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Bott manifolds with vanishing Futaki invariants for all Kähler classes

Hajime Ono, Yuji Sano, Naoto Yotsutani

TL;DR

The paper classifies stage $n$ Bott manifolds with vanishing Futaki invariants for all Kähler classes, proving that such manifolds are precisely the $n$-fold products $(\mathbb{P}^1)^n$. The authors deploy toric geometry techniques, analyzing the moment polytope and its slices, and use the Brunn–Minkowski inequality to deduce a splitting of the polytope and, hence, the manifold itself. Through an induction on dimension, they show that the Bott matrix must be the identity, establishing the product structure; this also ties to Calabi dream phenomena and relative K-stability concepts. The work further connects to the FY24 result via a detailed discussion of $\xi_L(D)$ and its relation to slope semistability and the Donaldson–Futaki framework, demonstrating equivalence under complex coefficients. Overall, the paper provides a sharp toric-geometric criterion that singles out products of $\mathbb{P}^1$ as the only Bott manifolds with universal Futaki vanishing across Kähler classes, with implications for canonical metrics and stability theory.

Abstract

We prove that the only Bott manifolds such that the Futaki invariant vanishes for any Kähler class are isomorphic to the products of the projective lines.

Bott manifolds with vanishing Futaki invariants for all Kähler classes

TL;DR

The paper classifies stage Bott manifolds with vanishing Futaki invariants for all Kähler classes, proving that such manifolds are precisely the -fold products . The authors deploy toric geometry techniques, analyzing the moment polytope and its slices, and use the Brunn–Minkowski inequality to deduce a splitting of the polytope and, hence, the manifold itself. Through an induction on dimension, they show that the Bott matrix must be the identity, establishing the product structure; this also ties to Calabi dream phenomena and relative K-stability concepts. The work further connects to the FY24 result via a detailed discussion of and its relation to slope semistability and the Donaldson–Futaki framework, demonstrating equivalence under complex coefficients. Overall, the paper provides a sharp toric-geometric criterion that singles out products of as the only Bott manifolds with universal Futaki vanishing across Kähler classes, with implications for canonical metrics and stability theory.

Abstract

We prove that the only Bott manifolds such that the Futaki invariant vanishes for any Kähler class are isomorphic to the products of the projective lines.
Paper Structure (10 sections, 14 theorems, 88 equations)

This paper contains 10 sections, 14 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Toric geometry and convex polytopes
  3. Bott manifolds
  4. Definition and basic properties
  5. A Bott manifold and its associated fan
  6. The proof of Theorem $\ref{['thm:main']}$
  7. Volume preserving constraints for each slice of $P$
  8. The Brunn-Minkowski inequality for convex polytopes
  9. The induction on the dimension of an SCD Bott manifold
  10. Appendix

Key Result

Theorem 1.1

Let $X$ be a stage $n$ Bott manifold. Then the Futaki invariant of $X$ vanishes for any Kähler class if and only if $X$ is isomorphic to an $n$-fold product of the projective line i.e., $X\cong (\mathbb P^1)^n$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3: FY24
  • Proposition 2.1: Theorem $3.1.19$ in CLS11
  • Definition 2.2
  • Proposition 2.3: Proposition $2.1.6$ in Bat98, Proposition $1.3.4$ in Sat02
  • Definition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • proof
  • ...and 15 more