Rational curves and MBM classes on hyperkahler manifolds: a survey

TL;DR

This survey reviews recent advances in understanding rational curves, extremal rays, and the Kähler cone structure on hyperkähler manifolds, highlighting their role in birational geometry.

Contribution

It compiles and discusses recent results on minimal rational curves, their deformations, and the geometry of the Kähler cone in hyperkähler manifolds.

Findings

Characterization of minimal rational curves on hyperkähler manifolds

Description of extremal rays associated with rational curves

Insights into the structure of the Kähler cone

Abstract

This paper deals with rational curves and birational contractions on irreducible holomorphically symplectic manifold. We survey some recent results about minimal rational curves, their deformations, extremal rays associated with these curves, and the geometry of the K\"ahler cone.