Paper · Jan 25, 2024

Optimal Degenerations of K-unstable Fano threefolds

Minghao Miao, Linsheng Wang

math.AGarXiv:2401.13999v2
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Abstract

We explicitly determine the optimal degenerations of Fano threefolds in family No 2.23 of Mori-Mukai's list as predicted by the Hamilton-Tian conjecture. More precisely, we find a special degeneration of such that is weighted K-polystable, which is equivalent to admitting a Kähler-Ricci soliton (KRS) by and. Furthermore, we study the moduli spaces of . The -invariant of divides the natural parameter space into two strata, which leads to different moduli spaces of KRS Fano varieties. We show that one of them is isomorphic to the GIT-moduli space of biconic curves , and the other one is a single point.

Comments: 32 pages, 1 table. Comments are very welcome! v2: Section 2.3 added and minor changes

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”, “unstable”, “admitting”

  7. The Miyaoka-Yau inequality and the delta invariant for Fano varieties II -soliton case-

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  8. Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

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

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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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