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´¡²ú²õ³Ù°ù²¹³¦³Ù:ÌýIn this talk, we study the base loci of nef and big line bundles on K3 surfaces and discuss their higher-dimensional analogues on hyperkähler manifolds.

We begin with classical results of Saint-Donat and Mayer on linear systems on K3 surfaces. In particular, we describe the movable and fixed components of nef and big line bundles that are not base point free. We also discuss the role of Reider’s theorem and its consequences for the base-point freeness of multiples of such line bundles.

We then turn to higher-dimensional hyperkähler manifolds. After reviewing the Beauville–Bogomolov–Fujiki quadratic form and the associated Riemann–Roch formula, we explain how positivity properties of line bundles can be expressed in terms of this quadratic form. Finally, we present results of Riess and others describing the fixed divisorial components of nef and big line bundles on hyperkähler manifolds. These results may be viewed as higher-dimensional analogues of the classical theory for K3 surfaces.