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## Delta-invariants for Fano varieties with large automorphism groups

International Journal of Mathematics. 2020. Vol. 31. No. 10. P. 2050077.

For a variety *𝑋*, a big *ℚ*-divisor *𝐿* and a closed connected subgroup *𝐺*⊂Aut(*𝑋*,*𝐿*) we define a *𝐺*-invariant version of the *𝛿*-threshold. We prove that for a Fano variety (*𝑋*,−*𝐾_**𝑋*) and a connected subgroup *𝐺*⊂Aut(*𝑋*) this invariant characterizes *𝐺*-equivariant uniform *𝐾*-stability. We also use this invariant to investigate *𝐺*-equivariant *𝐾*-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of *𝐺* being a finite group.

Keywords: многообразия ФаноFano varietiesметрики Кэлера-Эйнштейнаautomorphism groupгруппы автоморфизмовK-stability

Publication based on the results of:

Gorinov A., Nikolay Konovalov, / Cornell University. Series "Working papers by Cornell University". 2020. No. 1712.02578.

Let $G$ be a complex connected reductive algebraic group that acts on a smooth complex algebraic variety $X$, and let $E$ be a $G$-homogeneous algebraic vector bundle over $X$. A section of $E$ is {\it regular} if it is transversal to the zero section. Let $U\subset\Gamma(X,E)$ be the subset of regular sections. We give a ...

Added: March 16, 2020

Przyjalkowski V., Известия РАН. Серия математическая 2013 Т. 77 № 4 С. 135-160

We consider Landau–Ginzburg models for smooth Fano threefolds of the principal series and prove that they can be represented by Laurent polynomials. We check that these models can be compactified to open Calabi–Yau varieties. In the spirit of Katzarkov's programme we prove that the numbers of irreducible components of the central fibres of compactifications of ...

Added: February 6, 2013

Nikolay Konovalov, / Cornell University. Series "Working papers by Cornell University". 2022. No. 2202.07507.

Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence ...

Added: September 12, 2022

Sheina K., / Cornell University. Series arXiv "math". 2020. No. 04348v1.

The basic automorphism group of a Cartan foliation (M, F) is the quotient group of the automorphism group of (M, F) by the normal subgroup, which preserves every leaf invariant. For Cartan foliations covered by fibrations, we find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in their basic automorphism groups. Estimates ...

Added: December 9, 2020

Shramov K., Prokhorov Y., / Cornell University. Series arXiv "math". 2019.

We classify compact complex surfaces whose groups of bimeromorphic selfmaps have bounded finite subgroups. We also prove that the stabilizer of a point in the automorphism group of a compact complex surface of zero Kodaira dimension, as well as the stabilizer of a point in the automorphism group of an arbitrary compact Kaehler manifold of ...

Added: November 19, 2019

Tokyo : American Mathematical Society, World Scientific, 2017

Preface
The workshop “Algebraic Varieties and Automorphism Groups” was held at the Research Institute of Mathematical Sciences (RIMS), Kyoto University during July 7-11, 2014. There were over eighty participants including twenty people from overseas Canada, France, Germany, India, Korea, Poland, Russia, Singapore, Switzerland, and USA.
Recently, there have been remarkable developments in algebraic geometry and related fields, ...

Added: July 12, 2017

Shramov K., European Journal of Mathematics 2019

We show that automorphism groups of Hopf and Kodaira surfaces have unbounded
finite subgroups. For elliptic fibrations on Hopf, Kodaira, bielliptic, and K3 surfaces,
we make some observations on finite groups acting along the fibers and on the base
of such a fibration. ...

Added: December 11, 2019

Kuyumzhiyan K., Proceedings of the American Mathematical Society 2020 No. 148 P. 3723-3731

We prove the conjecture of Berest-Eshmatov-Eshmatov by showing that the group of automorphisms of a product of Calogero-Moser spaces C_n_i, where the n_i are pairwise distinct, acts m-transitively for each m. ...

Added: August 18, 2020

Prokhorov Y., Труды Математического института им. В.А. Стеклова РАН 2019 Т. 307 С. 230-253

Классифицированы некоторые специальные классы трехмерных нерациональных многообразий Фано с терминальными особенностями. В частности, найдены все такие гиперэллиптические и тригональные многообразия. ...

Added: May 10, 2020

Prokhorov Y., Shramov K., / Cornell University. Series arXiv "math". 2018.

We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan. ...

