Explicit Gromov–Hausdorff compactifications of moduli spaces of Kähler–Einstein Fano manifolds

Abstract

We exhibit the first non-trivial concrete examples of Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds in all complex dimensions bigger than two (Fano K-moduli spaces). We also discuss potential applications to explicit study of moduli spaces of K-stable Fano manifolds with large anti-canonical volume. Our arguments are based on recent progress about the geometry of metric tangent cones and on related ideas about the algebro-geometric study of singularities of K-stable Fano varieties.

Some Recent Developments in Kähler Geometry and Exceptional Holonomy

“…• For explicit examples of interest such as cubic 3-folds in P 4 [67], and intersections of two quadrics [75], K-stability is completely understood (including for the singular varieties that are added to compactify moduli spaces).…”

Birational Superrigidity and K-Stability of Singular Fano Complete Intersections

“…(a) If each singularity (x i ∈ X i ) lives on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds, then by [LX17a, Corollary 5.7] we know that vol(x i , X i ) = n n i i Θ(x i , X i ) where Θ(·, ·) is the volume density of a singularity (see e.g. [HS17,SS17]). It is well known that Θ((x 1 , x 2 ), X 1 × X 2 ) = Θ(x 1 , X 1 ) · Θ(x 2 , X 2 ), so Conjecture 4.7 holds in this case.…”

“…As for K-stability, very little is known except in small dimensions or small degrees, e.g. log del Pezzo surfaces [OSS16], quasi-smooth 3-folds [JK01], complete intersection of two quadric hypersurfaces in all dimensions [SS17], and cubic threefolds with isolated A k (k ≤ 4) singularities [LX17b].…”

Kähler-Einstein metrics: Old and New

“…For n = 4 the conjecture follows by (very recent!) works [SS17,LX17]. Thus for 3 ≤ d ≤ n − 1 and n ≥ 5 the existence problem of KE metrics on all smooth hypersurfaces is open.…”

“…[Li15a,Liu16]) based on better understanding of the metric tangent cones in higher dimension and their relations with algebraic geometry, made possible to study in concrete situations such moduli spaces in higher dimensions too (e.g. [SS17,LX17]).…”