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## Rational Curves and MBM Classes on Hyperkähler Manifolds: A Survey

P. 75-96.

Abasheva A., / Cornell University. 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

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., Advances in Mathematics 2016 Vol. 298 No. 6 August P. 473-483

We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori. ...

Added: June 2, 2016

Kurnosov N., Ясинский Е., / Cornell University. 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

Verbitsky M., Pure and Applied Mathematics Quarterly 2014 Vol. 10 No. 2 P. 325-354

Let S be a smooth rational curve on a complex manifold M. It is called ample if its normal bundle is positive: NS=⨁O(i_k),i_k<0. We assume that M is covered by smooth holomorphic deformations of S. The basic example of such a manifold is a twistor space of a hyperkähler or a 4–dimensional anti-selfdual Riemannian manifold ...

Added: January 23, 2015

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

Boston : Birkhäuser, 2013

This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the ...

Added: February 14, 2013

Amerik E., , in : Birational Geometry and Moduli Spaces. Vol. 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

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

Bogomolov F. A., McQuillan M., , in : Proceedings of the Simons Conference "Foliation theory in Algebraic Geometry". : Springer, 2015. P. 1-22.

This article represents a study of ample subbundles of the tangent sheaf of a variety in a formal neighbourhood of a curve. With the added hypothesis of integrability it is best possible. A particular corollary is Mori’s cone theorem for foliations by curves. ...

Added: November 24, 2015