Added: June 8, 2019

Vladimir L. Popov, Transformation Groups 2014 Vol. 19 No. 2 P. 549-568

We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given. ...

Added: March 17, 2014

Popov V. L., Zarhin Y., / Cornell University. Series math "arxiv.org". 2018. No. 1808.01136.

We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $\mathcal L(K)$ generated by ${\rm Aut} (K)$ and multipli\-ca\-tions by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are ...

Added: August 8, 2018

Zhukova N., Moscow Mathematical Journal 2018

We introduce a category of rigid geometries on singular spaces which
are leaf spaces of foliations and are considered as leaf manifolds. We
single out a special category F_0 of leaf manifolds containing the orbifold
category as a full subcategory. Objects of F_0 may have non-Hausdorff
topology unlike the orbifolds. The topology of some objects of F_0 does
not satisfy ...

Added: April 2, 2018

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2014. No. 1401.0278.

We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given. ...

Added: January 3, 2014

Vladimir L. Popov, Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 185-213

This is an expanded version of my talk at the workshop
``Groups of Automorphisms in Birational and Affine Geometry'',
October 29–November 3, 2012, Levico Terme, Italy.
The first section is focused on Jordan groups in abstract setting,
the second on that in the settings of automorphisms groups and
groups of birational self-maps of algebraic varieties.
The appendix is an expanded version ...

Added: April 28, 2014

Avilov A., Известия РАН. Серия математическая 2019 Т. 83 № 3 С. 5-14

Трехмерные многообразия дель Пеццо степени 22 являются двулистными накрытиями P^3 с ветвлением в квартике. В этой заметке мы показываем, что для многообразий дель Пеццо степени 22 с 15 обыкновенными двойными точками соответствующая квартика является гиперплоским сечением квартики Игусы. Само многообразие дель Пеццо является элементом конкретной линейной системы на четырехмерном многообразии Кобла, а его группа автоморфизмов индуцирована с группы автоморфизмов многообразия Кобла. Кроме того, мы классифицируем бирационально жесткие ...

Added: June 4, 2019

Przyjalkowski V., Cheltsov I., Shramov K., Известия РАН. Серия математическая 2019 Т. 83 № 4 С. 226-280

We classify smooth Fano threefolds with infinite automorphism groups. ...

Added: October 8, 2019

Cheltsov I., Kuznetsov A., Shramov K., Algebra & Number Theory 2020 Vol. 14 No. 1 P. 213-274

We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all 𝔖6-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman–Edge pencil. As an application, ...

Added: May 10, 2020

Prokhorov Y., Cheltsov I., / Cornell University. Series arXiv "math". 2020.

We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups. ...

Added: August 19, 2020

Avilov A., Sbornik Mathematics 2016 Vol. 307 No. 3 P. 315-330

We prove that any G-del Pezzo threefold of degree 4, except for a one-parameter family and four distinguished cases, can be equivariantly reconstructed to the projective space ℙ3, a quadric Q ⊂ ℙ4 , a G-conic bundle or a del Pezzo fibration. We also show that one of these four distinguished varieties is birationally rigid ...

Added: July 6, 2016

Avilov A., Математические заметки 2020 Т. 107 № 1 С. 3-10

The forms of the Segre cubic over non-algebraically closed fields, their automorphisms groups, and equivariant birational rigidity are studied. In particular, it is shown that all forms of the Segre cubic over any field have a point and are cubic hypersurfaces. ...

Added: May 11, 2020

Nina I. Zhukova, Anna Yu. Dolgonosova .., Central European Journal of Mathematics 2013 Vol. 11 No. 12 P. 2076-2088

The category of foliations is considered. In this category
morphisms are differentiable mappings transforming leaves of one
foliation into leaves of the other foliation.
We proved that the automorphism group of the foliations
admitting a transverse linear connection is an infinite-dimensional
Lie group modeled on $LF$-spaces. This result extends the corresponding
result of Macias-Virgos and Sanmartin for Riemannian foliations.
In particular, our ...

Added: September 28, 2014

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2013. No. 1307.5522.

This is an expanded version of my talk at the workshop ``Groups of Automorphisms in Birational and Affine Geometry'', October 29–November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. ...

Added: July 21, 2013

Shramov K., Przyjalkowski V., Proceedings of the Steklov Institute of Mathematics 2019 Vol. 307 P. 198-209

We show that smooth well-formed weighted complete intersections have finite automorphism groups, with several obvious exceptions. ...

Added: August 12, 2020