The Miyaoka-Yau inequality and the delta invariant for Fano varieties

2026/07/28 by Tomoyuki Hisamoto, Masataka Iwai
#math.AG #math.CV #math.DG

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Abstract

We establish the following Miyaoka-Yau inequality for any n-dimensional klt Fano variety X, possibly K-unstable, in terms of its delta invariant: (2(n+1)\widehatc₂(X)-n c₁(X)²)⋅ c₁(X)^n-2 ≥ -n (1-min\1,δ(X)\)² ⋅ c₁(X)ⁿ. Furthermore, inspired by recent work of Inoue and Hallam-Lahdili, we formulate and prove an equivariant version of this inequality in the soliton setting, in which the Chern classes and the delta invariant are replaced by the equivariant Chern classes and the weighted delta invariant, respectively. As a consequence, we obtain the equivariant Miyaoka-Yau inequality for every smooth Fano manifold admitting a Kähler-Ricci soliton.

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