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Pluriclosed 3-folds with vanishing Bismut Ricci form: General theory in the quasi-regular case

Vestislav Apostolov, Abdellah Lahdili, Kuan-Hui Lee

TL;DR

The paper analyzes compact non-Kähler Bismut–Hermitian–Einstein 3-folds through a quasi-regular reduction to a 6th-order nonlinear PDE on a 2–dimensional Kähler base, revealing a formal infinite-dimensional momentum-map framework akin to the cscK problem. It establishes the Mabuchi functional, Futaki character, and Calabi–Lichnerowicz–Matsushima obstructions in this setting, and proves convexity and conditional uniqueness results for extremal solutions, with a LeBrun–Simanca openness result enabling perturbative existence. Classification and rigidity results show that, in the regular case, Samelson geometries arise precisely when the base has 2-dimensional Bott–Chern cohomology and the reduced space is smooth, while explicit toric constructions yield infinitely many non-Samelson BHE 3-folds, including on $S^3 imes S^3$ and $S^1 imes S^2 imes S^3$. The work also provides explicit orthotoric solutions on toric orbifold surfaces with $b_2=2$, expanding the known landscape of non-Kähler canonical metrics in dimension three. Overall, the paper develops a robust variational and momentum-map apparatus for 6th-order Kähler-type equations in the quasi-regular BHE setting and derives substantive existence, obstruction, and explicit example results with connections to supersymmetric AdS backgrounds and complex geometric rigidity phenomena.

Abstract

We study compact complex $3$-dimensional non-Kähler Bismut Ricci flat pluriclosed Hermitian manifolds (BHE) via their dimensional reduction to a special Kähler geometry in complex dimension $2$, recently obtained by Barbaro, Streets and the first and third authors. We show that in the quasi-regular case, the reduced geometry satisfies a 6th order non-linear PDE which has infinite dimensional momentum map interpretation, similar to the much studied Kähler metrics of constant scalar curvature (cscK). We use this to associate to the reduced manifold or orbifold Mabuchi and Calabi functionals, as well as to obtain obstructions for the existence of solutions in terms of the authomorphism group, paralleling results by Futaki and Calabi-Lichnerowicz-Matsushima in the cscK case. This is used to characterize the Samelson locally homogeneous BHE geometries in complex dimension 3 as the only non-Kähler BHE $3$-folds with $2$-dimensional Bott-Chern $(1,1)$-cohomology group, for which the reduced space is a smooth Kähler surface. We also discuss explicit solutions of the PDE on orthotoric Kähler orbifold surfaces, extending examples found by Couzens-Gauntlett-Martelli-Sparks in the framework of supersymmetric ${\rm AdS}_3 \times Y_7$ type IIB supergravity. Our construction yields infinitely many non-Kähler BHE structures on $S^3\times S^3$ and $S^1\times S^2 \times S^3$, which are not locally isometric to a Samelson geometry. These appear to be the first such examples.

Pluriclosed 3-folds with vanishing Bismut Ricci form: General theory in the quasi-regular case

TL;DR

The paper analyzes compact non-Kähler Bismut–Hermitian–Einstein 3-folds through a quasi-regular reduction to a 6th-order nonlinear PDE on a 2–dimensional Kähler base, revealing a formal infinite-dimensional momentum-map framework akin to the cscK problem. It establishes the Mabuchi functional, Futaki character, and Calabi–Lichnerowicz–Matsushima obstructions in this setting, and proves convexity and conditional uniqueness results for extremal solutions, with a LeBrun–Simanca openness result enabling perturbative existence. Classification and rigidity results show that, in the regular case, Samelson geometries arise precisely when the base has 2-dimensional Bott–Chern cohomology and the reduced space is smooth, while explicit toric constructions yield infinitely many non-Samelson BHE 3-folds, including on and . The work also provides explicit orthotoric solutions on toric orbifold surfaces with , expanding the known landscape of non-Kähler canonical metrics in dimension three. Overall, the paper develops a robust variational and momentum-map apparatus for 6th-order Kähler-type equations in the quasi-regular BHE setting and derives substantive existence, obstruction, and explicit example results with connections to supersymmetric AdS backgrounds and complex geometric rigidity phenomena.

Abstract

We study compact complex -dimensional non-Kähler Bismut Ricci flat pluriclosed Hermitian manifolds (BHE) via their dimensional reduction to a special Kähler geometry in complex dimension , recently obtained by Barbaro, Streets and the first and third authors. We show that in the quasi-regular case, the reduced geometry satisfies a 6th order non-linear PDE which has infinite dimensional momentum map interpretation, similar to the much studied Kähler metrics of constant scalar curvature (cscK). We use this to associate to the reduced manifold or orbifold Mabuchi and Calabi functionals, as well as to obtain obstructions for the existence of solutions in terms of the authomorphism group, paralleling results by Futaki and Calabi-Lichnerowicz-Matsushima in the cscK case. This is used to characterize the Samelson locally homogeneous BHE geometries in complex dimension 3 as the only non-Kähler BHE -folds with -dimensional Bott-Chern -cohomology group, for which the reduced space is a smooth Kähler surface. We also discuss explicit solutions of the PDE on orthotoric Kähler orbifold surfaces, extending examples found by Couzens-Gauntlett-Martelli-Sparks in the framework of supersymmetric type IIB supergravity. Our construction yields infinitely many non-Kähler BHE structures on and , which are not locally isometric to a Samelson geometry. These appear to be the first such examples.
Paper Structure (22 sections, 30 theorems, 210 equations)

This paper contains 22 sections, 30 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Quasi-regular BHE 3-folds: reduction to a 6th order PDE
  3. The transversal Kähler geometry of a BHE and the reverse construction
  4. Regular BHE's from compact Kähler surfaces
  5. Kodaira's classification of complex surfaces possibly admitting solutions to \ref{['PDEA']}
  6. Higher-dimensional setup
  7. A momentum map interpretation
  8. The Calabi--Lichnerowicz--Matsushima obstruction
  9. The Mabuchi functional and Futaki character
  10. The Futaki character
  11. The Calabi--Lichnerowicz--Matsushima obstruction revisited
  12. Convexity of the Mabuchi functional along smooth geodesics and conditional uniqueness
  13. The linearization of the 6th order PDE
  14. Relaxing \ref{['PDEAn']}: the extremal Kähler metrics for \ref{['PDEAn']} and the LeBrun--Simanca openness result
  15. Applications to BHE 3-folds: Proof of Theorem \ref{['thm:main2']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

ABLS Let $(N, J_N, g_N, \omega_N)$ be a compact complex 3-dimensional non-Kähler Hermitian manifold such that the Kähler form $\omega_N$ is pluriclosed and the Ricci form of the Bismut connection is identically zero. Then the Bott--Chern cohomology group $H^{1,1}_{\rm BC}(N,J_N)$ is at least $2$- di where $R(\omega_K^T)$ and $\rho(\omega_K^T)$ stand, respectively, for the scalar curvature and Ricc

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 66 more