Paper · Sep 25, 2023

Kähler-Ricci solitons on Fano threefolds with non-trivial moduli

Minghao Miao, Linsheng Wang

math.AGarXiv:2309.14212v4Math. Ann. 392 (2025), 4483-4523
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Abstract

We find Fano threefolds admitting Kähler-Ricci solitons (KRS) with non-trivial moduli, which are -varieties of complexity two. More precisely, we show that the weighted K-stability of (where is the soliton candidate) is equivalent to certain GIT-stability. In particular, this provides the first examples of strictly weighted K-semistable Fano varieties. On the other hand, we generalize Koiso's theorem to the log Fano setting. Indeed, we show that the K-stability of a log Fano pair is equivalent to the weighted K-stability of a cone over it. This also leads to new examples of KRS Fano varieties with non-trivial moduli and small automorphism groups. To achieve these, we establish the weighted Abban-Zhuang estimate generalizing the work of, which gives a lower bound of the weighted stability threshold . This is an effective way to check the weighted K-semistablity of a log Fano triple . This estimate is also useful in testing (weighted) K-polystability based on the work of.

Comments: 36 pages. Comments are very welcome. v4: exposition improved and the statement of Theorem 4.6 is corrected

Ricci solitonsKähler–Ricci solitons3-manifoldsKähler geometryStability & uniquenessKähler–Einstein & K-stability

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    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, 3-manifolds · Same authors Minghao Miao, Linsheng Wang · Shared terms “Fano threefold”, “KRS”, “Fano varieties”

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    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “Fano varieties”, “admitting Kähler”

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  7. On the Existence of Weighted-cscK Metrics

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  8. Kähler-Ricci flow for deformed complex structures

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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