Existence and deformations of Kähler–Einstein metrics on smoothable Q-Fano varieties

Abstract

In this article we prove the existence of Kähler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings. P N by sections of K −λ X . A weak Kähler-Einstein metric on X is a Kähler current in 2πc 1 (X) with locally continuous potential, and that is a smooth Kähler-Einstein metric on the smooth part X reg of X. Note there are different definitions in the literature, but they are all equivalent in this context [23]. A Q-Fano variety X is called Q-Gorenstein smoothable 1 if there is a flat family π : X → ∆, over a disc ∆ in C so that X ∼ = X 0 , X t is smooth for t = 0, and X admits a relatively Q-Cartier anti-canonical divisor −K X /∆ (in this case π : X → ∆ is called a Q-Gorenstein smoothing of X 0 ). It is well-known that, by possibly shrinking ∆, we may assume that X t is a Fano manifold for t = 0, and that there exists an integer λ > 0 such that K −λ Xt are very ample line bundles with vanishing higher cohomology for all t ∈ ∆. Moreover, the dimension, denoted by N (λ), of the corresponding linear systems | − λK Xt | is constant in t. Thus, when needed, we may assume that the family X is relatively very ample, i.e. there is a smooth embedding i :Xt . The main theorem of this paper is the following result, which extends the results of [12,13,14,15] to Q-Gorenstein smoothable Q-Fano varieties and simultaneusly gives some understanding on the way such singular metric spaces are approached by smooth KE metrics on the (analytically) nearby Fano manifolds.Theorem 1.1. Let π : X → ∆ be a Q-Gorenstein smoothing of a Q-Fano variety X 0 . If X 0 is K-polystable then X 0 admits a weak Kähler-Einstein metric ω 0 .Moreover, assuming that the automorphism group Aut(X 0 ) is discrete, X t admit smooth Kähler-Einstein metrics ω t for all |t| sufficiently small and (X 0 , ω 0 ) is the limit in the Gromov-Hausdorff topology of (X t , ω t ), in the sense of [21].Few remarks are in place. First, by the generalized Bando-Mabuchi uniqueness theorem [6] the above weak Kähler-Einstein metric ω 0 is unique up to Aut 0 (X 0 ), the identity component of Aut(X 0 ), thus can be viewed as a canonical metric on X 0 . Second, the above theorem does not just state that there is a sequence of nearby KE Fano manifolds which converges, in the Gromov-Hausdorff sense, to the weak KE metric, but that all the nearby KE Fano manifolds actually converge to the unique singular limit (X 0 , ω 0 ). Thus this property provides a good topological correspondence between complex analytic deformations (alias flat-families) and the notion of Gromov-Hausdorff convergence. For example, by the Bando-Mabuchi theorem we have the following immediate corollary: Corollary 1.2. Let π : X → ∆ and π ′ : X ′ → ∆ be two Q-Gorenstein smoothings of Q-Fano varieties X 0 and X ′ 0 . Suppose X t and X ′ t are bi-holomorphic for all t = 0, and X 0 and X ′ 0 are both K-polystable with discrete automorphism group, then X 0 and X ′ 0 are isomorphic varieties.Notice t...

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