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Hermitian metrics on complex non-Kähler manifolds

Daniele Angella

TL;DR

This survey analyzes analytic problems for the Chern connection on Hermitian manifolds, emphasizing non-Kähler geometry through three pillars: constant Chern curvature problems, conformal (Chern) Yamabe-type questions, and evolution flows. It develops the Gauduchon framework of Hermitian connections, clarifying how different curvature traces and torsion components influence existence and uniqueness results, including non-Kähler Calabi–Yau (first-Chern-Ricci flat) metrics when $c_1^{BC}(X)=0$, and the Chern–Yamabe problem via Gauduchon representatives and the Gauduchon degree. The paper also discusses Chern–Einstein problems, their conformal reduction, and obstructions, with Hopf and Inoue–Bombieri surfaces as key compact examples illustrating both positive and degenerate regimes. Finally, it surveys Hermitian flows—notably the Chern–Ricci flow, pluriclosed flow, and Hull–Strominger system—documenting convergence phenomena on compact non-Kähler surfaces and highlighting the deep connections to generalized geometry and string-theoretic contexts.

Abstract

In this survey, we consider various analytic problems related to the geometry of the Chern connection on Hermitian manifolds, such as the existence of metrics with constant Chern-scalar curvature, generalizations of the Kähler-Einstein condition to the non-Kähler setting, and the convergence of the Chern-Ricci flow on compact complex surfaces.

Hermitian metrics on complex non-Kähler manifolds

TL;DR

This survey analyzes analytic problems for the Chern connection on Hermitian manifolds, emphasizing non-Kähler geometry through three pillars: constant Chern curvature problems, conformal (Chern) Yamabe-type questions, and evolution flows. It develops the Gauduchon framework of Hermitian connections, clarifying how different curvature traces and torsion components influence existence and uniqueness results, including non-Kähler Calabi–Yau (first-Chern-Ricci flat) metrics when , and the Chern–Yamabe problem via Gauduchon representatives and the Gauduchon degree. The paper also discusses Chern–Einstein problems, their conformal reduction, and obstructions, with Hopf and Inoue–Bombieri surfaces as key compact examples illustrating both positive and degenerate regimes. Finally, it surveys Hermitian flows—notably the Chern–Ricci flow, pluriclosed flow, and Hull–Strominger system—documenting convergence phenomena on compact non-Kähler surfaces and highlighting the deep connections to generalized geometry and string-theoretic contexts.

Abstract

In this survey, we consider various analytic problems related to the geometry of the Chern connection on Hermitian manifolds, such as the existence of metrics with constant Chern-scalar curvature, generalizations of the Kähler-Einstein condition to the non-Kähler setting, and the convergence of the Chern-Ricci flow on compact complex surfaces.

Paper Structure

This paper contains 18 sections, 5 theorems, 22 equations.

Table of Contents

  1. The holomorphic landscape
  2. Almost-complex structures
  3. Complex cohomologies
  4. Hermitian metrics
  5. Complex algebraic geometry and Kähler geometry
  6. Riemannian metrics on differentiable manifolds
  7. Space forms
  8. Einstein metrics
  9. Yamabe problem
  10. Hermitian metrics on holomorphic manifolds
  11. Gauduchon connections
  12. Curvatures
  13. Special metrics
  14. Analytic problems for the Chern connection
  15. Constant holomorphic sectional curvature
  16. ...and 3 more sections

Key Result

Theorem 1

On a compact holomorphic manifold, any conformal class of Hermitian metrics contains a unique Gauduchon representative of volume $1$.

Theorems & Definitions (5)

  • Theorem 1: gauduchon-cras
  • Theorem 2: angella-calamai-spotti
  • Theorem 3: tosattitosatti-weinkove
  • Theorem 4: angella-calamai-spotti-2
  • Theorem 5: fang-tosatti-weinkove-zhengangella-tosatti