[Paper Review] Complex deformation of critical Kähler metrics

Haozhao Li|arXiv (Cornell University)|Jun 5, 2012

Geometry and complex manifolds16 references3 citations

TL;DR

This paper provides a new analytic approach to the deformation of constant scalar curvature Kähler (cscK) metrics and Kähler-Ricci solitons under complex structure variations, using a functional method inspired by Pacard-Xu for constant mean curvature surfaces. It establishes the existence of cscK metrics in nearby Kähler classes under traceless cohomology classes and extends this to varying complex structures when symmetries are preserved, proving existence of extremal solitons in deformed families with group invariance.

ABSTRACT

In this paper, we use Pacard-Xu's methods to discuss the complex deformation of constant scalar curvature metrics in the case of fixed and varying complex structures. Moreover, we also discuss the complex deformation of Kähler Ricci solitons.

Motivation & Objective

• To provide an alternative proof for the deformation of constant scalar curvature Kähler (cscK) metrics under fixed and varying complex structures, using a novel functional method.
• To extend previous results on cscK and Kähler-Ricci soliton deformations by removing nondegeneracy assumptions on the Futaki invariant.
• To establish existence of cscK metrics in Kähler classes perturbed by traceless (1,1)-classes under small deformations of the complex structure.
• To prove existence of extremal solitons in deformed Kähler classes when the deformation preserves a maximal compact subgroup of the isometry group.

Proposed method

• Introduces a functional Φ(t, β) constructed via the Futaki invariant to detect existence of cscK metrics in perturbed Kähler classes [ωg + tβ] for small t.
• Applies the Pacard-Xu method from constant mean curvature problems to overcome kernel obstructions in the linearized equation, avoiding reliance on implicit function theorem with nontrivial kernel.
• Uses projection operators Πg⊥ and Πφ⊥ to project scalar curvature onto the orthogonal complement of Killing potentials, ensuring solvability of the deformation equation.
• Imposes G-invariance on complex deformations (Jt, gt, ωt) to preserve symmetry and ensure invertibility of the linearized operator in the deformation problem.
• Employs a modified scalar curvature equation S(t, φ) = 0 with orthogonal projections to handle nontrivial holomorphic vector fields and maintain solvability.
• Applies implicit function theorem techniques to the projected equation S(t, φ) = 0 in the space of G-invariant Kähler potentials, proving existence of solutions for small t.

Experimental results

Research questions

• RQ1 Can cscK metrics be deformed under small complex structure variations without assuming nondegeneracy of the Futaki invariant?
• RQ2 Does the Pacard-Xu functional method successfully resolve kernel obstructions in the deformation of cscK metrics?
• RQ3 Under what conditions on the complex deformation (Jt, gt, ωt) do Kähler-Ricci solitons persist in nearby Kähler classes?
• RQ4 Can extremal solitons be constructed in deformed Kähler classes when the isometry group acts holomorphically on the deformation?

Key findings

• For any traceless β ∈ H^{1,1}(M) with unit norm, there exists ε₀ > 0 such that if Φ(t, β) = 0 for t ∈ (0, ε₀), then M admits a cscK metric in [ωg + tβ].
• The functional Φ has a t² expansion: Φ(t, β) = t²∫_M (Πg(Ri¯jβj¯i))² ωgⁿ + O(t³), showing the leading-order obstruction is quadratic in t.
• When ωg is Kähler-Einstein and β is traceless, the obstruction becomes t⁴: Φ(t, β) = t⁴∫_M (Πg(βi¯jβj¯i))² ωgⁿ + O(t⁵).
• If the first m coefficients a₁(β), ..., aₘ(β) of Φ(t, β) vanish, then M admits a Kähler metric ωt,β in [ωg + tβ] with scalar curvature close to average: ||s(ωt,β) - s̄(t)||_{Cᵏ} ≤ C t^{(m+1)/2}.
• For G-invariant complex deformations (Jt, gt, ωt), the equation S(t, φ) = 0 has a solution φ ∈ Cᵏ⁺², ensuring existence of a G-invariant extremal soliton in [ωt] for small t.
• An example shows that the blowup of ℂℙ² at a point admits extremal solitons in 2πc₁(ℳ) - t[E] for small t > 0, despite lacking Kähler-Einstein metrics.