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Bismut-Torsion-Parallel Hermitian Manifolds With Constant Chern Holomorphic Sectional Curvature

Haohao Wang

arXiv:2608.13386v1math.DG

TL;DR

A longstanding conjecture asks whether constant Chern holomorphic sectional curvature forces compact Hermitian manifolds to be Kähler in the nonzero case. This paper proves the nonzero balanced Bismut-torsion-parallel case in every complex dimension and deduces Kählerness for compact BTP Hermitian manifolds.

  • Problem

    The balanced BTP case of the conjecture remained open in complex dimensions at least four after results for non-balanced metrics and balanced threefolds, with only a dimension-specific fourfold result.

  • Method

    The proof encodes Bismut-parallel torsion as a Lie bracket, uses balancedness as unimodularity, and reconstructs Bismut curvature from the bracket and the constant Chern curvature.

  • Results

    For nonzero constant Chern holomorphic sectional curvature, the torsion vanishes and the metric is Kähler; consequently, compact BTP Hermitian manifolds in this setting are Kähler.

  • Takeaways & Limitations

    The paper establishes the nonzero balanced BTP case in arbitrary complex dimension and confirms the corresponding compact BTP consequence.

  • Takeaways & Limitations

    The rigidity result gives no conclusion for zero Chern holomorphic sectional curvature and does not directly apply to other holomorphic sectional curvature notions.

Abstract

from arXiv · show

A well-known conjecture in complex geometry states that a compact Hermitian manifold with constant Chern holomorphic sectional curvature must be Kähler when the constant is nonzero and Chern flat when the constant is zero. The conjecture is known in complex dimension two and in several special classes in higher dimensions. For Hermitian metrics with Bismut-parallel torsion, the non-balanced case and the balanced threefold case were established by Chen--Zheng, while the balanced fourfold case was settled recently by Wang--Zheng. In this article, we prove the nonzero case for balanced Bismut-torsion-parallel Hermitian manifolds in every complex dimension. As a corollary, we confirm that a compact BTP Hermitian manifold with Chern holomorphic sectional curvature is a nonzero constant, then $g$ is Kähler.

1. Introduction

The paper proves the nonzero-curvature case of the conjecture for balanced BTP Hermitian manifolds in every complex dimension, showing that the metric is Kähler. Together with earlier non-balanced results, this confirms the compact BTP case, while the proof proceeds through a pointwise Lie-algebraic analysis.

  • Conjectural context: The result advances the conjecture that compact Hermitian manifolds with constant Hc are Kähler when c ≠ 0 and Chern flat when c = 0.The conjecture was known in complex dimension two and in several higher-dimensional special classes before this work.
  • Main result: Theorem 1.2 shows that a balanced Hermitian manifold with ∇bT = 0 and constant nonzero Chern holomorphic sectional curvature has T = 0, hence is Kähler.The argument requires neither compactness, completeness, homogeneity, a special unitary frame, nor a classification theorem.
  • Main result: Combining this theorem with Chen–Zheng’s non-balanced result [8] yields that every compact BTP Hermitian manifold with nonzero constant Chern holomorphic sectional curvature is Kähler.Chen and Zheng had already treated all non-balanced BTP metrics and balanced BTP threefolds [8], while Wang and Zheng settled the balanced fourfold case [14].
  • Proof strategy: The proof is pointwise: BTP identities define a torsion Lie algebra, Bismut curvature acts by derivations, balancedness gives unimodularity, and constant Chern curvature reconstructs Bismut curvature from c and bracket quadratics.The algebraic argument excludes semisimple quotients, then solvable non-nilpotent algebras, forcing nilpotence; central-direction curvature finally yields vanishing torsion.
  • Limitation: The nonzero assumption is essential because non-Kähler Chern-flat BTP examples exist in complex dimension at least three [17], so the zero-curvature case requires different ideas.In the zero-curvature case, Chern flatness cannot generally be strengthened to Kählerness; compact non-Kähler Chern-flat manifolds already exist in complex dimension at least three.

2. Preliminaries

The preliminaries fix torsion and curvature conventions, characterize balanced BTP geometry through pointwise torsion Lie algebras, and establish the algebraic curvature identities used later. They also record Lie-theoretic consequences of BTP parallelism, including derivation and unimodularity properties.

