ahler pairs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YNMN5AID}},
  note         = {Machine review of arXiv:2505.04303}
}
abstract

We introduce the Bott-Chern complexity of a compact K\"ahler pair $(X,B)$. This invariant compares $\dim(X)$, $\dim H^{1,1}_{\rm BC}(X)$ and the sum of the coefficients of $B$. When $(X,B)$ is Calabi-Yau, we show that its Bott-Chern complexity is non-negative. We prove that the Bott-Chern complexity of a Calabi-Yau compact K\"ahler pair $(X,B)$ is at least three whenever $X$ is not projective. Furthermore, we show this value is optimal and is achieved by certain singular non-projective K3 surfaces.

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