# Two conjectures on Ricci-flat Kaehler metrics

## TL;DR

This paper proposes and verifies two conjectures relating Ricci-flat Kähler metrics to flatness, using specific conditions like radial symmetry, stability, and ALE properties, and demonstrates the necessity of Ricci-flatness.

## Contribution

It introduces two conjectures linking Ricci-flatness to flatness in Kähler metrics and verifies them under various geometric conditions.

## Key findings

- Conjecture 1 verified for radial, stable-projectively induced metrics.
- Conjecture 2 verified for radial or complete ALE complex surfaces.
- Ricci-flatness cannot be replaced by scalar-flatness in these conjectures.

## Abstract

We propose two conjectures about Ricci-flat metrics: Conjecture 1: A Ricci-flat projectively induced metric is flat. Conjecture 2: A Ricci-flat metric on an $n$-dimensional complex manifold such that the $a_{n+1}$ coefficient of the TYZ expansion vanishes is flat. We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the metric is radial and stable-projectively induced and Conjecture 2 (see Theorem 1.2) for complex surfaces whose metric is either radial or complete and ALE. We end the paper by showing, by means of the Simanca metric, that the assumption of Ricci-flatness in Conjecture 1 and in Theorem 1.2 cannot be weakened to scalar-flatness (see Theorem 1.3).