### ?

## Cohomology Rings and Algebraic Torus Actions on Hypersurfaces in the Product of Projective Spaces and Bounded Flag Varieties

Arnold Mathematical Journal. 2022. P. 1-46.

Solomadin G.

In this paper, for any Milnor hypersurface, we find the largest dimension of effective

algebraic torus actions on it. The proof of the corresponding theorem is based on

the computation of the automorphism group for any Milnor hypersurface. We find

all generalized Buchstaber–Ray and Ray hypersurfaces that are toric varieties. We

compute the Betti numbers of these hypersurfaces and describe their integral singular

cohomology rings in terms of the cohomology of the corresponding ambient varieties

Bodzenta-Skibinska A., DG quivers of smooth rational surfaces / Cornell University. Series math "arxiv.org". 2013.

Let X be a smooth rational surface. We calculate a DG quiver of a full exceptional collection of line bundles on X obtained by an augmentation from a strong exceptional collection on the minimal model of X. In particular, we calculate canonical DG algebras of smooth toric surfaces. ...

Added: November 5, 2014

Alexander I. Efimov, Journal of London Mathematical Society 2014 Vol. 90 No. 2 P. 350-372

In this paper, we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant $c>\frac 34$ there exist infinitely many toric Fano varieties $Y$ with ...

Added: January 28, 2015

Shabanov D. A., Akhmejanova M., Discrete Mathematics 2020 Vol. 343 No. 4 P. 1-11

The paper deals with an extremal problem concerning colorings of hypergraphs with bounded edge degrees. Consider the family of b-simple hypergraphs, in which any two edges do not share more than b common vertices. We prove a new lower bound for the maximum edge degree in a n-uniform b-simple non-r-colorable hypergraph. We also establish some ...

Added: October 31, 2019

Buchstaber V.M., Terzic S., Moscow Mathematical Journal 2016 Vol. 16 No. 2 P. 237-273

We consider the canonical action of the compact torus T4 on the complex Grassmann manifold G4,2 and prove that the orbit space G4,2/T4 is homeomorphic to the sphere S5. We prove that the induced map from G4;2 to the sphere S5 is not smooth and describe its smooth and singular points. We also consider the action of T4 on ...

Added: June 17, 2021

Boldyrev I., Gayfullin S., Математические заметки 2021 Т. 110 № 6 С. 837-855

Criteria for the flexibility, rigidity, and almost rigidity of nonnormal affine toric varieties are obtained. For rigid and almost rigid toric varieties, automorphism groups are explicitly calculated. ...

Added: February 6, 2022

Ornea L., Verbitsky M., Vuletescu V., International Mathematics Research Notices 2013 No. 12 P. 2809-2821

A locally conformally Khler (LCK) manifold is a complex manifold which admits a covering endowed with a Kähler metric with respect to which the covering group acts through homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally ...

Added: October 10, 2013

Kravtsov D., Krokhmal N., Shabanov D. A., Russian Mathematical Surveys 2018 Vol. 73 No. 4 P. 731-733

The paper deals with the problem of finding the probability threshold for the
existence of a panchromatic colouring for a random hypergraph in the binomial
model. ...

Added: November 15, 2018

Ye L., Faisceau Automorphe Unipotent pour G2,Nombres de Franel, et Stratification deThom-Boardman / Cornell University. Series math "arxiv.org". 2020.

Thesis of the author. ...

Added: December 16, 2019

Galkin S., Nagaraj D. S., Projective bundles and blow-ups of Projective spaces / Cornell University. Series math "arxiv.org". 2020. No. 2006.12112.

The aim of this note is to investigate the relation between two types of non-singular projective varieties of Picard rank 2, namely the Projective bundles over Projective spaces and certain Blow-up of Projective spaces. ...

Added: April 15, 2021

Shabanov D. A., Discrete Mathematics 2015 Vol. 338 No. 11 P. 1976-1981

The work deals with combinatorial problems concerning colorings of non-uniform hypergraphs. Let $H=(V,E)$ be a hypergraph with minimum edge-cardinality $n$. We show that if $H$ is a simple hypergraph (i.e. every two distinct edges have at most one common vertex) and
$$
\sum_{e\in E}r^{1-|e|}\leqslant c\sqrt{n},
$$
for some absolute constant $c>0$, then $H$ is $r$-colorable. We also obtain ...

Added: October 6, 2015

Shafarevich A., Moscow University Mathematics Bulletin 2019 Vol. 74 No. 5 P. 209-211

Let X be an affine toric variety over an algebraically closed field of characteristic zero. Orbits of connected component of identity of automorphism group in terms of dimensions of tangent spaces of the variety X are described. A formula to calculate these dimensions is presented. ...

Added: September 10, 2021

Shabanov D. A., Graphs and Combinatorics 2014 Vol. 30 No. 5 P. 1249-1260

The work deals with a generalization of Erdos-Lovasz problem concerning colorings of non-uniform hypergraphs. We establish a new sufficient condition for r-colorability of a non-unifrom hypergraph with large edge sizes and girth at leats 4 in terms of expectation of the number of monochromatic edges in a random coloring. ...

