[Paper Review] Kaehler-Einstein Fano threefolds of degree 22

Ivan Cheltsov, Constantin Shramov|arXiv (Cornell University)|Mar 7, 2018

Geometry and complex manifolds39 references4 citations

TL;DR

This paper resolves the existence of Kähler–Einstein metrics on smooth Fano threefolds of degree 22 with a faithful $\mathbb{C}^*$-action (type $V_{22}^*$), proving that all such threefolds are Kähler–Einstein except possibly two explicitly described cases. Using equivariant alpha-invariants and log canonical threshold analysis, the authors establish the log canonicity of certain pairs, confirming the existence of Kähler–Einstein metrics via Tian’s criterion.

ABSTRACT

We study the problem of existence of Kähler--Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree $22$ that admit a faithful action of the multiplicative group $\mathbb{C}^\ast$. We prove that, except possibly two explicitly described cases, all such smooth Fano threefolds are Kähler--Einstein.

Motivation & Objective

• To resolve the existence of Kähler–Einstein metrics on smooth Fano threefolds of degree 22 with a faithful $\mathbb{C}^*$-action, known as type $V_{22}^*$.
• To compute the $G$-equivariant $\alpha$-invariant for the group $G = \mathbb{C}^* \rtimes \boldsymbol{\mu}_2$ acting on these threefolds.
• To verify Donaldson's conjecture that all such threefolds are Kähler–Einstein, except possibly two exceptional cases.
• To establish log canonicity of log pairs $(V_u, \varepsilon(u)D)$ for $D \sim_{\mathbb{Q}} -K_{V_u}$, which implies Kähler–Einstein existence via Tian's criterion.

Proposed method

• Computes the $G$-equivariant $\alpha$-invariant $\alpha_G(V_u)$ for $G = \mathbb{C}^* \rtimes \boldsymbol{\mu}_2$ using log canonical threshold analysis on blow-ups of the threefold.
• Analyzes the behavior of log pairs $(\widehat{V}_u, \epsilon \widehat{D} + (\epsilon m - 1)E_\sigma)$ after a $G$-equivariant blow-up along a curve $\mathcal{C}_4$, tracking multiplicities and singularities.
• Applies Corollary 7.11 to show that if a log pair is log canonical at a general point of a curve, it is log canonical everywhere under certain numerical conditions.
• Uses the relation $D' \sim_{\mathbb{Q}} -(2 - m)K_{V_{u'}}$ and computes $\mathrm{mult}_{\mathcal{C}_4'}(D') = 3 - 2m$ to bound the log pair's singularities.
• Employs the fact that $\varepsilon(u) \leq \frac{2}{3}$ for $u=2$ and $\varepsilon(u) = \frac{5}{6}$ for $u \neq 2$, and verifies log canonicity at the curve $\mathcal{C}_4$.
• Applies the result that K-semistability is an open condition and uses degeneration from the Mukai–Umemura threefold to infer K-semistability of $X^a$.

Experimental results

Research questions

• RQ1 Are all smooth Fano threefolds of type $V_{22}^*$ with a faithful $\mathbb{C}^*$-action Kähler–Einstein, except possibly two cases?
• RQ2 What is the $G$-equivariant $\alpha$-invariant $\alpha_G(V_u)$ for $G = \mathbb{C}^* \rtimes \boldsymbol{\mu}_2$ acting on $V_u$?
• RQ3 Does the log pair $(V_u, \varepsilon(u)D)$ remain log canonical at a general point of the curve $\mathcal{C}_4$ for $D \sim_{\mathbb{Q}} -K_{V_u}$?
• RQ4 Can the log canonicity of $(V_u, \varepsilon(u)D)$ at $\mathcal{C}_4$ be extended to global log canonicity using known criteria?
• RQ5 Is the $\alpha$-invariant sufficient to guarantee the existence of a Kähler–Einstein metric on $V_u$ when $\varepsilon(u) > \frac{3}{4}$?

Key findings

• The $G$-equivariant $\alpha$-invariant $\alpha_G(V_u)$ is at least $\frac{5}{6}$ for all threefolds $V_u$ of type $V_{22}^*$, with equality holding for $u \neq 2$, and $\frac{2}{3}$ for $u=2$.
• The log pair $(V_u, \frac{5}{6}D)$ is log canonical at a general point of the curve $\mathcal{C}_4$ for $u \neq 2$, and $(V_u, \frac{2}{3}D)$ is log canonical for $u=2$.
• The log canonicity of $(V_u, \varepsilon(u)D)$ at $\mathcal{C}_4$ implies global log canonicity due to the open nature of K-semistability and the application of Corollary 7.11.
• The authors prove that all threefolds of type $V_{22}^*$ are Kähler–Einstein except possibly two explicitly described cases, confirming Donaldson’s conjecture in all but two instances.
• The Mukai–Umemura threefold is Kähler–Einstein, and its degeneration to $X^a$ (with $\mathbb{C}^+$-action) preserves K-semistability, supporting the existence of Kähler–Einstein metrics.
• The paper resolves Problem 1.5 from Dinew, Kapustka, and Kapustka by computing $\alpha_G(V_u)$ for $G = \mathbb{C}^* \rtimes \boldsymbol{\mu}_2$, showing it is $\frac{5}{6}$ for $u \neq 2$ and $\frac{2}{3}$ for $u=2$.