[Paper Review] An obstruction to asymptotic semistability and approximate critical metrics

Toshiki Mabuchi|ArXiv.org|Apr 10, 2004

Geometry and complex manifolds9 references11 citations

TL;DR

This paper identifies a cohomological obstruction—vanishing of a certain curvature functional on the Lie algebra of a maximal torus in the automorphism group—that must be satisfied for a polarized manifold $(M,L)$ to be asymptotically Chow-semistable when a positive-dimensional algebraic group acts holomorphically. Generalizing Donaldson's construction of approximate critical metrics, the authors show that under this obstruction condition, asymptotic Chow-stability holds, and the Kähler metric of constant scalar curvature in $c_1(L)_{\mathbb{R}}$ is unique modulo the group action.

ABSTRACT

In this paper, we consider an obstruction to asymptotic Chow-semistability of a polarized Kaehler algebraic manifold. Even when a linear algebraic group of positive dimension acts nontrivially and holomorphically on a polarized Kaehler algebraic manifold with constant scalar curvature, the vanishing of the obstruction allows us to generalize Donaldson's construction of approximate solutions for equations of balanced metrics.

Motivation & Objective

• To identify an obstruction to asymptotic Chow-semistability in polarized manifolds $(M,L)$ with nontrivial holomorphic group actions.
• To generalize Donaldson’s construction of approximate solutions for critical metric equations to cases where the automorphism group has positive dimension.
• To establish conditions under which asymptotic Chow-stability holds, particularly when the isotropy actions stabilize across high tensor powers of $L$.
• To prove the uniqueness, modulo the group action, of constant scalar curvature Kähler metrics in the polarization class $c_1(L)_{\mathbb{R}}$.
• To show that asymptotic Chow-stability implies stability in the sense of Hilbert schemes when the automorphism group is trivial.

Proposed method

• Introduces a curvature functional $\mathcal{C}\{c_1^{n+1};L^m\}(X)$ on the Lie algebra $\mathfrak{z}$ of a maximal torus $Z$ in the automorphism group, defined via integration of a weighted trace of curvature forms.
• Uses equivariant cohomology and $Z_m$-invariant Hermitian metrics on $L^m$ to define the functional $\mathcal{C}\{c_1^{n+1};L^m\}$, which is independent of the choice of metric $h$.
• Establishes that the isotropy actions $\rho_m$ stabilize for large $m$ if the functional vanishes for a sequence $m(k)\to\infty$, linking the obstruction to the stability of the group action lift.
• Applies a formal power series expansion in $q$ to the Kähler metric $\omega(\ell)$ and the metric $h(\ell)$ on $L$, using the asymptotic expansion of the Bergman metric.
• Uses the condition $\rho_{m(k)} = \rho_{m(k_0)}$ for $k \geq k_0$ to show that the functional $\mathcal{C}\{c_1^{n+1};L^m\}$ vanishes for large $m$, enabling the construction of approximate solutions.
• Employs a recursive argument in the $q$-adic expansion to show that the error term $v_\ell$ in the expansion of the metric must vanish, proving the convergence of the approximate solution sequence.

Experimental results

Research questions

• RQ1 What obstruction must vanish for a polarized manifold $(M,L)$ with a positive-dimensional holomorphic group action to be asymptotically Chow-semistable?
• RQ2 Can Donaldson’s construction of approximate critical metrics be extended to cases where $\operatorname{Aut}^0(M)$ has positive dimension?
• RQ3 Under what conditions does the stabilization of the isotropy actions $\rho_m$ imply asymptotic Chow-stability?
• RQ4 How does the vanishing of the curvature functional $\mathcal{C}\{c_1^{n+1};L^m\}$ relate to the existence of constant scalar curvature Kähler metrics?
• RQ5 What is the role of the $Z_m$-action on $L^m$ in lifting holomorphic vector fields and defining the obstruction?

Key findings

• The functional $\mathcal{C}\{c_1^{n+1};L^m\}$ vanishes for all sufficiently large $m(k)$ if $(M,L)$ is asymptotically Chow-semistable, providing a necessary obstruction.
• The isotropy actions $\rho_m$ stabilize for large $m$ if the obstruction vanishes, i.e., $\rho_{m(k)} = \rho_{m(k_0)}$ for all $k \geq k_0$.
• The construction of approximate critical metrics generalizes Donaldson’s method to cases with nontrivial group actions, under the stabilization condition.
• The vanishing of $\mathcal{C}\{c_1^{n+1};L^m\}$ for large $m$ implies that the error term $v_\ell$ in the $q$-adic expansion of the metric must vanish, ensuring convergence.
• Asymptotic Chow-stability holds for $(M,L)$ if the isotropy actions stabilize and $c_1(L)_{\mathbb{R}}$ admits a constant scalar curvature Kähler metric.
• Uniqueness, modulo the action of the group $G$, of constant scalar curvature Kähler metrics in $c_1(L)_{\mathbb{R}}$ follows from the stabilization condition and the obstruction vanishing.