• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site
Of all publications in the section: 8
Sort:
by name
by year
Article
Kuznetsov A. G. Communications in Contemporary Mathematics. 2010. Vol. 12. No. 3. P. 373-416.

It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed p-form with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y of degree n. We provide several definitions of this form - via the Abel-Jacobi map, via Hochschild homology, and via the linkage class, and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Y we show that the Fano scheme is birational to a certain moduli space of sheaves on a p-dimensional Calabi--Yau variety X arising naturally in the context of homological projective duality, and that the constructed form is induced by the holomorphic volume form on X. This remains true for a general non Pfaffian hypersurface but the dual Calabi-Yau becomes non commutative.

Added: Oct 4, 2013
Article
Bogomolov F. A., Kamenova L., Verbitsky M. Communications in Contemporary Mathematics. 2020. Vol. 22. No. 2. P. 1950003.

A projective manifold M is algebraically hyperbolic if there exists a positive constant A such that the degree of any curve of genus g on M is bounded from above by A(g−1). A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here, we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.

Added: May 10, 2020
Article
Sobolevski A. Communications in Contemporary Mathematics. 1999. Vol. 1. No. 4. P. 517-533.
Added: Jan 2, 2012
Article
Grantcharov G., Verbitsky Misha. Communications in Contemporary Mathematics. 2013. Vol. 15. No. 2. P. 1-27.

We describe a family of calibrations arising naturally on a hyper-Kähler manifold M. These calibrations calibrate the holomorphic Lagrangian, holomorphic isotropic and holomorphic coisotropic subvarieties. When M is an HKT (hyper-Kähler with torsion) manifold with holonomy SL(n, H), we construct another family of calibrations Φi, which calibrates holomorphic Lagrangian and holomorphic coisotropic subvarieties. The calibrations Φi are (generally speaking) not parallel with respect to any torsion-free connection on M.

Added: Mar 27, 2013
Article
Arzhantsev I., Bragin S., Zaitseva Y. Communications in Contemporary Mathematics. 2020. Vol. 22. No. 8. P. 1950064: 1.

We study commutative associative polynomial operations A^n×A^n→A^n with unit on the affine space A^n over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties

Added: Sep 19, 2019
Article
Bobkov S., Goetze F., Sambale H. Communications in Contemporary Mathematics. 2019. Vol. 21. No. 3. P. 1-31.

We study sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order d-1 for any d \in N. The bounds are based on dth order derivatives or difference operators. In particular, we consider deviations of functions of independent random variables and differentiable functions over probability measures satisfying a logarithmic Sobolev inequality, and functions on the unit sphere. Applications include concentration inequalities for U-statistics as well as for classes of symmetric functions via polynomial approximations on the sphere (Edgeworth-type expansions).

Added: Oct 15, 2018
Article
Verbitsky M., Kamenova L. Communications in Contemporary Mathematics. 2019. Vol. 21. No. 8. P. 1850065- .

Let p: M -> B be a Lagrangian fibration on a hyperkahler manifold of maximal holonomy (also known as IHS), and H the generator of the Picard group of B. We prove that the pullback p∗(H) is a primitive class on M.

 

Added: Oct 15, 2018
Article
Fang X., Feigin E., Fourier G. et al. Communications in Contemporary Mathematics. 2019. Vol. 21. No. 1. P. 1-27.

We study algebraic, combinatorial and geometric aspects of weighted Poincaré–Birkhoff–Witt (PBW)-type degenerations of (partial) flag varieties in type A. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.

Added: Oct 8, 2019