RECORD
Existence and deformations of Kahler-Einstein metrics on smoothable Q-Fano varieties
Differential Geometry · Submitted 6 Nov 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove the existence of Kahler-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.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We prove the existence of Kahler-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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We prove the existence of Kahler-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. | result 0.54 | 0.96 | 0.95 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We prove the existence of Kahler-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.
Human review
How the literature uses it
8 arXiv papers cite this in 29 sentences
Papers citing it · 17 sentences without a confident label
Uses its method or tool 2 papers
-
Our proof essentially depends on the recent development for one-parameter deformations cases in [ref] and [ref] .
Compact moduli space of Kahler-Einstein Fano varieties · Introduction · 0.87
-
We will overcome these issues arguing by contradiction, using Lemma [ref] (hence [ref] via [ref] , see also [ref] ) and what we have just proved.
Calabi-Yau manifolds with isolated conical singularities · Construction of a unique metric tangent cone at $t = t_∞$ · 0.90
Cites it as support 1 paper
-
The quotient space $Hilb^KE/SL$ satisfies the Hausdorff axiom due to the separated-ness theorem [ref] proved by ( [ref] + [ref] ) while $M̅^GH$ is compact due to the Gromov compactness theorem.
Compact moduli space of Kahler-Einstein Fano varieties · Local GIT polystability · 0.96
Cites it as background 3 papers
-
Although we do not have any contribution on it in this paper, we would like to recall the result as we need a comment (on how to combine [ref] , [ref] , [ref] , as written below) on the proof to make things rigorous.
Compact moduli space of Kahler-Einstein Fano varieties · Local GIT polystability · 0.89
-
Further the theory of Cheeger, Colding and Tian was used to prove Yau-Tian-Donaldson conjecture by Chen-Donaldson-Sun [ref] , Tian [ref] , and to study the compactification of the moduli space of Kähler-Einstein manifolds in [ref] , [ref] , [ref] , [ref] .
Fano-Ricci limit spaces and spectral convergence · Introduction · 0.81
-
In such studies related to the canonical Kähler metrics, we have occasionally encountered the problem of constructing compactification of moduli spaces (“K-moduli”[i.e. vaguely speaking, moduli of K-stable (polarized) varieties cf., [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , and the last year's settlement of (smoothable) Fano case [ref] , [ref] , [ref] .
Canonical Kahler metrics and Arithmetics -- Generalising Faltings heights · Introduction · 0.99
What it cites · 32 sentences without a confident label
Uses the method or tool of 4 papers
-
for some positive integers $a_i$, then we have [equation] From this equation and the definition of relative canonical line bundle one can calculate, as in [ref] , the behavior of $μ^*_h_t$ in a neighborhood of a point $x̃∈𝒳_0'∪⋃_i E_i$.
Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds · Lower bound of log-Mabuchi-functional · 0.96
-
By well-known approximation theorem for plurisubharmic functions on Kähler manifolds (for example, Proposition 2.1 (2) in [ref] ), for any fixed $ϵ$, we can find a sequence of smooth hermitian metrics, written as $e^-Ψ_i_h_t$ where $Ψ_i$ is a function on $π^-1(Δ')$ that decreasingly converges to $r(β)Φ+(1-β)log|S_t|^2_h_t$, and with $Ω+√(-1)Ψ_i≥0$.
A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem · Lower bound of log-Ding-functional · 0.94
-
The openness part makes use of the idea in [ref] and some of the topological arguments required are somehow reminiscent of the ones in Odaka-Spotti-Sun [ref] .
Compact Moduli Spaces of Del Pezzo Surfaces and K\"ahler-Einstein metrics · Introduction · 0.94
Compares with 4 papers
-
The condition in the previous Proposition [ref] is just the condition in Proposition 4.13 in [ref] in our conventions.
K\"ahler-Einstein metrics and the K\"ahler-Ricci flow on log Fano varieties · An alpha-invariant estimate and uniform small angle of existence · 0.99
-
While we are finishing this paper, we learned that Chi Li, Xiaowei Wang, and Chenyang Xu [ref] have independently proved a result that is closely related to the results obtained here.
On the proper moduli spaces of smoothable K\"ahler-Einstein Fano varieties · Introduction · 0.99
-
Theorem [ref] should be also compared with the result on the existence of KE metrics and K-polystability on semi-log-canonical varieties with ample canonical $ℚ$-bundle obtained respectively by Berman-Guenancia in [ref] and by Odaka in [ref] and [ref] (although the reason for the KE/K-polystability correspondence in this latter case is not fully clear).
K\"ahler-Einstein metrics on stable varieties and log canonical pairs · Introduction · 1.00
Cites as support 3 papers
-
According to [ref] , $(X_t,(1-β)D_t)$ is K-polystable for $t∈Δ^*, β∈(0,β”]$.
K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics · An alpha-invariant estimate and uniform small angle of existence · 0.98
-
By [ref] and [ref] , the automorphism $Aut(X_0, D_0)$ is discrete, the openness part of this continuity method holds for $β<1$ (we do not need to assume that $Aut(X_0)$ is discrete).
K\"ahler-Einstein metrics and the K\"ahler-Ricci flow on log Fano varieties · Finishing Proof of Theorem [ref]: · 0.99
-
By [ref] and [ref] , the automorphism $Aut(X_0, D_0)$ is discrete, the openness part of this continuity method holds for $β<1$ (we do not need to assume that $Aut(X_0)$ is discrete).
The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality · Finishing Proof of Theorem [ref]: · 0.99
Cites as background 2 papers
-
We first remark that we have different notions of being Kähler-Einstein depending on the situation we are considering (we refer to [ref] Section 4, for the precise definitions): a smooth Kähler-Einstein metric is the standard one on a smooth manifold; a (genuinely) conical Kähler-Einstein metric is defined for a pair $(X, (1-β)D)$ of a smooth Kähler manifold and a smooth divisor (called Kähler-Einstein metrics with cone singularities in [ref] ); a weak Kähler-Einstein metric is defined on a $$-Fano variety $X$ [ref] ; a weak conical Kähler-Einstein metric is defined on a KLT pair $(X, (1-β)D)$ (see [ref] ).
K\"ahler-Einstein metrics and the K\"ahler-Ricci flow on log Fano varieties · Small angle existence and $L^∞$-bound on the potentials · 0.97
-
The line bundle $ℒ=K_𝒳/Δ^*^-1$ over $𝒳$ gives rise to a line bundle $L=<ℒ, ⋯, ℒ>$ over $Δ^*$, called the Deligne pairing [ref] , [ref] , [ref] , depending multi-linearly on the $n+1$ components.
K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics · Lower bound of log-Ding-functional · 0.95
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.