On balanced Hermitian threefolds with parallel Bismut torsion

Methods and scope

• The analysis is restricted to dimension 3 because balanced BTP manifolds lack admissible frames, while special local unitary frames exist for balanced threefolds.The B tensor is globally defined with constant eigenvalues and rank under the BTP condition.Source: §1 · p. 2

Theorem 1.1

• Rank 3 forces the metric to be Chern flat, with universal cover SL(2, C) carrying a scaled standard metric and holomorphically trivial canonical bundle.These manifolds are compact quotients of SL(2, C).Source: §1 · p. 2
• Rank 1 identifies the Wallach threefold, namely the flag threefold P(TP2) with its invariant Killing-form metric.Source: §1 · p. 2
• Rank 2 defines the middle type, whose kernel line bundle L satisfies L⊗2 ∼= OM and is primary when L is trivial, otherwise secondary.The secondary case admits a double cover that is primary.Source: §1 · p. 2
• Primary middle-type manifolds have abelian Bismut holonomy U(1)×U(1)×1 and admit Vaisman companions sharing the same Bismut connection.Secondary manifolds have a non-abelian Z2-extension of this holonomy group.Source: §1 · p. 2

Further consequences

• The first two cases are BAS, while middle type is BAS exactly when its curvature-determining functions s and σ2 are constant.BAS means that both Bismut torsion and curvature are parallel.Source: §1 · p. 2
• Every compact balanced BTP threefold of middle type cannot admit a pluriclosed metric, so the Fino-Vezzoni and Streets-Tian conjectures hold for compact BTP threefolds.The Fano case is already Kählerian, leaving middle type as the relevant case for the balanced setting.Source: §1 · p. 3

Proof mechanism

• The proof uses the BTP equivalence ∇bT = 0, the balanced condition, and frame changes to derive torsion and connection constraints.The B tensor is globally defined and Hermitian symmetric, while special frames diagonalize its torsion data.Source: §2 · pp. 3, 5

Special frames

• Balanced threefolds admit local special unitary frames simplifying Chern torsion, and BTP makes the B tensor’s eigenvalues and rank globally constant.The special-frame construction applies wherever the B tensor has constant rank, which BTP guarantees globally.Source: §2 · pp. 4–5
• In a special frame, only three torsion components can be nonzero; they become global constants a1 ≥ a2 ≥ a3, and the Bismut equations impose relations among them.The constants encode the possible ranks of the B tensor.Source: §2 · pp. 4–5

Rank 3

• Rank 3 yields a compact quotient of SL(2, C) with a scaled Killing metric, holomorphically trivial canonical bundle, and restricted Bismut holonomy contained in SO(3) ⊆ U(3).The standard metric on SL(2, C) is balanced, non-Kähler, Chern flat, and BTP.Source: §2 · pp. 7–8

• The Kähler form induced by the first Chern Ricci form is Kähler-Einstein with positive Ricci curvature 2, so M3 is a Fano threefold.Source: §3 · p. 9
• The rank B = 1 splitting produces a holomorphic distribution E and a complementary foliation F, yielding the exact sequence involving TM/E.Source: §3 · p. 10
• Chern-class identities imply that the anticanonical bundle is L⊗2 for an ample line bundle L, so the Fano index is either 4 or 2.Source: §3 · p. 10
• The classification of Fano threefolds leaves only P3 and the flag threefold P(TP2) after Chern-number constraints exclude the other del Pezzo possibilities.Source: §3 · p. 10
• A nowhere-zero holomorphic section ξ of Ω(L) has constant norm under the Kähler-Einstein metric, but P3 would force that norm to be nonconstant, excluding P3.Source: §3 · pp. 11–12
• Thus the only rank B = 1 compact balanced BTP threefold is the flag threefold X3 = P(TP2), whose Wallach metric is analyzed next.Source: §3 · pp. 12–13

