Stability thresholds for big classes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929692385214464 |
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| author | Jin, Chenzi Rubinstein, Yanir A. Tian, Gang |
| author_facet | Jin, Chenzi Rubinstein, Yanir A. Tian, Gang |
| contents | In 1987, the $α$-invariant theorem gave a fundamental criterion for existence of Kahler-Einstein metrics on smooth Fano manifolds. In 2012, Odaka-Sano extended the framework to $\mathbb{Q}$-Fano varieties in terms of K-stability, and in 2017 Fujita related this circle of ideas to the $δ$-invariant of Fujita-Odaka. We introduce new invariants on the big cone and prove a generalization of the Tian-Odaka-Sano Theorem to all big classes on varieties with klt singularities, and moreover for all volume quantiles $τ\in[0,1]$. The special degenerate (collapsing) case $τ=0$ on ample classes recovers Odaka-Sano's theorem. This leads to many new twisted Kahler-Einstein metrics on big classes. Of independent interest, the proof involves a generalization to sub-barycenters of the classical Neumann-Hammer Theorem from convex geometry. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_18150 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability thresholds for big classes Jin, Chenzi Rubinstein, Yanir A. Tian, Gang Differential Geometry Algebraic Geometry Functional Analysis In 1987, the $α$-invariant theorem gave a fundamental criterion for existence of Kahler-Einstein metrics on smooth Fano manifolds. In 2012, Odaka-Sano extended the framework to $\mathbb{Q}$-Fano varieties in terms of K-stability, and in 2017 Fujita related this circle of ideas to the $δ$-invariant of Fujita-Odaka. We introduce new invariants on the big cone and prove a generalization of the Tian-Odaka-Sano Theorem to all big classes on varieties with klt singularities, and moreover for all volume quantiles $τ\in[0,1]$. The special degenerate (collapsing) case $τ=0$ on ample classes recovers Odaka-Sano's theorem. This leads to many new twisted Kahler-Einstein metrics on big classes. Of independent interest, the proof involves a generalization to sub-barycenters of the classical Neumann-Hammer Theorem from convex geometry. |
| title | Stability thresholds for big classes |
| topic | Differential Geometry Algebraic Geometry Functional Analysis |
| url | https://arxiv.org/abs/2501.18150 |