Semicontinuity of Kodaira dimension

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with the convention K(X, L) = °° if L has no nontrivial sections for all i > 0. In the particular case when X is nonsingular and L = ^ is the canonical bundle, the invariant K(X) = K(X, £2) is called the canonical (or Kodaira) dimension of X and is the fundamental invariant in the classification of surfaces. Recent works by Ueno [4] and Iitaka [1], [2] have studied K(X, L) for higher dimensional varieties. A fundamental open question is the behavior of K(X, L) under deformations of (X, L). When X is a smooth surface the plurigenera (and hence the Kodaira dimension) are deformation invariant [1], and Iitaka has constructed a family of threefolds Xt with K(X0) = 0 and K(Xt) = ~ t* 0. Our main result is