法诺簇的宫冈-丘不等式与δ不变量
The Miyaoka-Yau inequality and the delta invariant for Fano varieties
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中文总结 AI 辅助
研究针对\(n\)维klt法诺簇建立宫冈 - 丘不等式，还受他人启发在孤子情形下证明其等变形式，用等变陈类和加权δ不变量取代相关量，得出允许凯勒 - 里奇孤子的光滑法诺流形的等变宫冈 - 丘不等式。
中文摘要 AI 辅助
我们针对任意\(n\)维klt法诺簇\(X\)（可能\(K\)不稳定），依据其δ不变量建立了如下宫冈-丘不等式：\(\left(2(n + 1)\widehat{c}_2(X) - n c_1(X)^2\right)\cdot c_1(X)^{n - 2} \geq -n \left(1 - \min\{1, \delta(X)\}\right)^2 \cdot c_1(X)^n\)。此外，受井上和哈勒姆 - 拉迪利近期工作启发，我们在孤子情形下表述并证明了该不等式的等变形式，其中陈类和δ不变量分别被等变陈类和加权δ不变量取代。结果，我们得到了每个允许凯勒 - 里奇孤子的光滑法诺流形的等变宫冈 - 丘不等式。
英文摘要
We establish the following Miyaoka-Yau inequality for any $n$-dimensional klt Fano variety $X$, possibly K-unstable, in terms of its delta invariant: $$ \left(2(n+1)\widehat{c}_2(X)-n c_1(X)^2\right)\cdot c_1(X)^{n-2} \ge -n \left(1-\min\{1,δ(X)\}\right)^2 \cdot c_1(X)^n. $$ 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.
