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A rigid cohomology class on a complex manifold is a class that is representedby a unique closed positive current. The positive current representing a rigidclass is also called rigid. For a compact Kahler manifold $X$ all eigenvectorsof hyperbolic automorphisms acting on $H^{1,1}(X)$ that have non-uniteigenvalues are rigid classes. Such classes are always parabolic, namely, theybelong to the boundary of the Kahler cone and have vanishing volume. We studyparabolic $(1,1)$-classes on compact hyperkahler manifolds with $b_2 \geq 7$.We show that a parabolic class is rigid if it is not orthogonal to a rationalvector with respect to the BBF form. This implies that a general parabolicclass on a hyperkahler manifold is rigid.

Rigid currents on compact hyperkahler manifolds