On the K-stability of complete intersections in polarized manifolds

TL;DR

This paper investigates the K-stability of complete intersections in polarized manifolds, providing formulas relating Futaki invariants and demonstrating examples of Fano manifolds with specific stability properties.

Contribution

It introduces formulas connecting Futaki invariants of complete intersections to ambient manifolds and offers new examples and evidence regarding K-stability.

Findings

A new example of a Fano 5-fold without Kähler-Einstein metrics.

Evidence supporting K-stability of complete intersections in Grassmannians.

Formulas relating Futaki invariants of complete intersections to their ambient spaces.

Abstract

We consider the problem of existence of constant scalar curvature Kaehler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight of sections defining it and to the Futaki invariant of the ambient manifold. As applications we give a new Mukai-Umemura-Tian like example of Fano 5-fold admitting no Kaehler-Einstein metric and a strong evidence of K-stability of complete intersections on Grassmannians.