Futaki invariant on Hopf manifolds

TL;DR

The paper investigates the Futaki invariant, an obstruction to volume forms tied to the Ricci form, in the context of Hopf manifolds within locally conformally Kähler (LCK) geometry. It develops a reduction to diagonal Hopf manifolds via Poincaré–Dulac normalization, analyzes holomorphic vector fields through resonant structures, and constructs a gamma-equivariant volume form to compare with the diagonal Vaisman model. The main result is that the Futaki invariant vanishes on Hopf manifolds in all dimensions: $F_H(V)=0$ for every holomorphic vector field $V$. This extends known Vaisman cases and strengthens the understanding of canonical-volume obstructions in LCK geometry, illustrating that Hopf quotients do not obstruct the existence of volume forms proportional to the determinant of their Ricci form.

Abstract

The Futaki invariant is a fundamental tool in Kähler geometry representing an obstruction to the existence of Kähler-Einstein metrics. Recently, it was generalized to compact complex manifolds. In this paper, we prove that it vanishes on Hopf manifolds.

Theorems & Definitions (13)

• Definition 2.1
• Remark 2.1
• Definition 2.2
• Definition 2.3
• Example 2.1
• Theorem 2.1
• Remark 2.2
• Lemma 3.1
• proof
• Lemma 3.2