[Paper Review] Note on Poincaré type Kähler metrics and Futaki characters

Hugues Auvray|arXiv (Cornell University)|Dec 31, 2013

Geometry and complex manifolds10 references3 citations

TL;DR

This paper introduces a Poincaré-type Futaki character for extremal Kähler metrics on the complement of a simple normal crossing divisor in a compact Kähler manifold, relating it to the classical Futaki character via a correction term involving integrals of the gradient potential over divisor components. The key result is a numerical obstruction to the existence of extremal Poincaré-type metrics, expressed as a scalar curvature inequality involving mean curvatures and Futaki invariants on the divisor components.

ABSTRACT

A Poincaré type Kähler metric on the complement X\D of a simple normal crossing divisor D, in a compact Kähler manifold X, is a Kähler metric on X\D with cusp singularity along D. We relate the Futaki character for holomorphic vector fields parallel to the divisor, defined for any fixed Poincaré type Kähler class, to the classical Futaki character for the relative smooth class. As an application we express a numerical obstruction to the existence of extremal Poincaré type Kähler metrics, in terms of mean scalar curvatures and Futaki characters.

Motivation & Objective

• To define and study a Poincaré-type Futaki character for holomorphic vector fields tangent to a simple normal crossing divisor in a compact Kähler manifold.
• To relate this new Futaki character to the classical smooth Futaki character for the ambient Kähler class.
• To derive a numerical obstruction to the existence of extremal Poincaré-type Kähler metrics using the Poincaré-type Futaki character.
• To generalize the Yau-Tian-Donaldson conjecture framework to singular Kähler metrics with cusp singularities along divisors.

Proposed method

• Define Poincaré-type Kähler metrics on $X \setminus D$ as metrics with cusp singularities along a simple normal crossing divisor $D$, asymptotically modeled on a standard cusp metric.
• Restrict attention to holomorphic vector fields in $\mathfrak{h}_{\!/\!/}^{D}$, i.e., those tangent to $D$ and bounded under the model metric.
• Define the Poincaré-type Futaki character $\mathscr{F}^{D}_{[\omega_X]}$ using the $L^2$-inner product of the gradient potential of such vector fields with the Ricci form of the Poincaré-type metric.
• Establish a precise formula relating the Poincaré-type Futaki character to the classical smooth Futaki character: $\mathscr{F}^{D}_{[\omega_X]}(\mathsf{Z}) = \mathscr{F}_{[\omega_X]}(\mathsf{Z}) + \sum_{j=1}^{N} \int_{D_j} f \frac{(\omega_X|_{D_j})^{m-1}}{(m-1)!}$, where $f$ is the gradient potential of $\mathsf{Z}$.
• Use asymptotic product structure near the divisor and limit arguments on tubular neighborhoods to relate the scalar curvature of the metric on $X \setminus D$ to the scalar curvature on the divisor components.
• Apply the formula to extremal metrics by analyzing the Riemannian gradient of the scalar curvature and deriving a scalar curvature inequality involving the mean scalar curvature on divisor components and the Futaki invariants on $X \setminus D$ and $X \setminus (D \setminus D_j)$.

Experimental results

Research questions

• RQ1 How does the Futaki character for Poincaré-type Kähler metrics differ from the classical smooth Futaki character on the same holomorphic vector fields?
• RQ2 What is the precise correction term relating the Poincaré-type Futaki character to the smooth Futaki character?
• RQ3 What numerical obstruction arises for the existence of extremal Poincaré-type Kähler metrics in terms of scalar curvatures and Futaki invariants?
• RQ4 How does the asymptotic behavior of the metric near a divisor component affect the scalar curvature and the Futaki invariant?
• RQ5 Can the Yau-Tian-Donaldson conjecture framework be extended to include metrics with cusp singularities along divisors?

Key findings

• The Poincaré-type Futaki character differs from the classical smooth Futaki character by a correction term involving the integral of the gradient potential of the vector field over each irreducible component of the divisor.
• The correction term is explicitly given by $\sum_{j=1}^{N} \int_{D_j} f \frac{(\omega_X|_{D_j})^{m-1}}{(m-1)!}$, where $f$ is the Riemannian gradient potential of the holomorphic vector field.
• For an extremal Poincaré-type metric, the mean scalar curvature $\overline{\mathbf{s}}^{D}$ on $X \setminus D$ satisfies the inequality $\overline{\mathbf{s}}^{D} < \overline{\mathbf{s}}_{D_j}^{D^j} + \frac{1}{4\pi \operatorname{Vol}(D_j)} \left( \mathscr{F}^{D-D_j}_{[\omega_X]}(\mathsf{K}) - \mathscr{F}^{D}_{[\omega_X]}(\mathsf{K}) \right)$ for each component $D_j$.
• The inequality is derived from the positivity of the parameter $a_j > 0$ in the asymptotic model metric near $D_j$, which controls the cusp behavior.
• The result holds in both the smooth divisor case and the simple normal crossing divisor case, with the latter requiring a more refined analysis using tubular neighborhoods and limit behavior of the potential functions.
• The derivation relies on the fact that the gradient potential of the scalar curvature on the divisor $D_j$ is normalized to have zero average, which allows rewriting the Futaki character difference in terms of scalar curvature differences and the $a_j$ parameter.