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Balanced Bismut torsion-parallel fourfolds with constant holomorphic sectional curvature
Qingsong Wang, Fangyang Zheng
TL;DR
The paper addresses the largely open constant holomorphic sectional curvature conjecture for balanced BTP manifolds in complex dimension four. Using curvature identities and a detailed case analysis, it proves that every compact balanced BTP fourfold is Kähler or Chern flat, while identifying the non-Kähler universal-cover model and leaving higher dimensions open.
Problem
The constant holomorphic sectional curvature conjecture remains largely open in dimensions n ≥3, especially for balanced BTP manifolds beyond threefolds.
Source: §1 · pp. 1–2Method
The paper analyzes Bismut torsion and curvature identities through a detailed rank-based case analysis for balanced BTP fourfolds.
Source: §1 · pp. 2–3, §4 · p. 9Results
Every compact balanced BTP fourfold with constant Chern holomorphic sectional curvature is either Kähler or Chern flat; in the non-Kähler case its universal cover is (SL(2, C), λgκ) × (C, g0).
Source: §1 · pp. 2–3Takeaways & Limitations
The result confirms the conjecture for all compact balanced BTP manifolds in complex dimension 4 and provides structural information about their Chern-flat case.
Source: §1 · pp. 2–3Extending the argument to balanced BTP manifolds in dimensions 5 or higher remains open because the case analysis does not seem to extend easily.
Source: §1 · p. 4
Abstract
from arXiv · showhide
An old conjecture in non-Kähler geometry states that, if a compact Hermitian manifold has constant holomorphic sectional curvature, then the metric must be Kähler (when the constant is non-zero) or Chern flat (when the constant is zero). It is known to be true in complex dimension $2$ by the work of Balas and Gauduchon in 1985 (when the constant is negative or zero) and Apostolov, Davidov and Muskarov in 1996 (in general). In dimension $3$ or higher, the conjecture is only known in some special cases, such as for all twistor spaces by the work of Davidov, Grantcharov, and Muskarov, or for the locally conformally Kähler case (when the constant is negative or zero) by the work of H. Chen, L. Chen and Nie, or for all non-balanced Bismut torsion parallel (BTP) manifolds by the work of S. Chen and Zheng, where they also showed that the conjecture holds for all balanced BTP threefolds, utilizing a classification result by Zhao and Zheng. In this article, we show that the conjecture is valid for all balanced BTP manifolds in complex dimension $4$. The interesting part is that while balanced BTP manifolds are highly restrictive, the classification is still lacking in dimensions $4$ or higher, and this study might shed some light on the structure of balanced BTP manifolds in dimension $4$.
1. Introduction and statement of results
The paper studies the constant holomorphic sectional curvature conjecture for compact balanced BTP Hermitian fourfolds, extending known results beyond previously treated special cases. It proves that such metrics are either Kähler or Chern flat and gives an explicit description of the non-Kähler case.
Related work
- Known higher-dimensional cases include twistor spaces, locally conformally Kähler manifolds under c ≤0, and non-balanced BTP manifolds.S. Chen and Zheng also proved the conjecture for balanced BTP threefolds using Zhao and Zheng’s classification.Source: §1 · p. 2
Main result
- The main theorem proves the conjecture for every compact balanced BTP Hermitian fourfold with constant Chern holomorphic sectional curvature.Under these assumptions, the metric is either Kähler or Chern flat.Source: §1 · p. 2
Proof strategy
- The proof addresses the harder balanced non-Kähler setting, where the Gauduchon torsion form vanishes and no flat Bismut-curvature direction is available.Its strategy uses torsion identities, simultaneous diagonalization, and a case analysis involving the rank of the B tensor and the Bismut Ricci tensor.Source: §1 · pp. 2–3
Explicit classification
- In the Chern-flat non-Kähler case, the universal cover is holomorphically isometric to (SL(2, C), λgκ) × (C, g0).The Kähler case is a complex space form.Source: §1 · p. 3
Open directions
- Extending the theorem to balanced BTP manifolds in dimensions 5 or higher remains open because the four-dimensional case analysis does not extend easily.The classification of balanced BTP manifolds is also still wide open in dimensions 4 or higher.Source: §1 · pp. 3–4
2. Preliminaries
The preliminaries translate the BTP and constant-curvature assumptions into curvature and torsion identities. These identities produce globally defined tensors that can be simultaneously diagonalized and used to organize the proof.
Curvature identities
- For a BTP metric with constant Chern holomorphic sectional curvature c, the curvature tensor satisfies a specific identity under every local unitary frame.The Chern holomorphic sectional curvature is computed from the Chern connection.Source: §2 · p. 4
Balanced reduction
- Balancedness is equivalent to vanishing Gauduchon torsion 1-form, and this condition yields an explicit formula for the first Bismut Ricci tensor S.The tensors A and B are globally defined because they are independent of the unitary frame.Source: §2 · p. 4
Algebraic consequences
- The resulting diagonal-frame identities constrain the kernel and orthogonal complement of B and provide the algebraic framework for the later rank-based case analysis.A curvature component vanishes when one index lies in the positive-eigenspace complement of ker(B).Source: §2 · pp. 4–5
Simultaneous diagonalization
- BTP curvature identities imply that B and the first Bismut Ricci tensor S can be simultaneously diagonalized in a local unitary frame.In the balanced constant-curvature setting, A can also be diagonalized simultaneously with B and S.Source: §2 · p. 5
3. Balanced BTP fourfolds with rB < 4
The rB < 4 analysis eliminates ranks 1 and 2, while rank 0 is Kähler and rank 3 yields a Chern-flat product classification. These cases establish the theorem for balanced BTP fourfolds with rB < 4.
