On the modified Futaki invariant of complete intersections in projective spaces

TL;DR

This paper presents a method to compute the modified Futaki invariant for Fano complete intersections in projective spaces, which is crucial for understanding the existence of Kähler-Ricci solitons and stability conditions.

Contribution

It introduces a novel computational approach for the modified Futaki invariant specifically for Fano complete intersections in projective spaces.

Findings

Provides explicit formulas for the modified Futaki invariant.

Connects the invariant with algebro-geometric stability.

Facilitates stability analysis for Fano complete intersections.

Abstract

It is known that a necessary condition for the existence of K\"ahler-Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian-Zhu. In a recent work of Berman-Nystr\"om, it was generalized for (singular) Fano varieties and the notion of algebro-geometric stability of the pair $(M,V)$ of a Fano manifold $M$ and a holomorphic vector field $V$ was introduced. In this paper, we propose a method of computing the modified Futaki invariant for Fano complete intersections in projective spaces.