On the blow-up formula of weighted stability for polarized toric manifolds

TL;DR

This paper derives explicit combinatorial formulas for the Chow weight and Futaki-Ono invariant of blow-ups of polarized toric manifolds, linking stability properties to the geometry of associated polytopes and symplectic cuts.

Contribution

It introduces a new combinatorial approach to compute stability invariants for blow-ups of toric varieties, including weighted stability and K-stability.

Findings

Explicit formula for Chow weight of blow-ups in terms of Delzant polytope

Comparison of Chow stability between toric and general point blow-ups

Blow-up formula for Futaki-Ono invariant as an obstruction to stability

Abstract

Let $X$ be a smooth projective toric variety, and let $\widetilde{X}$ denote the blow-up of $X$ at finitely many distinct tours-invariant points. This paper provides an explicit combinatorial formula for the Chow weight of $\widetilde{X}$ in terms of the base toric manifold $X$ and the symplectic cuts of its associated Delzant polytope. We apply this blow-up formula to the projective plane and compare Chow stability of toric blow-ups with that of blow-ups at general points. Furthermore, we derive the blow-up formula of the Futaki-Ono invariant, which serves an obstruction to asymptotic Chow semistability of a polarized toric manifold. In the final section, we extend our approach to study blow-up formulas for weighted (Chow/K-) stability using our combinatorial framework.