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Абсолютно трианалитические торы в обобщённом многообразии Куммера
С. 57–58.
Kurnosov N.
Keywords: hyperkahler manifoldsгиперкэлеровы многообразияtrianalytic subvaritiesтрианалитические подмногообразия
Publication based on the results of:
In book
Математический институт им. В. А. Стеклова РАН, 2015.
Amerik E., Verbitsky M., Soldatenkov A., / Series arXiv "math". 2025.
Wierzba and Wisniewski proved that in dimension 4, every bimeromorphic map of hyperkahler manifolds is represented as a composition of Mukai flops. Hu and Yau conjectured that this result can be generalized to arbitrary dimension. They defined ``Mukai's elementary transformation'' as the blow-up of a subvariety ruled by complex projective spaces, composed with the contraction ...
Added: December 1, 2025
Amerik E., Verbitsky M., Algebraic Geometry 2022 Vol. 9 No. 3 P. 252–265
We describe the extremal rays and the exceptional loci of extremal contractions on a hyperk ̈ahler manifold of K3[n] type for small n by deforming to the Hilbert scheme of a non-algebraic K3 surface. ...
Added: November 29, 2022
Amerik E., Verbitsky M., / Series arXiv "math". 2021.
An MBM locus on a hyperkahler manifold is the union of all deformations of a minimal rational curve with negative self-intersection. MBM loci can be equivalently defined as centers of bimeromorphic contractions. It was shown that the MBM loci on deformation equivalent hyperkahler manifolds are diffeomorphic. We determine the MBM loci on a hyperkahler manifold ...
Added: April 7, 2022
Amerik E., Verbitsky M., / Series arXiv "math". 2021.
A parabolic automorphism of a hyperkahler manifold is a holomorphic automorphism acting on H2(M) by a non-semisimple quasi-unipotent linear map. We prove that a parabolic automorphism which preserves a Lagrangian fibration acts on its fibers ergodically. The invariance of a Lagrangian fibration is automatic for manifolds satisfying the hyperkahler SYZ conjecture; this includes all known examples of ...
Added: April 6, 2022
Verbitsky M., Kamenova L., / Series arXiv "math". 2021.
Let M be a hyperkahler manifold of maximal holonomy (that is, an IHS manifold), and let K be its Kahler cone, which is an open, convex subset in the space H1,1(M,R) of real (1,1)-forms. This space is equipped with a canonical bilinear symmetric form of signature (1,n) obtained as a restriction of the Bogomolov-Beauville-Fujiki form. The set of vectors of positive square in ...
Added: November 25, 2021
Abasheva A., / Series math "arxiv.org". 2020. No. arXiv:2007.05773.
In this paper we study the geometry of the total space Y of a cotangent bundle to a Kähler manifold N where N is obtained as a Kähler reduction from Cn. Using the hyperkähler reduction we construct a hyperkähler metric on Y and prove that it coincides with the canonical Feix-Kaledin metric. This metric is in general non-complete. We show that the metric completion Y~ of ...
Added: July 21, 2020
Verbitsky M., Amerik E., / Series arXiv "math". 2019.
We study the exceptional loci of birational (bimeromorphic) contractions of a hyperkähler manifold M. Such a contraction locus is the union of all minimal rational curves in a collection of cohomology classes which are orthogonal to a wall of the Kähler cone. Homology classes which can possibly be orthogonal to a wall of the Kähler cone ...
Added: June 9, 2019
Amerik E., Verbitsky M., / Series arXiv "math". 2018.
An MBM class on a hyperkahler manifold M is a second cohomology class such that its orthogonal complement in H^2(M) contains a maximal dimensional face of the boundary of the Kahler cone for some hyperkahler deformation of M. An MBM curve is a rational curve in an MBM class and such that its local deformation ...
Added: December 4, 2018
Tomberg A., Математические заметки 2019 Т. 105 № 6 С. 949–954
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Added: November 11, 2018
Verbitsky M., Entov M., Selecta Mathematica, New Series 2018 Vol. 24 No. 3 P. 2625–2649
Let M be a closed symplectic manifold of volume V. We say that M admits an unobstructed symplectic packing by balls if any collection of symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds ...
Added: September 13, 2018
Verbitsky M., Markman E., Mehrotra S., / Series arXiv "math". 2017.
Let S be a K3 surface and M a smooth and projective 2n-dimensional moduli space of stable coherent sheaves on S. Over M x M there exists a rank 2n-2 reflexive hyperholomorphic sheaf E_M, whose fiber over a non-diagonal point (F,G) is Ext^1(F,G). The sheaf E_M can be deformed along some twistor path to a ...
Added: October 10, 2017
Kamenova L., Verbitsky M., / Series arXiv "math". 2016.
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: April 10, 2017
Kamenova L., Verbitsky M., New York Journal of Mathematics 2017 Vol. 23 P. 489–495
A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which follows from Kobayashi hyperbolicity. We prove that hyperkähler manifolds are not algebraically hyperbolic when the Picard rank is at least 3, or ...
Added: April 10, 2017
Verbitsky M., Selecta Mathematica, New Series 2017 Vol. 23 No. 3 P. 2203–2218
The transcendental Hodge lattice of a projective manifold M is the smallest Hodge substructure in pth cohomology which contains all holomorphic p-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkähler manifold. As an application, we obtain a theorem about ...
Added: February 6, 2017
Kurnosov N., В кн.: Международная конференция по алгебраической геометрии, комплексному анализу и компьютерной алгебре.: М.: Математический институт им. В. А. Стеклова РАН, 2016. С. 46–47.
В нашей работе, используя инварианты Розанского-Виттена, мы доказываем, что если M компактное неприводимое гиперкэлеровое многообразие размерности шесть, то выполнено некоторое неравенство на размерности чисел Бетти M. ...
Added: October 24, 2016
Amerik E., Verbitsky M., Annales Scientifiques de l'Ecole Normale Superieure 2017 Vol. 50 No. 4 P. 973–993
Let M be a simple hyperk¨ahler manifold, that is, a simply connected compact holomorphically symplectic manifold of K¨ahler type with h 2,0 = 1. Assuming b2(M) 6= 5, we prove that the group of holomorphic automorphisms of M acts on the set of faces of its K¨ahler cone with finitely many orbits. This statement is ...
Added: September 8, 2016
Amerik E., Campana F., Journal of London Mathematical Society 2017 Vol. 95 No. 1 P. 115–127
We prove that the characteristic foliation F on a nonsingular divisor D in an irreducible projective hyperk¨ahler manifold X cannot be algebraic, unless the leaves of F are rational curves or X is a surface. More generally, we show that if X is an arbitrary projective manifold carrying a holomorphic symplectic 2-form, and D and ...
Added: September 8, 2016
Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperkahler geometry
Amerik E., Verbitsky M., / Series arXiv "math". 2016.
Let M be a hyperk\"ahler manifold with b2(M)≥5. We improve our earlier results on the Morrison-Kawamata cone conjecture by showing that the Beauville-Bogomolov square of the primitive MBM classes (i.e. the classes whose orthogonal hyperplanes bound the K\"ahler cone in the positive cone, or, in other words, the classes of negative extremal rational curves on ...
Added: September 7, 2016