# Reductivity of the automorphism group of K-polystable Fano varieties

@article{Alper2019ReductivityOT, title={Reductivity of the automorphism group of K-polystable Fano varieties}, author={Jarod Alper and Harold Blum and Daniel Halpern-Leistner and Xu Chen}, journal={arXiv: Algebraic Geometry}, year={2019} }

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $\Theta$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space.

#### 38 Citations

Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups

- Mathematics
- 2019

It is known that the automorphism group of a K-polystable Fano manifold is reductive. Codogni and Dervan construct a canonical filtration of the section ring, called Loewy filtration, and conjecture… Expand

On properness of K-moduli spaces and optimal degenerations of Fano varieties

- Mathematics
- 2020

We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable Fano varieties.… Expand

G-uniform stability and Kähler-Einstein metrics on Fano varieties

- 2019

Let X be any Q-Fano variety and Aut(X)0 be the identity component of the automorphism group of X . Let G denote a connected reductive subgroup of Aut(X)0. We prove that if X is G-uniformly K-stable,… Expand

G-uniform stability and Kähler-Einstein metrics on Fano varieties

- 2019

Let X be any Q-Fano variety and Aut(X)0 be the identity component of the automorphism group of X. Let G denote a connected reductive subgroup of Aut(X)0. We prove that if X is G-uniformly K-stable,… Expand

Openness of K-semistability for Fano varieties

- Mathematics
- 2019

In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also prove that any special… Expand

Log K-stability of GIT-stable divisors on Fano manifolds

- Mathematics
- 2021

For a given K-polystable Fano manifold X and a natural number l, we show that there exists a rational number 0 < c1 < 1 depending only on the dimension of X, such that D ∈ | − lKX | is… Expand

Positivity of the CM line bundle for K-stable log Fanos

- Mathematics
- 2019

We prove the bigness of the Chow-Mumford line bundle associated to a $\mathbb{Q}$-Gorenstein family of log Fano varieties of maximal variation with uniformly K-stable general geometric fibers. This… Expand

Effective semi-ampleness of Hodge line bundles on curves I

- Mathematics
- 2021

In this note, we prove effective semi-ampleness conjecture due to Prokhorov and Shokurov for a special case, more concretely, for Q-Gorenstein klt-trivial fibrations over smooth projective curves… Expand

On deformation spaces of toric varieties

- Mathematics
- 2021

Firstly, we see that the bases of the miniversal deformations of isolated Q-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension ≤… Expand

Equivariant K-stability under finite group action

- Mathematics
- 2020

We show that G-equivariant K-semistability (resp. G-equivariant K-polystability) implies K-semistability (resp. K-polystability) for log Fano pairs when G is a finite group.

#### References

SHOWING 1-10 OF 56 REFERENCES

Uniqueness of K-polystable degenerations of Fano varieties.

- Mathematics
- 2018

We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,BL18], the latter… Expand

Almost Proper GIT-Stacks

- Mathematics
- 2010

We prove that the classifying stack of an reductive group scheme over a field is very close to being proper. Using this we prove a result about isotrivial families of varieties. Fix a polarized… Expand

On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties

- Mathematics
- 2014

In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano K\"ahler-Einstein manifolds inside an appropriate Hilbert scheme. In… Expand

A Luna étale slice theorem for algebraic stacks

- Mathematics
- 2015

We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is etale-locally a quotient stack in a neighborhood of a point with a linearly… Expand

Algebraicity of the metric tangent cones and equivariant K-stability

- Mathematics
- 2018

We prove two new results on the K-polystability of Q-Fano varieties based on purely algebro-geometric arguments. The first one says that any K-semistable log Fano cone has a special degeneration to a… Expand

Openness of K-semistability for Fano varieties

- Mathematics
- 2019

In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also prove that any special… Expand

Existence of moduli spaces for algebraic stacks.

- Mathematics
- 2018

We provide necessary and sufficient conditions for when an algebraic stack admits a good moduli space. This theorem provides a generalization of the Keel--Mori theorem to moduli problems whose… Expand

Boundedness of Q-Fano varieties with degrees and alpha-invariants bounded from below

- Mathematics
- 2020

We show that $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable $\mathbb{Q}$-Fano… Expand

A minimizing valuation is quasi-monomial

- Mathematics
- 2019

We prove a version of Jonsson-Mustaţǎ's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we… Expand

Asymptotic Chow stability of toric Del Pezzo surfaces

- Mathematics
- 2017

In this short note, we study the asymptotic Chow polystability of toric Del Pezzo surfaces appear in the moduli space of K\"ahler-Einstein Fano varieties constructed in [OSS16].