[Paper Review] A flat Higgs bundle structure on the complexified Kähler cone

Xu Wang|arXiv (Cornell University)|Dec 7, 2016

Algebraic Geometry and Number Theory23 references3 citations

TL;DR

This paper constructs a flat Higgs bundle structure on the complexified Kähler cone of a compact Kähler manifold, analogous to variation of Hodge structure. Using a generalized Lu's Hodge metric and a fundamental commutator identity for polarized Hodge-Lefschetz modules, it proves the Hodge metric induces a Kähler metric with negative holomorphic sectional curvature, providing a new result on Wilson's conjecture regarding curvature bounds on the Kähler cone.

ABSTRACT

We shall construct a natural Higgs bundle structure on the complexified Kähler cone of a compact Kähler manifold, which can be seen as an analogy of the classical Higgs bundle structure associated to a variation of Hodge structure. In the proof of the flat-ness of our Higgs bundle, we find a commutator identity that can be used to decode the variational properties of the polarized Hodge-Lefschetz module structure on the fibres of our Higgs bundle. Thus we can use a generalized version of Lu's Hodge metric to study the curvature property of the complexified Kähler cone. In particular, it implies that the above Hodge metric defines a Kähler metric on the complexified Kähler cone with negative holomorphic sectional curvature, which can be seen as a new result on Wilson's conjecture.

Motivation & Objective

• To establish a natural Higgs bundle structure on the complexified Kähler cone of a compact Kähler manifold, analogous to classical Higgs bundles in variation of Hodge structure.
• To prove the flatness of this Higgs bundle using a novel commutator identity governing the variational properties of the polarized Hodge-Lefschetz module structure on fibers.
• To apply the generalized Lu's Hodge metric to the complexified Kähler cone and derive curvature properties, particularly negative holomorphic sectional curvature.
• To provide a new geometric insight into Wilson's conjecture on curvature bounds of the Kähler cone, especially for Calabi-Yau manifolds.
• To generalize curvature inequalities such as Alexandrov-Fenchel and Khovanskii-Teissier to the context of the complexified Kähler cone via Higgs bundle techniques.

Proposed method

• Constructs a trivial vector bundle $ H $ over the complexified Kähler cone $ \mathcal{K}_{\mathbb{C}} $ with fiber $ \bigoplus_{p,q} H^{p,q}(X,\mathbb{C}) $, equipped with a Hermitian metric $ h $ defined via the Lefschetz decomposition and Hodge star operator.
• Defines a Higgs field $ \theta: T\mathcal{K}_{\mathbb{C}} \to \mathrm{End}(H) $ by wedge multiplication with $ \alpha \in H^{1,1}(X,\mathbb{C}) $, satisfying $ \theta^2 = 0 $, thus forming a Higgs bundle $ (H, \theta) $.
• Proves flatness of the Higgs bundle by deriving a fundamental commutator identity involving $ \Lambda $, $ \theta_{z^j} $, and $ \partial/\partial z^j $, which encodes the variational behavior of the Hodge-Lefschetz structure.
• Applies the generalized Lu's Hodge metric to the complexified Kähler cone, showing it defines a Kähler metric with negative holomorphic sectional curvature.
• Uses the commutator identity to decode the curvature properties of the Hodge metric, linking it to classical inequalities like Griffiths' formula and Lefschetz decomposition.
• Establishes the equivalence of the Higgs field adjoint formula $ \theta_{z^k}^* = -\frac{1}{2}[\Lambda, [\Lambda, \overline{\theta_{z^k}}]] $, which underpins the curvature analysis.

Experimental results

Research questions

• RQ1 Can a natural Higgs bundle structure be constructed on the complexified Kähler cone of a compact Kähler manifold, analogous to the Higgs bundle in variation of Hodge structure?
• RQ2 Does the Hodge metric defined via the Lefschetz decomposition on the complexified Kähler cone yield a Kähler metric with negative holomorphic sectional curvature?
• RQ3 What is the role of the commutator identity $ [\Lambda, \theta_{z^j}] $ in governing the variational properties of the polarized Hodge-Lefschetz module structure on the fibers of the Higgs bundle?
• RQ4 How does the generalized Lu's Hodge metric behave on the complexified Kähler cone, and what curvature properties does it induce?
• RQ5 Can this Higgs bundle framework provide new insights into Wilson's conjecture on the curvature bounds of the Kähler cone, particularly for Calabi-Yau manifolds?

Key findings

• The Higgs bundle $ (H, \theta) $ over the complexified Kähler cone is flat, as established by a novel commutator identity involving $ \Lambda $, $ \theta_{z^j} $, and $ \partial/\partial z^j $.
• The generalized Lu's Hodge metric on the complexified Kähler cone defines a Kähler metric with negative holomorphic sectional curvature.
• The curvature result provides a new geometric resolution to Wilson's conjecture, showing that the Hodge metric on the Kähler cone has negative holomorphic sectional curvature, which is a stronger condition than the previously conjectured sectional curvature bounds.
• The fundamental commutator identity $ [\Lambda, \theta_{z^j}] $ encodes the variational properties of the polarized Hodge-Lefschetz module structure and generalizes the classical Kähler identity.
• The adjoint formula $ \theta_{z^k}^* = -\frac{1}{2}[\Lambda, [\Lambda, \overline{\theta_{z^k}}]] $ is proven, which is essential for the curvature analysis and generalizes known identities in Hodge theory.
• The construction and curvature results extend classical inequalities such as Alexandrov-Fenchel and Khovanskii-Teissier to the setting of the complexified Kähler cone via Higgs bundle techniques.