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Absolutely trianalytic tori in the generalized Kummer variety
Advances in Mathematics. 2016. Vol. 298. No. 6 August . P. 473–483.
Kurnosov N.
We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori.
Amerik E., Verbitsky M., Journal de Mathématiques Pures et Appliquées 2023 Vol. 179 P. 232–252
A parabolic automorphism of a hyperkähler manifold M is a holomorphic automorphism acting on by a non-semisimple quasi-unipotent linear map. We prove that a parabolic automorphism which preserves a Lagrangianfibration acts on almost all fibers ergodically. The existence of an invariant Lagrangian fibration is automatic for manifolds satisfying the hyperkähler SYZ conjecture; this includes all known examples of hyperkähler manifolds. When there are two parabolic automorphisms preservingtwo distinct ...
Added: November 7, 2023
Amerik E., Verbitsky M., , in: Rationality of Varieties.: Birkhäuser, 2021. P. 75–96.
Added: April 6, 2022
Amerik E., Verbitsky M., Selecta Mathematica, New Series 2021 Vol. 27 Article 60
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. Cohomology classes which can possibly be orthogonal to a wall of the Kähler cone ...
Added: October 10, 2021
Amerik E., , in: Birational Geometry and Moduli SpacesVol. 39.: Springer, 2020. Ch. 1 P. 1–11.
We survey some results about rational curves on hyperkähler manifolds, explaining how to prove a certain deformation-invariance statement for loci covered by rational curves with negative Beauville–Bogomolov square. ...
Added: August 14, 2020
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
Kurnosov N., Soldatenkov A., Verbitsky M., Advances in Mathematics 2019 Vol. 351 P. 275–295
Let M be a simple hyperkähler manifold. Kuga-Satake
construction gives an embedding of H^2(M, C) into the
second cohomology of a torus, compatible with the Hodge
structure. We construct a torus T and an embedding of the
graded cohomology space H^•(M, C) → H^{•+l}(T, C) for some
l, which is compatible with the Hodge structures and the
Poincaré pairing. Moreover, this ...
Added: June 3, 2019
Kurnosov N., Ясинский Е., / Series arXiv "math". 2018.
We study groups of bimeromorphic and biholomorphic automorphisms of projective hyperkähler manifolds. Using an action of these groups on some non-positively curved space, we deduce many of their properties, including finite presentation, strong form of Tits' alternative and some structural results about groups consisting of transformations with infinite order. ...
Added: December 2, 2018