[Paper Review] On compact Hermitian manifolds with flat Gauduchon connections

Bo Yang, Fangyang Zheng|arXiv (Cornell University)|Sep 8, 2017

Geometry and complex manifolds8 references3 citations

TL;DR

This paper investigates compact Hermitian manifolds admitting a flat s-Gauduchon connection, a one-parameter family of Hermitian connections interpolating between the Chern and Bismut connections. Using curvature identities and the Bochner formula for the minimal Gauduchon connection (s = 2/3), the authors prove that any compact Hermitian surface with a flat s-Gauduchon connection must be Kähler unless s = 2, in which case non-Kähler examples (isosceles Hopf surfaces) exist. For higher dimensions, flatness for s ≥ 4 + 2√3 or s ≤ 4 − 2√3 (s ≠ 0) also implies the metric is Kähler.

ABSTRACT

Given a Hermitian manifold $(M^n,g)$, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call $ abla^s = (1-\frac{s}{2}) abla^c + \frac{s}{2} abla^b$ the $s$-Gauduchon connection of $M$, where $ abla^c$ and $ abla^b$ are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat $s$-Gauduchon connection. This is related to a question asked by Yau \cite{Yau}. The cases with $s=0$ (a flat Chern connection) or $s=2$ (a flat Bismut connection) are classified respectively by Boothby \cite{Boothby} in the 1950s or by Q. Wang and the authors recently \cite{WYZ}. In this article, we observe that if either $s\geq 4+2\sqrt{3} \approx 7.46$ or $s\leq 4-2\sqrt{3}\approx 0.54$ and $s eq 0$, then $g$ is Kähler. We also show that, when $n=2$, $g$ is always Kähler unless $s=2$. Note that non-Kähler compact Bismut flat surfaces are exactly those isosceles Hopf surfaces by \cite{WYZ}.

Motivation & Objective

• To classify compact Hermitian manifolds that admit a flat s-Gauduchon connection, a one-parameter family of Hermitian connections joining the Chern and Bismut connections.
• To resolve a conjecture by Yau on holonomy reduction in Hermitian geometry by determining when flat Hermitian connections force a manifold to be Kähler.
• To extend known classifications of flat Chern (s=0) and flat Bismut (s=2) connections to intermediate values of s.
• To investigate whether non-Kähler compact Hermitian manifolds can admit flat s-Gauduchon connections for s ≠ 0,2.
• To establish sharp bounds on s for which non-Kähler s-Gauduchon flat manifolds can exist in higher dimensions.

Proposed method

• Derives curvature identities for the s-Gauduchon connection using the torsion tensor T and its dual η, particularly analyzing the structure of the curvature components.
• Applies the Bochner formula to the function f = log(λ + ε), where λ = |T|², to study the behavior of the torsion tensor under flatness of the s-Gauduchon connection.
• Uses the identity ∂∂̄f ∧ ω = (sum of curvature terms) ω² to derive a differential inequality that forces λ ≡ 0 when s = 2/3, implying the metric is Kähler.
• Analyzes the coefficient of ∫|T|²ωⁿ in curvature identities to determine when flatness forces T ≡ 0, thus implying the metric is Kähler.
• Applies the identity (3s−2)(s−2)∫|T|²ω² = 0 for complex surfaces to show that flatness for s ≠ 2 and s ≠ 2/3 implies Kähler metric.
• Specializes the general curvature identities to the case n=2 and s=2/3 to prove the minimal Gauduchon connection case via the Bochner formula.

Experimental results

Research questions

• RQ1 For which values of s does a compact Hermitian manifold with a flat s-Gauduchon connection necessarily have to be Kähler?
• RQ2 Can non-Kähler compact Hermitian manifolds admit a flat s-Gauduchon connection for s ≠ 0,2?
• RQ3 What are the sharp bounds on s for which non-Kähler s-Gauduchon flat manifolds can exist in dimensions n ≥ 3?
• RQ4 Does the flatness of the minimal Gauduchon connection (s = 2/3) on a compact Hermitian surface force the metric to be Kähler?
• RQ5 How do curvature identities involving torsion and the Gauduchon parameter s constrain the geometry of s-Gauduchon flat manifolds?

Key findings

• For compact Hermitian surfaces (n=2), if the s-Gauduchon connection is flat and s ≠ 2, then the metric must be Kähler, so the surface is a finite quotient of a complex torus.
• The only non-Kähler compact Hermitian surfaces with flat s-Gauduchon connection occur precisely when s = 2, corresponding to isosceles Hopf surfaces.
• For n ≥ 3, if s ≥ 4 + 2√3 ≈ 7.46 or s ≤ 4 − 2√3 ≈ 0.54 (and s ≠ 0), then flatness of the s-Gauduchon connection implies the metric is Kähler.
• The interval (4−2√3, 4+2√3) is the only range where non-Kähler s-Gauduchon flat manifolds could potentially exist in higher dimensions.
• For n=3, the critical interval for potential non-Kähler examples is approximately [0.56, 4.77], based on the roots bₙ±.
• In the locally conformally Kähler case, flatness of the s-Gauduchon connection forces T ≡ 0 (hence Kähler) unless s is one of the two roots bₙ±, which are the only values allowing non-Kähler solutions.