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## Log canonical thresholds of certain Fano hypersurfaces

Mathematische Zeitschrift. 2014. No. 276. P. 51-79.

Cheltsov Ivan, Park J., Won J.

We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an important application, we show that they have Kähler–Einstein metrics if they are general.

Cheltsov Ivan, Wilson A., Journal of Geometric Analysis 2013 Vol. 23 No. 3 P. 1257-1289

We classify smooth del Pezzo surfaces whose α-invariant of Tian is bigger than 1. ...

Added: November 14, 2013

Cheltsov I., Zhang K., European Journal of Mathematics 2019 Vol. 5 P. 729-762

We prove that 𝛿δ-invariants of smooth cubic surfaces are at least 6/5. ...

Added: May 10, 2020

Cheltsov Ivan, Shramov Constantin, Experimental Mathematics 2013 Vol. 22 No. 3 P. 313-326

We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose α-invariant of Tian is greater than 2/3. ...

Added: January 27, 2014

Kharlamov V., Viktor Kulikov, / Cornell University. Series math "arxiv.org". 2013.

In this article, we investigate some properties of cyclic coverings of complex surfaces of general type branched along smooth curves that are numerically equivalent to a multiple of the canonical class. The main results concern coverings of surfaces of general type with p_g=0 and Miyaoka--Yau surfaces; in particular, they provide new examples of multicomponent moduli ...

Added: December 27, 2013

Lee K., Shabalin T., / Cornell University. Series math "arxiv.org". 2014.

We construct exceptional collections of maximal length on four families of
surfaces of general type with $p_g=q=0$ which are isogenous to a product of
curves. From these constructions we obtain new examples of quasiphantom
categories as their orthogonal complements. ...

Added: October 17, 2014

Lev Soukhanov, / Cornell University. Series math "arxiv.org". 2014.

We consider the systems of diffusion-orthogonal polynomials, defined in the
work [1] of D. Bakry, S. Orevkov and M. Zani and (particularly) explain why
these systems with boundary of maximal possible degree should always come from
the group, generated by reflections. Our proof works for the dimensions $2$ (on
which this phenomena was discovered) and $3$, and fails in ...

Added: September 19, 2014

Victor Kulikov, Shustin E., / Cornell University. Series math "arxiv.org". 2014.

We study the geometry of equiclassical strata of the discriminant in the space of plane curves of a given degree, which are families of curves of given degree, genus and class (degree of the dual curve). Our main observation is that the use of duality transformation leads to a series of new sufficient conditions for ...

Added: February 2, 2015

Ivan Cheltsov, Park J., Won J., / Cornell University. Series math "arxiv.org". 2013.

For each del Pezzo surface $S$ with du Val singularities, we determine
whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we
present an effective divisor $D$ that is $\mathbb{Q}$-linearly equivalent to
$-K_S$ and such that the open set $S\setminus\mathrm{Supp}(D)$ is a cylinder.
As a corollary, we classify all the del Pezzo surfaces with du ...

Added: December 27, 2013

Fedor Bogomolov, De Oliveira B., / Cornell University. Series math "arxiv.org". 2014.

In the authors's previous work on symmetric differentials and their
connection to the topological properties of the ambient manifold, a class of
symmetric differentials was introduced: closed symmetric differentials
([BoDeO11] and [BoDeO13]). In this article we give a description of the local
structure of closed symmetric 2-differentials on complex surfaces, with an
emphasis towards the local decompositions as products of ...

Added: November 21, 2014

Rybakov S., / Cornell University. Series math "arxiv.org". 2014.

A k-isogeny class of abelian varieties over a finite field k is uniquely determined by the Weil polynomial f of any variety from this class. When we consider classification problems concerning abelian varieties inside an isogeny class, the classification can be given in terms of the corresponding Weil polynomial. In this paper we improve our ...

Added: January 21, 2014

Positselski L., Efimov A., / Cornell University. Series math "arxiv.org". 2013. No. arXiv:1102.0261.

We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations ...

