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Bismut torsion parallel metrics with constant holomorphic sectional curvature
Shuwen Chen, Fangyang Zheng
TL;DR
It remains an open question whether a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler for c ≠ 0 or Chern flat for c = 0. The paper studies compact Hermitian manifolds whose Bismut connection has parallel torsion, a condition defining BTP manifolds. The conjectures hold for all non-balanced BTP manifolds and balanced BTP threefolds, and therefore for all compact Vaisman manifolds.
Problem
It remains an open question whether a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler for c ≠ 0 or Chern flat for c = 0.
Source: §1 · p. 1Method
The paper studies compact Hermitian manifolds whose Bismut connection has parallel torsion, a condition defining BTP manifolds.
Source: §1 · p. 2Results
The conjectures hold for all non-balanced BTP manifolds and balanced BTP threefolds, and therefore for all compact Vaisman manifolds.
Source: §1 · p. 3Takeaways & Limitations
The result extends constant-holomorphic-sectional-curvature rigidity to BTP classes that include Bismut Kähler-like and Vaisman manifolds.
Source: §1 · pp. 2–3Balanced BTP manifolds in dimensions n ≥ 4 remain unresolved, and the zero-curvature Bismut case has counterexamples for n ≥ 3.
Source: §1 · pp. 3–4
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 (when the constant is positive). In dimension $3$ or higher, the conjecture is only known in some special cases, such as the locally conformally Kähler case (when the constant is negative or zero) by the work of Chen, Chen and Nie, or for complex nilmanifolds with nilpotent $J$ by the work of Li and the second named author. In this note, we confirm the above conjecture for all non-balanced Bismut torsion parallel (BTP) manifolds. Here the BTP condition means that the Bismut connection has parallel torsion. In particular, the conjecture is valid for all Vaisman manifolds.
1. Introduction and statement of results
The paper studies longstanding constant holomorphic sectional curvature conjectures for compact Hermitian manifolds and proves them for broad classes of BTP manifolds. Its main scope includes all non-balanced BTP manifolds, balanced BTP threefolds, and consequently all compact Vaisman manifolds.
Conjecture and prior work
- The Constant Holomorphic Sectional Conjecture asserts that constant curvature forces a compact Hermitian metric to be Kähler when c ≠ 0 and Chern flat when c = 0.The conjecture is known in complex dimension two but remains largely open in dimensions n ≥ 3, with only several special-case results.Source: §1 · pp. 1–2
Main result
- The paper confirms Conjectures 1 and 2 for all non-balanced BTP manifolds and all balanced BTP threefolds.BTP means that the Bismut connection has parallel torsion; the class includes Bismut Kähler-like and Vaisman manifolds.Source: §1 · pp. 2–3
- Because every Vaisman manifold is non-balanced BTP, the conjectures hold for all compact Vaisman manifolds.This consequence is especially new for the positive-constant case, since the locally conformally Kähler case was already known for c ≤ 0.Source: §1 · pp. 2–3
Extensions
- The paper also proves the Bismut and t = −1 Gauduchon conjectures for compact non-balanced BTP manifolds and balanced BTP threefolds.For other Gauduchon connections, including Chern, and for the Riemannian connection, the constant-curvature conjecture holds on the same classes.Source: §1 · pp. 3–4
Scope and limitations
- For the Bismut and t = −1 connections, the result applies when c ≠ 0, while c = 0 has counterexamples in dimensions n ≥ 3.The zero-curvature Bismut case includes standard metrics on isosceles Hopf manifolds with Hb = 0 but nonvanishing Bismut curvature.Source: §1 · pp. 3–4
2. Preliminaries
The preliminaries establish curvature conventions and relate the Chern, Bismut, Riemannian, and t-Gauduchon connections. They also define BTP geometry and identify admissible frames available on non-balanced BTP manifolds.
- Holomorphic sectional curvature is obtained by restricting a metric connection’s sectional curvature to J-invariant 2-planes, represented by X = x − iJx of type (1, 0).The setup uses complex multilinear extensions of the metric and curvature tensors.Source: §2 · p. 4
- The Chern, Bismut, and Riemannian curvature tensors are compared through formulas involving torsion and its covariant derivatives, with Bismut derivatives used for BTP manifolds.These relations culminate in formulas connecting Riemannian curvature to Chern curvature, Bismut curvature, and t-Gauduchon curvature.Source: §2 · pp. 5–6
- Bismut torsion-parallel (BTP) means that the Bismut connection has parallel torsion, equivalently ∇bT = 0 because Bismut torsion can be expressed through Chern torsion and vice versa.The preliminaries explicitly introduce BTP to support later replacement of Chern derivatives by Bismut derivatives.Source: §2 · p. 5
- The t-Gauduchon connections form the affine family ∇(t) = (1 − t)∇ + t∇b joining the Chern and Bismut connections.Their curvature is related to Bismut curvature by a formula used later in the proof.Source: §2 · p. 6
- Every non-balanced BTP manifold admits local admissible unitary frames with constants λ > 0 and torsion-form coefficients satisfying an = 0 and a1 + · · · + an−1 = λ.The torsion 1-form η vanishes exactly in the balanced case, so non-balanced metrics have η ≠ 0 and satisfy the stated frame condition.Source: §2 · p. 7
3. BTP manifolds with constant holomorphic sectional curvature
For BTP manifolds, constant holomorphic sectional curvature constrains the full Bismut curvature through Chern torsion, ruling out such constants broadly in the non-balanced case and in balanced threefolds.
Curvature reduction
- Constant Riemannian or t-Gauduchon holomorphic sectional curvature lets the full Bismut curvature be expressed using Chern torsion components.This curvature reduction is the main mechanism used to test the conjectures.Source: §3 · pp. 7–8
Non-balanced case
- Non-balanced BTP manifolds cannot have constant Riemannian holomorphic sectional curvature.The constant-curvature assumption forces c = 0 and all torsion parameters to vanish, contradicting their positive sum.Source: §3 · p. 8
- For non-balanced BTP manifolds, every t-Gauduchon connection with t ≠ ±1 has nonconstant holomorphic sectional curvature.At t = ±1, the curvature can only be constant when c = 0, so the Bismut and mirror connections admit no non-zero constant.Source: §3 · pp. 8–9
Balanced threefolds
- Balanced BTP manifolds remain difficult in general, but balanced BTP threefolds can be handled using their classification into explicit torsion-rank cases.The classification includes the Chern-flat SO(3, C) quotient and the Wallach threefold, among other cases.Source: §3 · p. 9
- Balanced non-Kähler BTP threefolds have no constant Riemannian holomorphic sectional curvature.The curvature expression derived from the constant-curvature assumption contradicts every classified torsion-rank case.Source: §3 · pp. 9–10