• The Wallach metric is defined on X3 = P(TP2) by subtracting a global (1,1)-form σ from the restricted product Fubini-Study Kähler-Einstein metric.Source: §4 · p. 13
• The computed Chern bisectional curvature is nonnegative, while the Chern holomorphic sectional curvature is positive for every nonzero tangent vector.Source: §4 · p. 15
• The three Chern Ricci forms are positive definite, with the first and third Ricci forms equal.Source: §4 · p. 15
• At the chosen origin, the torsion computation gives vanishing Gauduchon torsion 1-form, hence the metric is balanced.Source: §4 · p. 15
• The remaining component checks establish that the Wallach threefold is non-Kähler, balanced, and BTP.Source: §4 · pp. 15–16
• As a Riemannian manifold, the Wallach metric is Einstein with constant Ricci curvature 3 and has nonnegative sectional curvature, lying on the boundary of Wallach's positively curved region.Source: §4 · p. 16

• A holomorphic line bundle L satisfies L⊗2 ∼= O_M, and its square-triviality makes L globally defined from local special frames.Source: §5.1 · p. 16
• The Bismut curvature is determined by the global functions s and σ2; constancy makes the metric Bismut Ambrose–Singer and gives a homogeneous universal cover.Source: §5.1 · p. 18
• Middle-type threefolds do not admit pluriclosed metrics, and therefore establish both the Fino–Vezzoni and Streets–Tian conjectures for compact BTP threefolds.Source: §5.1 · p. 18
• At the Lie-algebra level, every balanced BTP structure of middle type belongs to one of the families A_s,t or B_z,t, with vanishing Bismut Ricci form and specified nonzero curvature components.Source: §5.2 · pp. 19–20
• The Bismut holonomy is abelian exactly in the primary case, where it equals U(1)×U(1)×1; secondary cases have a non-abelian Z2-extension and admit a primary double cover.Source: §5.1 · p. 17, §5.3 · pp. 24–25
• Primary structures possess holomorphic foliations and integrable companion complex structures whose Vaisman metrics share the original Bismut connection.Source: §5.3 · pp. 25–27

6. Generalization to higher dimensions

• Classification of compact balanced BTP manifolds in dimensions 4 and higher is currently unresolved.The authors suggest investigating higher-dimensional analogues of the Chern flat, Fano, and middle-type threefold cases.Source: §6 · p. 29
• The same rigidity and existence pattern extends to certain Kähler C-spaces K/H whose isotropy representation has at most two direct summands.The Killing metric is Kähler exactly for compact Hermitian symmetric spaces; the remaining listed spaces yield non-Kähler balanced BTP examples.Source: §6 · p. 29
• The construction relies on a Bismut-connection-preserved orthogonal splitting TM = L1 ⊕ L2 ⊕ L, with L generated by the vector dual to Gauduchon’s torsion form.The proof verifies integrability, balancedness, and equality of the Bismut connections using adapted frames.Source: §6 · pp. 30–31
• A splitting-type Vaisman manifold yields a balanced BTP Hermitian manifold after reversing the complex structure on one preserved subbundle.The resulting complex structure is integrable, and the two Hermitian manifolds share the same Bismut connection.Source: §6 · pp. 30–31

7. Appendix

• The Wallach threefold’s Ricci curvature is constantly 3, so its metric is Einstein.The appendix derives Ric(x,x) = 3|x|^2.Source: §7 · pp. 32–34
• For the Wallach threefold, the relevant curvature components vanish at the origin under the displayed torsion normalization.The appendix then uses these components in the Riemannian curvature computation.Source: §7 · p. 32
• The curvature calculation reduces sectional curvature to sums of squares involving imaginary parts and antisymmetric products of complex tangent components.This sum-of-squares form directly gives nonnegativity.Source: §7 · p. 33
• The Wallach threefold has nonnegative Riemannian sectional curvature, but some nonzero two-planes have zero sectional curvature.The curvature expression is a sum of nonnegative terms, and the authors exhibit a configuration where it vanishes.Source: §7 · p. 33