Delta invariants of projective bundles and projective cones of Fano type

TL;DR

This work provides explicit, computable formulas for delta-invariants of two natural constructions in Fano geometry: projective bundles and projective cones over Fano-type bases. By exploiting a natural ${\mathbb C}^*$-action and Calabi symmetry, the authors derive matching analytic and algebraic bounds, yielding sharp thresholds for delta-invariants and hence for K-stability and the existence of conical Kähler–Einstein metrics. The main contributions include a closed formula for $\delta(\tilde{Y})$ in terms of the base data, a precise expression for $\delta(Y,cV_\infty)$, and an optimal-angle result for K-stability of pairs $(V,aS)$, with extensive examples such as cones over smooth Fano hypersurfaces and cones arising from ample $\mathbb{Q}$-line bundles. The methods blend $\mathbb{C}^*$-invariant valuation analyses with purely algebraic arguments, extendable to log Fano settings, and illuminate how symmetry constrains delta-invariants and stability in higher-dimensional Fano geometry.

Abstract

In this paper, we will give a precise formula to compute delta invariants of projective bundles and projective cones of Fano type.

Theorems & Definitions (72)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Corollary 1.6
• Theorem 1.7
• Corollary 1.8
• Lemma 2.1
• proof