  • Conventions: The paper fixes the Chern torsion normalization, emphasizing that it is twice some earlier conventions and that squared torsion sums are ordered.This normalization matches the curvature formula of Chen–Zheng and affects numerical coefficients in curvature reconstruction.
  • Torsion Lie algebra: For BTP metrics, the torsion bracket defines a complex Lie algebra, and Bismut curvature acts by derivations of each pointwise torsion Lie algebra.The Jacobi identity follows from the BTP quadratic torsion identity [16, Proposition 1.5], while torsion parallelism yields the derivation property.
  • Balanced metrics: Balancedness is equivalent to vanishing Lee form and, equivalently, to closure of ω^(n−1); pointwise, it is also equivalent to unimodularity of the torsion Lie algebra.The unimodularity characterization connects the differential-geometric balanced condition to the algebraic structure used in the proof.
  • Curvature identities: For BTP metrics with constant Chern holomorphic sectional curvature H_c ≡ c, Chen–Zheng’s identity reconstructs the full Bismut curvature from c and the Chern torsion.The reconstruction is obtained algebraically from polarization and the Chern–Bismut comparison formula, with no derivatives of c, integration, or global topology required.
  • Lie-theoretic tools: The preliminaries collect standard finite-dimensional complex Lie-algebra facts about derivations, radicals, semisimple algebras, nilpotent centers, and solvable or nilpotent structure.They also invoke Lie’s theorem and Engel’s theorem [10] for later structural arguments.

3. The pointwise torsion algebra and two quadratic operators

The section models the torsion at each point as an algebra on a Hermitian vector space and introduces two quadratic operators, A and B. These operators are nonnegative Hermitian, have kernels given by the center and orthogonal complement of the derived algebra, and share the same trace.

  • The torsion defines an algebra on the Hermitian vector space V via the bracket [x,y] = T_p(x,y).
  • The operators A and B are nonnegative Hermitian endomorphisms of V.
  • Their kernels satisfy ker A = Z(g) and ker B = [g,g]⊥.Thus ker A consists of vectors annihilating all torsion inputs, while ker B consists of vectors orthogonal to every bracket output.
  • The operators A and B have equal traces, given by the same complete ordered sum of squared torsion components after relabeling indices.
  • The two kernel conditions are generally distinct: a central vector can still occur as a bracket value, as in the Heisenberg algebra.

4. The contracted curvature derivation

The section derives the contracted curvature identities for balanced metrics with Bismut-parallel torsion. Balancedness removes two contracted torsion contributions, while the resulting scalar term creates the rigidity obstruction used later.

  • 4. The contracted curvature derivation: Lemma 4.1 derives the contracted curvature formula and its traced identity under the balancedness assumption.The derivation combines all curvature and quadratic torsion contributions, then takes the trace using (3.3).
  • 4. The contracted curvature derivation: Balancedness enters precisely through the vanishing of the two contracted −3/4 terms in formula (4.3).The derivation also uses the reconstruction formula, whereas the derivation property of S requires only ∇bT = 0.
  • 4. The contracted curvature derivation: The scalar part of (4.3) supplies the first rigidity mechanism because a nonzero scalar multiple of the identity cannot derive a nonabelian Lie algebra.The derivation identity has one scalar copy on the left-hand side but two on the right-hand side.

5. Excluding a nonzero semisimple quotient

Decomposing the torsion Lie algebra into its radical and orthogonal complement restricts torsion to four types and yields two identities. These identities exclude any nonzero semisimple quotient, proving that c ≠ 0 forces solvability.

  • 5. Excluding a nonzero semisimple quotient: Writing r = Rad(g), U = r⊥, p = dim_C r, and q = dim_C U = n − p, the ideal property gives T(r, V) ⊆ r.The torsion therefore has exactly four possible types: Λ^2r → r, U ∧ r → r, Λ^2U → r, and Λ^2U → U.
  • 5. Excluding a nonzero semisimple quotient: The induced derivation on the semisimple quotient is traceless, yielding 2q(n + 1)c = M + N.Although the orthogonal complement U need not be invariant, its induced quotient block has the same trace, and the Λ^2U → U contribution cancels.
  • 5. Excluding a nonzero semisimple quotient: For c ≠ 0, the torsion Lie algebra g is solvable.The proof rules out both the semisimple case r = 0 and every proper radical 0 < p < n.
  • 5. Excluding a nonzero semisimple quotient: Combining the two identities gives N + 2E = 0, forcing c = 0 unless r = g.Because the relevant coefficients are positive and M, N, E ≥ 0, a proper radical contradicts c ≠ 0.
  • 5. Excluding a nonzero semisimple quotient: The argument retains arbitrary bracket components internal to the radical because the pure-radical quantity P does not enter either lemma.P is absent because the curvature component includes an input from U, while the second identity traces only over U.