rB = 0
- When rB = 0, the torsion tensor vanishes, so the metric is Kähler and the manifold is a complex space form.This is the Kähler branch of the conjectured dichotomy.Source: §3 · p. 6
rB = 1
- The rank rB = 1 case is impossible for balanced BTP manifolds with constant Chern holomorphic sectional curvature in every dimension.The argument derives E = 0 and hence T = 0, contradicting the assumed rank.Source: §3 · pp. 6, 8
rB = 2
- For rB = 2 in dimension 4, the constant-curvature equations force c = 0 and then E = F = 0, contradicting rB = 2.The contradiction follows after showing that c > 0 is impossible and using the resulting identities for the torsion matrices.Source: §3 · pp. 6–8
rB = 3
- For rB = 3, compactness implies that g is Chern flat and its universal cover is holomorphically isometric to (SL(2, C), λgκ) × (C, g0).The flat factor arises from preservation of ker(B), while the three-dimensional factor is Chern flat by the local rank-three argument.Source: §3 · pp. 8–9
Conclusion
- Propositions 1–3 therefore prove Theorem 1 for all balanced BTP fourfolds with rB < 4, leaving only rB = 4 for subsequent sections.The rank decomposition covers rB = 0, 1, 2, and 3.Source: §3 · pp. 6, 9
4. Balanced BTP fourfolds with rB = 4 and S̸ = 0
The section proves that a balanced BTP fourfold with constant Chern holomorphic sectional curvature cannot have rB = 4 when the first Bismut Ricci curvature S is not identically zero. The proof excludes all possible numbers of zero eigenvalues of S through Claims 1–4.
Proposition 4
- rB = 4 and S ≠ 0 cannot occur for a balanced BTP fourfold with constant Chern holomorphic sectional curvature.This is Proposition 4, established by Claims 1–4.Source: §4 · pp. 9, 12
Claim 1
- S must have at least one zero eigenvalue, because assuming all four eigenvalues are nonzero leads to contradictions in each possible sign configuration.The argument orders the eigenvalues λ1 ≥ λ2 ≥ λ3 ≥ λ4 and rules out the resulting cases using torsion-component constraints and the positivity of B.Source: §4 · pp. 9–10
Claim 2
- S cannot have exactly one zero eigenvalue: its nonzero torsion component forces opposite nonzero eigenvalues, and the resulting trace contradiction rules out the case.The proof reaches c = 0 and hence tr S = 0, whereas the chosen eigenvalue form gives tr S = b ≠ 0.Source: §4 · p. 10
Claim 4
- S cannot have exactly three zero eigenvalues: the remaining torsion components and curvature identities contradict the balanced assumption.The proof isolates the only nontrivial torsion component involving the fourth index and derives incompatible identities before concluding the case is impossible.Source: §4 · pp. 10–11
Claim 3
- S cannot have exactly two zero eigenvalues, since both trace cases contradict the positive-definiteness condition implied by rB = 4.For x + y ≠ 0, the torsion identities yield B1̄1B2̄2 = |B1̄2|2; the x + y = 0 case similarly forces incompatible torsion conditions.Source: §4 · p. 11
5. Balanced BTP fourfolds with rB = 4 and S = 0
The section rules out every rB = 4, S = 0 configuration for balanced BTP fourfolds with constant Chern holomorphic sectional curvature. Consequently, the non-Kähler case has rB = 3 and is identified, under compactness, with a quotient of SL(2, C) × C.
Proposition 5
- The case rB = 4 and S = 0 cannot occur for balanced BTP fourfolds with constant Chern holomorphic sectional curvature.This is Proposition 5, established by excluding both possible eigenvalue structures of B.Source: §5 · pp. 12, 17
Eigenvalue cases
- B cannot have an eigenvalue of multiplicity 1, and two distinct eigenvalues of multiplicity 2 are also excluded.The multiplicity-one case is ruled out directly, while the multiplicity-two case forces T = 0, contradicting rB = 4.Source: §5 · pp. 12–13
Scalar case
- When B is scalar, Proposition 6 excludes S = 0 and B = λI with λ > 0 by deriving a contradiction from the torsion identities.After rescaling to B = 2I, the argument constructs a local frame and ultimately obtains T = 0, contradicting rB = 4.Source: §5 · pp. 13–14
- The scalar-case frame construction produces normalized torsion components, but the resulting identities force T = 0 and complete the contradiction.A unitary change of frame makes the relevant skew-symmetric matrix singular and yields the stated torsion normal form before the final contradiction.Source: §5 · pp. 14–15
Theorem 1
- Together with earlier propositions, the exclusions imply that a non-Kähler balanced BTP fourfold has rB = 3 and, if compact, is a quotient of SL(2, C) × C.The metric is the product of a constant multiple of the standard Killing metric on SL(2, C) and the flat Euclidean metric.Source: §5 · p. 17