Added: December 22, 2013

Fedor Bogomolov, Yuri Prokhorov, / Cornell University. Series math "arxiv.org". 2013.

We discuss the problem of stable conjugacy of finite subgroups of Cremona
groups. We show that the group $H^1(G,Pic(X))$ is a stable birational invariant
and compute this group in some cases. ...

Added: November 21, 2014

Positselski L., / Cornell University. Series math "arxiv.org". 2014. No. 1209.2995.

Contraherent cosheaves are globalizations of cotorsion (or similar) modules over commutative rings obtained by gluing together over a scheme. The category of contraherent cosheaves over a scheme is a Quillen exact category with exact functors of infinite product. Over a quasi-compact semi-separated scheme or a Noetherian scheme of finite Krull dimension (in a different version ...

Added: February 6, 2013

Galkin S., Shinder E., / Cornell University. Series math "arxiv.org". 2012. No. 1210.3339.

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface S. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on S. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections ...

Added: September 14, 2013

Brav C. I., Thomas H., Mathematische Annalen 2011 Vol. 351 No. 4 P. 1005-1017

We establish faithfulness of braid group actions generated by twists along an ADE configuration of 22-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity. ...

Added: September 29, 2014

F.A. Bogomolov, Vik.S. Kulikov, / Cornell University. Series math "arxiv.org". 2014.

In \cite{Ku0}, the ambiguity index $a_{(G,O)}$ was introduced for each
equipped finite group $(G,O)$. It is equal to the number of connected
components of a Hurwitz space parametrizing coverings of a projective line with
Galois group $G$ assuming that all local monodromies belong to conjugacy
classes $O$ in $G$ and the number of branch points is greater than some
constant. ...

Added: November 21, 2014

Romaskevich O. L., L'Enseignement Mathématique 2014

We consider 3 -periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have shown that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the complexification of the problem coupled with the complex law of reflection. ...

Added: December 25, 2014

Cheltsov Ivan, Kosta D., Journal of Geometric Analysis 2014 Vol. 24 No. October P. 798-842

We prove a new local inequality for divisors on surfaces and utilize it to compute α-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type A1 , A2 , A3 , A4 , A5 , or A6 are Kähler-Einstein. ...

Added: November 14, 2013

Campana F., Demailly J., Misha Verbitsky, / Cornell University. Series math "arxiv.org". 2013.

We prove that any compact K\"ahler 3-dimensional manifold which has no non-trivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of 'simple manifolds', central in the bimeromorphic classification of compact K\"ahler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of ...

Added: May 13, 2013

Aleksei Golota, / Cornell University. Series arXiv "math". 2019.

For a polarized variety (X,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,−KX) 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 ...

Added: October 7, 2019

Vikulova A., / Cornell University. Series arXiv "math". 2022.

In this paper we prove that for n-dimensional smooth l-Fano well formed weighted complete intersections, which is not isomorphic to a usual projective space, the upper bound for l is equal to ⌈log2(n+2)⌉−1. We also prove that the only l-Fano of dimension n among such manifolds with inequalities ⌈log3(n+2)⌉⩽l⩽⌈log2(n+2)⌉−1 is a complete intersection of quadrics in a usual projective space. ...

Added: November 27, 2022

Kishimoto T., Yuri Prokhorov, Zaidenberg M., Osaka Journal of Mathematics 2014 Vol. 51 No. 4 P. 1093-1113

We address the following question: When an affine cone over a smooth Fano threefold admits an effective action of the additive group? In this paper we deal with Fano threefolds of index 1 and Picard number 1. Our approach is based on a geometric criterion from our previous paper, which relates the existence of an ...

Added: October 10, 2013

Michael Finkelberg, Leonid Rybnikov, / Cornell University. Series math "arxiv.org". 2013.

Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic ...

Added: December 27, 2013

Bezrukavnikov R., Finkelberg M. V., / Cornell University. Series math "arxiv.org". 2012. No. 1208.3696.

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_n\ltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert ...

Added: February 6, 2013