Abstract

In this note we point out a simple application of a result by the authors in [2]. We show how to construct many families of strictly K-unstable polarized manifolds, destabilized by test configurations with smooth central fiber. The effect of resolving singularities of the central fiber of a given test configuration is studied, providing many new examples of manifolds which do not admit Kähler constant scalar curvature metrics in some classes.

Introduction

In this note we want to speculate about the following Conjecture due to Tian-Yau-Donaldson ([23], [24], [25], [7]):

Conjecture 2.1.1A polarized manifold (M, A) admits a Kähler metric of constant scalar curvature in the class c1(A) if and only if it is K-polystable.

The notion of K-stability will be recalled below. For the moment it suffices to say, loosely speaking, that a polarized manifold, or more generally a polarized variety (V, A), is K-stable if and only if any special degeneration or test configuration of (V,A) has an associated non positive weight, called Futaki invariant and that this is zero only for the product configuration, i.e. the trivial degeneration.

We do not even attempt to give a survey of results about Conjecture 2.1.1, but as far as the results of this note are concerned, it is important to recall the reader that Tian [24], Donaldson [7], Stoppa [22], using the results in [3] and [4], and Mabuchi [17] have proved the sufficiency part of the Conjecture.