6. The solvable case and adjoint weights

For solvable g, the orthogonal complement K=d⊥ realizes the abelianization while serving as the kernel of B, and adjoint weights constrain the Lie algebra. These constraints imply that c≠0 forces g to be nilpotent.

  • The solvable case: Set d=[g,g] and K=d⊥=ker B; K is a Hermitian representative of g/d but need not be a subalgebra.These identities organize the solvable case and identify the complementary space on which nonzero adjoint weights are detected.
  • Adjoint weights: Lie’s theorem yields upper-triangular adjoint maps whose distinct diagonal functionals are the adjoint weights.The finite set of distinct weights, rather than any ordering of repeated diagonal entries, is used throughout.
  • Adjoint weights: Every adjoint weight vanishes on d=[g,g], while if all adjoint weights vanish, Engel’s theorem implies that g is nilpotent.The first statement follows from commutators of upper-triangular matrices having zero diagonal; the second follows because all adjoint maps become strictly upper triangular.
  • The solvable case: If c≠0 and g is solvable, then g is nilpotent.The proof argues by contradiction: a nonzero adjoint weight produces incompatible curvature identities.

7. Central-direction rigidity

Under the reconstruction formula, if every curvature operator is a derivation and the center contains a nonzero direction, then nonzero c forces the bracket—and hence torsion—to vanish. The proof first removes central bracket outputs and then eliminates the entire bracket.

  • Central-direction rigidity: If c ≠ 0, Z(g) ≠ 0, and every curvature operator D_īj is a derivation, Lemma 7.1 concludes that the bracket of g is zero.The argument requires neither solvability nor nilpotency, only one nonzero central direction.
  • Central-direction rigidity: The first derivation identity, combined with centrality and the curvature relation, forces Ω_z = 0 and C_z = 0, eliminating bracket components in the central direction.Centrality removes z as a bracket input, while the derivation identity removes z-direction outputs.
  • Central-direction rigidity: After the central components vanish, the second derivation identity gives c^2T(u,v) = cT(u,v), so c ≠ 0 implies T(u,v) = 0 for all u,v.Thus the entire bracket and torsion vanish, completing the rigidity argument.
  • Central-direction rigidity: The proof distinguishes the two roles of derivation: the first removes central bracket outputs, while the second removes the remaining bracket.This distinction is necessary because centrality alone removes only bracket inputs, as illustrated by the complex Heisenberg algebra.

8. Proof of the main theorem and consequences

The proof shows that nonzero curvature forces the torsion bracket to vanish, hence the metric is Kähler, and derives the stated corollary and universal-cover classification. A finite-dimensional algebraic version isolates the same rigidity mechanism, while low-dimensional examples illustrate the successive exclusions and the argument’s limitations.

  • Proof of Theorem 1.2: Nonzero c forces the torsion algebra to be solvable and nilpotent; its central direction then forces the bracket to vanish, so T = 0 and g is Kähler.The equivalence T = 0 ⇔ ∂ω = 0, together with the reality of ω, gives dω = 0.
  • Consequences: For balanced metrics Theorem 1.2 proves the conclusion directly, while the non-balanced case follows from Chen–Zheng’s theorem.This establishes Corollary 1.3 across both balanced and non-balanced cases.
  • Consequences: If (M, g) is connected and complete, its universal cover is a suitably scaled complex projective space for c > 0 or complex hyperbolic space for c < 0, by the standard classification.The classification applies after Theorem 1.2 makes g Kähler.
  • Algebraic reformulation: Theorem 8.2 gives a finite-dimensional analogue: under the stated Lie-bracket, trace, and derivation conditions, c ≠ 0 implies μ = 0.The proof proceeds through solvability, nilpotency, and elimination of nonzero nilpotent brackets by the central-direction lemma.
  • Low-dimensional checks: The threefold consistency checks exclude the full-rank sl2(C) case by the semisimple trace argument, the rank-two case by adjoint weights, and the Heisenberg case by the central-direction lemma [17].Thus the pointwise argument recovers the low-dimensional case division without using the global classification.
  • Limitations: The rigidity argument requires c ≠ 0 and applies specifically to Chern holomorphic sectional curvature; c = 0 and other holomorphic sectional curvatures require separate analysis.The c = 0 limitation is consistent with non-Kähler Chern-flat examples, while the coefficients change for Riemannian, Bismut, and general Gauduchon curvatures.
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