Donaldson Ansatz for Ricci-flat Kähler Metrics

• Donaldson Ansatz is a framework for producing Ricci-flat Kähler metrics on complex surfaces with cone singularities using a Gibbons–Hawking ansatz and harmonic analysis.
• It employs a Neumann Green's function over a wedge-shaped domain in ℝ³ to encode singular geometry and yield locally hyperkähler metrics.
• The approach provides computable energy, canonical holomorphic coordinates, and asymptotic cone structures, enhancing the study of Kähler metrics with singularities.

The Donaldson ansatz refers to a construction producing explicit, complete Ricci-flat Kähler metrics with cone singularities on complex surfaces, specifically on $\mathbb{C}^2$ with a conical singularity along the conic $\{zw=1\}$. It utilizes the Gibbons–Hawking ansatz over a wedge-shaped domain in $\mathbb{R}^3$, encoding the singular geometry via a harmonic Green's function with Neumann boundary conditions. This framework yields model metrics that are locally hyperkähler, possess precise asymptotic behaviors, canonical holomorphic coordinates, and computable energy in terms of the cone angle parameter.

1. Geometric Framework and the Gibbons–Hawking Ansatz

The construction begins with the Gibbons–Hawking ansatz applied over a wedge in $\mathbb{R}^3$. Fix a parameter $0<\beta<1$; in cylindrical coordinates $(r, \tilde\theta, s)$, the wedge is defined by

$$W = \{ r \ge 0,\, -\pi\beta < \tilde\theta < \pi\beta,\, s\in\mathbb{R} \},$$

whose edge is $S = \{r=0\}$. A distinguished "pole" $p = (1, 0, 0) \in W$ is selected. The essential ingredient is $\tilde\Gamma_p$, the unique Green's function for the Laplacian with Neumann boundary condition: $\{zw=1\}$0 Modified angular coordinates $\{zw=1\}$1 are introduced so that the metric

$\{zw=1\}$2

is Euclidean with a cone angle $\{zw=1\}$3 along $\{zw=1\}$4. The Green's function in these coordinates is

$\{zw=1\}$5

solving $\{zw=1\}$6 in distributions.

The Gibbons–Hawking potential is thus

$\{zw=1\}$7

with the property that integrating $\{zw=1\}$8 over a sphere near $\{zw=1\}$9 yields $\mathbb{R}^3$0.

2. Model Metric on $\mathbb{R}^3$1 with Cone Singularities

Over $\mathbb{R}^3$2, one constructs the principal $\mathbb{R}^3$3-bundle $\mathbb{R}^3$4 of first Chern class $\mathbb{R}^3$5. A unique connection is defined by

$\mathbb{R}^3$6

with curvature $\mathbb{R}^3$7, and holonomy trivial around small loops near $\mathbb{R}^3$8. The Gibbons–Hawking metric on $\mathbb{R}^3$9 is

$\mathbb{R}^3$0

producing a locally hyperkähler metric. This metric extends smoothly over the lifted pole $\mathbb{R}^3$1 and defines a cone-angle $\mathbb{R}^3$2 singularity along the exceptional locus (the lift of $\mathbb{R}^3$3).

Holomorphic coordinates $\mathbb{R}^3$4 are globally defined as follows. Introducing $\mathbb{R}^3$5, $\mathbb{R}^3$6, and $\mathbb{R}^3$7, one sets

$\mathbb{R}^3$8

These coordinates yield a biholomorphism to $\mathbb{R}^3$9, with the locus $0<\beta<1$0 mapping to the conic $0<\beta<1$1. The explicit form of the metric in these coordinates is retained.

The Kähler form $0<\beta<1$2 is

$0<\beta<1$3

and the volume form, using $0<\beta<1$4, is

$0<\beta<1$5

3. Asymptotic Analysis at Infinity

The behavior of $0<\beta<1$6 at infinity matches that of $0<\beta<1$7 as $0<\beta<1$8, with the error $0<\beta<1$9. The resulting "model cone metric" on $(r, \tilde\theta, s)$0 is

$(r, \tilde\theta, s)$1

where $(r, \tilde\theta, s)$2 also arises from the Gibbons–Hawking ansatz using $(r, \tilde\theta, s)$3. There exists a diffeomorphism $(r, \tilde\theta, s)$4 such that

$(r, \tilde\theta, s)$5

with decay rate $(r, \tilde\theta, s)$6 for $(r, \tilde\theta, s)$7, and $(r, \tilde\theta, s)$8 for $(r, \tilde\theta, s)$9, where $$W = \{ r \ge 0,\, -\pi\beta < \tilde\theta < \pi\beta,\, s\in\mathbb{R} \},$$0. Thus, $$W = \{ r \ge 0,\, -\pi\beta < \tilde\theta < \pi\beta,\, s\in\mathbb{R} \},$$1 is asymptotic to the product cone $$W = \{ r \ge 0,\, -\pi\beta < \tilde\theta < \pi\beta,\, s\in\mathbb{R} \},$$2 at this rate.

4. Energy Calculation

For any complete Riemannian $$W = \{ r \ge 0,\, -\pi\beta < \tilde\theta < \pi\beta,\, s\in\mathbb{R} \},$$3-manifold $$W = \{ r \ge 0,\, -\pi\beta < \tilde\theta < \pi\beta,\, s\in\mathbb{R} \},$$4, the energy is defined as

$$W = \{ r \ge 0,\, -\pi\beta < \tilde\theta < \pi\beta,\, s\in\mathbb{R} \},$$5

In this context, a computation yields

$$W = \{ r \ge 0,\, -\pi\beta < \tilde\theta < \pi\beta,\, s\in\mathbb{R} \},$$6

and, due to the fibration structure, the $$W = \{ r \ge 0,\, -\pi\beta < \tilde\theta < \pi\beta,\, s\in\mathbb{R} \},$$7 norm reduces to an integral over $$W = \{ r \ge 0,\, -\pi\beta < \tilde\theta < \pi\beta,\, s\in\mathbb{R} \},$$8: $$W = \{ r \ge 0,\, -\pi\beta < \tilde\theta < \pi\beta,\, s\in\mathbb{R} \},$$9 Applying Stokes’ theorem to $S = \{r=0\}$0 and analyzing the contributions from small and large spheres shows the total integral is $S = \{r=0\}$1. It follows that

$S = \{r=0\}$2

This quantifies the energy entirely in terms of the cone parameter.

5. Structural Summary and Broader Context

The Donaldson ansatz constructs Ricci-flat metrics on $S = \{r=0\}$3 with prescribed cone singularities by encoding the geometry of a Neumann-Green's function on a wedge into a Gibbons–Hawking potential. The resulting metric is locally hyperkähler, globally Kähler, and exhibits a precise product cone structure at infinity. Global holomorphic coordinates $S = \{r=0\}$4 are constructed so that the singular locus is the conic $S = \{r=0\}$5. The approach yields metrics with finite, explicitly computable energy depending on the cone parameter $S = \{r=0\}$6, providing valuable model geometries for the study of Kähler metrics with singularities and their moduli (Borbon, 2017).