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Ограничения на когомологии гиперкэлеровых многообразий
С. 46–47.
Kurnosov N.
Keywords: гиперкэлеровы многообразия
In book
М.: Математический институт им. В. А. Стеклова РАН, 2016.
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
Tomberg A., Математические заметки 2019 Т. 105 № 6 С. 949–954
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Added: November 11, 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., 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
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
Bogomolov F. A., Kamenova L., Lu S. et al., / Series arXiv "math". 2016.
We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them and show that if a compact complex manifold has an automorphism whose order is infinite, then the fibers of this quotient map are nontrivial. We prove that the Kobayashi quotients associated to ergodic complex structures on a ...
Added: September 6, 2016
Amerik E., Verbitsky M., Research in the Mathematical Sciences 2016 Vol. 3 No. 7 P. 1–9
Let M be a compact hyperkähler manifold with maximal holonomy (IHS). The group H2(M,ℝ) is equipped with a quadratic form of signature (3,b2−3)(3,b2−3), called Bogomolov–Beauville–Fujiki form. This form restricted to the rational Hodge lattice H1,1(M,ℚ)has signature (1, k). This gives a hyperbolic Riemannian metric on the projectivization H of the positive cone in H1,1(M,ℚ). Torelli ...
Added: August 31, 2016
Kamenova L., Verbitsky M., / Series arXiv "math". 2016.
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 hyperk¨ahler manifolds are non algebraically hyperbolic when the Picard rank is at least 3, or ...
Added: April 21, 2016
Amerik E., Verbitsky M., / Series arXiv "math". 2016.
Let M be an irreducible holomorphic symplectic (hyperk¨ahler) manifold. If b2(M) >= 5, we construct a deformation M′ of M which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real (1, 1)-classes is hyperbolic. If b2(M) >= 14, similarly, we construct a deformation which ...
Added: April 13, 2016
Kurnosov N., / Series math "arxiv.org". 2015.
We prove that b2 is bounded for hyperk¨ahler manifolds with vanishing odd-Betti numbers. The explicit upper boundary is conjectured. Following the method described by Sawon we prove that b2 is bounded in dimension eight and ten in the case of vanishing odd-Betti numbers by 24 and 25 respectively. ...
Added: November 15, 2015
Amerik E., Verbitsky M., International Mathematics Research Notices 2015 Vol. 2015 No. 23 P. 13009–13045
Let M be an irreducible holomorphically symplectic manifold. We show that all faces of the Kähler cone of M are hyperplanes Hi orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kähler cone is a connected component of a complement of the positive cone to the union of all Hi. We ...
Added: October 28, 2015
Kurnosov N., / Series math "arxiv.org". 2015.
We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori. ...
Added: October 16, 2015
Kurnosov N., В кн.: V школа-конференция по алгебраической геометрии и комплексному анализу для молодых математиков России.: Математический институт им. В. А. Стеклова РАН, 2015. С. 57–58.
Подмногообразие называется абсолютно трианалитическим, если оно трианалитическое (комплексно-аналитическое относительно структур I, J, K) для любой индуцированной комплексной структуры, совместимой с
I. Доказано, что в обобщённом многообразии Куммера не содержится абсолютно трианалитических торов Z. ...
Added: October 16, 2015