[Paper Review] Generalized Kaehler manifolds with split tangent bundle

Vestislav Apostolov, Marco Gualtieri|ArXiv.org|May 12, 2006

Geometry and complex manifolds23 references3 citations

TL;DR

This paper classifies compact generalized Kähler four-manifolds with split tangent bundles where the induced complex structures $J_+$ and $J_-$ commute and yield opposite orientations. It proves that such manifolds must be complex surfaces with holomorphically split tangent bundles, and completely classifies them as either complex surfaces of Kodaira dimension 0 or 1, or specific Hopf surfaces, with explicit existence of generalized Kähler metrics established via conformal geometry and Lee form normalization.

ABSTRACT

We study generalized Kaehler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kaehler four-manifolds for which the induced complex structures yield opposite orientations.

Motivation & Objective

• To classify compact generalized Kähler four-manifolds where the induced complex structures $J_+$ and $J_-$ commute and induce opposite orientations.
• To determine which compact complex surfaces admit non-trivial generalized Kähler structures arising from a split tangent bundle $TM = T_-M igoplus T_+M$.
• To establish the existence of generalized Kähler metrics on these surfaces using conformal geometry and the normalization of Lee forms.
• To resolve the existence problem for ambihermitian generalized Kähler structures on complex surfaces with split holomorphic tangent bundles.

Proposed method

• Utilizes the equivalence between generalized Kähler structures and triples $(g, J_+, J_-)$ with commuting complex structures and $d^c_+F_+ + d^c_-F_- = 0$, where $F_ullet = gJ_ullet$.
• Applies the condition that $T_{ullet}M$ are holomorphic, integrable sub-bundles of $TM$, leading to a holomorphic foliation structure on $M$.
• Employs Beauville's conjecture on holomorphic tangent bundle splitting, confirming that $TM$ splits iff $M$ is locally a product of complex manifolds with diagonal $ ho_1(M)$-action.
• Uses Gauduchon’s result on $ard$-closed fundamental forms in conformal classes to construct a standard metric $g_0$ with $ heta_{g_0}^{g_0}$ harmonic.
• Leverages the fact that $ heta_+^g = heta_-^g$ for metrics compatible with the splitting, and normalizes $g_0$ so that $ard( heta_{g_0}^{g_0}) = 0$ to yield a generalized Kähler structure.
• Applies results from [8], [50], and [22] to classify surfaces with $b_1(M)$ even or odd, excluding non-split cases via monodromy and covering space arguments.

Experimental results

Research questions

• RQ1 Which compact complex four-manifolds admit a non-trivial generalized Kähler structure with commuting complex structures $J_+$ and $J_-$?
• RQ2 What are the necessary and sufficient conditions on a complex surface for its holomorphic tangent bundle to split into two holomorphic, integrable sub-bundles?
• RQ3 For such split tangent bundles, does there exist a Riemannian metric $g$ compatible with both $J_+$ and $J_-$ such that $(g, J_+, J_-)$ defines a generalized Kähler structure?
• RQ4 Which complex surfaces of Kodaira dimension 0 or 1, or Hopf surfaces, admit generalized Kähler metrics when $J_+$ and $J_-$ commute and induce opposite orientations?
• RQ5 Can the existence of such generalized Kähler structures be established via conformal geometry and normalization of the Lee form?

Key findings

• All compact generalized Kähler four-manifolds with split tangent bundle and commuting complex structures $J_+$, $J_-$ inducing opposite orientations are classified into six types: (a)–(f) in Theorem 1.
• The classification includes complex surfaces of Kodaira dimension 0 (K3, Enriques, primary/secondary Kodaira surfaces) and 1 (elliptic surfaces with $b_1$ even or odd), and specific Hopf surfaces.
• For surfaces with $b_1(M)$ even, the classification matches Beauville’s results: $M$ is either a product of Riemann surfaces or a ruled surface with projectively flat bundle.
• For $b_1(M)$ odd, only certain Hopf surfaces (e.g. with $ ho_1(M)$ acting diagonally on $bC imes bH$) admit such structures, excluding those with non-diagonal monodromy.
• The existence of generalized Kähler metrics is proven by constructing a standard metric $g_0$ in the conformal class $[g]$ such that $ard( heta_{g_0}^{g_0}) = 0$, ensuring the triple $(g_0, J_+, J_-)$ satisfies the generalized Kähler conditions.
• The family of such generalized Kähler metrics depends on one arbitrary smooth function, reflecting the freedom in choosing the conformal factor.