Rigid currents on compact hyperkahler manifolds

Abstract

A rigid cohomology class on a complex manifold is a class that is represented by a unique closed positive current. The positive current representing a rigid class is also called rigid. For a compact Kahler manifold $X$ all eigenvectors of hyperbolic automorphisms acting on $H^{1,1}(X)$ that have non-unit eigenvalues are rigid classes. Such classes are always parabolic, namely, they belong to the boundary of the Kahler cone and have vanishing volume. We study parabolic $(1,1)$-classes on compact hyperkahler manifolds with $b_2\ge 7$. We show that a parabolic class is rigid if it is not orthogonal to a rational vector with respect to the BBF form. This implies that a general parabolic class on a hyperkahler manifold is rigid.

42 pages, v. 3.1, minor corrections

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