Added: December 15, 2015

Shabanov D. A., Kozik J., Journal of Combinatorial Theory. Series B 2016 Vol. 116 P. 312-332

The paper deals with extremal problems concerning colorings of hypergraphs. By using a random recoloring algorithm we show that any n-uniform simple (i.e. every two distinct edges share at most one vertex) hypergraph H with not large maximum edge degree is r-colorable. As an application of our proof technique we establish a new lower bound ...

Added: December 15, 2015

Lebedeva A., Фундаментальная и прикладная математика 2014 Т. 19 № 2 С. 125-149

This paper deals with an extremal problem concerning hypergraph colorings. Let k be an integer. The problem is to find the value m(k,n) equal to the minimum number of edges in an n-uniform hypergraph not admitting two-colorings of the vertex set such that every edge of the hypergraph contains at least k vertices of each ...

Added: July 19, 2015

Shabanov D. A., Kupavskii A., Combinatorics Probability and Computing 2018 Vol. 27 No. 2 P. 245-273

This paper deals with a combinatorial problem concerning colourings of uniform hypergraphs
with large girth. We prove a new lower bound for the maximum edge degree for an n-uniform non-r-colourable simple hypergraph. As an application of our probabilistic technique we establish a lower bound for the classical
van der Waerden number W(n, r), the minimum natural N ...

Added: February 22, 2018

Galkin S., Belmans P., Mukhopadhyay S., Experimental Mathematics 2019

We give the first examples of smooth Fano and Calabi–Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarized varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have ...

Added: October 4, 2019

Shabanov D. A., European Journal of Combinatorics 2015 Vol. 43 P. 185-203

An equitable two-coloring of a hypergraph $H=(V,E)$ is a proper vertex two-coloring such that the cardinalities of color classes differ by at most one. In connection with the property B problem Radhakrishnan and Srinivasan proved that if $H$ is a $k$-uniform hypergraph with maximum vertex degree $\Delta(H)$ satisfying
$$
\Delta(H)\leqslant c\,\frac {2^{k-1}}{\sqrt{k\,\ln k}}
$$
for some absolute constant ...

Added: October 6, 2015

Shabanov D. A., Discrete Applied Mathematics 2020 Vol. 282 P. 168-183

The paper deals with estimating the r-colorability threshold for a random k-uniform hypergraph in the binomial model H(n,k,p). We consider the sparse case, when the expected number of edges is a linear function of n and prove a new lower bound for the sharp threshold of the property that H(n,k,p) is r-colorable. ...

Added: June 6, 2020

Shabanov D. A., Doklady Mathematics 2017 Vol. 96 No. 1 P. 321-325

The problem on the limit distribution of the chromatic number of a random uniform hypergraph in the sparse case is studied. It is shown that, for most parameters values, the limit distribution of the chromatic number is concentrated at precisely one point, which can be found explicitly. ...

Added: March 6, 2018

Shafarevich A., Results in Mathematics 2021 Vol. 76 No. 3 Article 145

Let KK be an algebraically closed field of characteristic zero and GaGa be the additive group of KK. We say that an irreducible algebraic variety X of dimension n over the field KK admits an additive action if there is a regular action of the group Gna=Ga×⋯×GaGan=Ga×⋯×Ga (n times) on X with an open orbit. In this paper we find all projective toric hypersurfaces admitting additive action. ...

Added: September 10, 2021

Shabanov D. A., Akolzin I., Discrete Mathematics 2016 Vol. 339 No. 12 P. 3020-3031

The paper deals with the classical extremal problem concerning colorings of hypergraphs. The problem is to find the value m(n,r), equal to the minimum number of edges in a n-uniform hypergraph with chromatic number greater than r. We obtain new upper and lower bounds for m(n,r) in the case when the parameter r is very ...

Added: September 4, 2016

Akhmejanova M., Shabanov D. A., Discrete Applied Mathematics 2020 Vol. 276 P. 2-12

The paper deals with an extremal problem concerning equitable colorings of uniform hypergraphs. Recall that a vertex coloring of a hypergraph is called proper if there are no monochromatic edges under this coloring. A hypergraph is said to be equitably r-colorable if there is a proper coloring with r colors such that the sizes of ...

Added: October 31, 2019

Panov T., Dolbeault cohomology of complex manifolds with torus action / Cornell University. Series arXiv "math". 2019.

We describe the basic Dolbealut cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds with a maximal holomorphic torus action. We also provide a dga model for the ordinary Dolbeault cohomology algebra. ...

Added: November 1, 2019

Katzarkov L., Abouzaid M., Auroux D., Publications Mathématiques de l'IHÉS 2016 Vol. 123 No. 1 P. 199-282

https://link.springer.com/article/10.1007/s10240-016-0081-9 ...

Added: October 23, 2017