In this paper we prove the following result. Main Theorem. lf u is a d-closed positive (k, k)-current on an open subset f2 of C", then for c>0 the set E c of points off2 where the Lelong number of u is > c is a subvariety of codimension >=k in fLThis result was conjectured by Harvey-King E6] and the following partial result was obtained by Skoda [15]" if O is Stein, then there exists a subvariety X c of codimension >k in f2 such that EccXccEk~. n The proof depends heavily on HSrmander's L 2 estimates of the c5 operator E8]. This relation between H~Srmander's result and the theory of closed positive currents was first used by Bombieri [t]. In this paper we will treat the case k = ! and the case of the general k separately. The proof of the case of the general k, so far as the basic ideas are concerned, is essentially the same as the case k = 1, but it is technically much more complicated and it necessitates the use of Federer's theory of slicing [4]. So, even though the case k = 1 follows from the case of a general k, in order to make the paper more easily understood and make it more accessible to people not familiar with Federer's theory, we treat the case k = 1 separately. The proof of the case of a general k depends, to a great extent, on the intermediate results in the proof of the case k = 1.In the last section of this paper we prove the following result on the extension of closed positive currents. Theorem 1. Suppose 0 is an open subset of ~", V is a subvariety off2 of codimension > k, and G is an open subset of (2 whose intersection with every branch of V of codimension k is nonempty and irreducible. If u is a d-closed positive (k, k)-current on (f2-V)wG, then u can be extended uniquely to a d-closed positive (k, k)-current on f2.

The seminar was made possible through the support of the Stiftung Volkswagenwerk.

A harmonic map f between two compact Kahler manifolds is shown to be either holomorphic or conjugate holomorphic under a suitable negativity condition on the curvature of the image manifold and a condition on the rank of df. As a consequence, a compact Kahler manifold of dimension >2 that is of the same homotopy type as a compact Kahler manifold with suitable negative curvature condition or as a compact quotient of an irreducible classical bounded symmetric domain must be either biholomorphic or conjugate biholomorphic to it.Mostow (1) proved the strong rigidity of locally symmetric Riemannian manifolds of nonpositive curvature not admitting any closed one-or two-dimensional geodesic submanifolds as locally direct factors. In particular, two compact quotients of the ball of complex dimension _2 with isomorphic fundamental groups are either biholomorphic or conjugate biholomorphic. S. T. Yau (personal communication) conjectured that this phenomenon of strong rigidity should hold also for compact Kahler manifolds of complex dimension _2 with negative sectional curvature. That is, any two such manifolds are biholomorphic or conjugate biholomorphic if they are of the same homotopy type. We confirm this conjecture if the curvature tensor is strongly negative according to the following definition.Definition: The curvature tensor Rj3a of a Kahler manifold is said to be strongly negative (respectively strongly seminegative) if at every point E R"/37s4X(T > 0 (respectively _ 0) for any nonzero matrix (t'/3) of the form And = A'tB4-The curvature tensor of the ball is strongly negative. So is the curvature tensor of the compact Kahler surface recently constructed by Mostow and Siu,* which has negative sectional curvature and whose universal covering is not biholomorphic to the ball. THEOREM 1. Suppose M and N are compact Kahler manifolds and the curvature tensor of M is strongly negative.

In this paper we give a proof of the following long conjectured result on the invariance of the plurigenera.Main Theorem. Let π : X → ∆ be a projective family of compact complex manifolds parametrized by the open unit 1-disk ∆. Assume that the family π : X → ∆ is of general type. Then for every positive integer m the plurigenus dim C Γ(X t , mK Xt ) is independent of t ∈ ∆, where X t = π −1 (t) and K Xt is the canonical line bundle of X t .Notations and Terminology. The canonical line bundle of a complex manifold Y is denoted by K Y . The coordinate of the open unit 1-disk ∆ is denoted by t. Let n be the complex dimension of each X t for t ∈ ∆. In the assumption of the Main Theorem the property of the family π : X → ∆ being of general type means that for every t ∈ ∆ there exist a positive integer m t and a point P t ∈ X t with the property that one can find elements s 0 , s 1 , . . . , s n+1 ∈ Γ(X, m t K X ) such that s 0 is nonzero at P t and s 1 s 0 , . . . , s n+1 s 0 form a local coordinate system of X at P t . By the family π : X → ∆ being projective we mean that there exists a positive holomorphic line bundle on X.Let K X,π be the line bundle on X whose restriction to X t is K Xt for each t ∈ ∆. Since the normal bundle of X t in X is trivial, the two line bundles K X and K X,π are naturally isomorphic. Under this natural isomorphism a local section s of K X,π corresponds to the local section s ∧ π * (dt) of K X . Unless there is some risk of confusion, in this paper we will, without any further explicit mention, always identify K X,π with K X by this natural isomorphism. Under this identification the Main Theorem is equivalent to the statement that for every t ∈ ∆ and every integer m every element of Γ(X t , mK Xt ) can be extended to an element of Γ(X, mK X ).The Hermitian metrics of holomorphic line bundles in this paper are allowed to have singularities and may not be smooth. For a Hermitian metric

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