SEMICONTINUITY OF KODAIRA DIMENSION

Let X be a compact analytic space (or a complete algebraic variety) and let L be a line bundle on X and denote by ft: X — • P ^ the rational map de-fined by the global sections of L®'. The L-dimension of X, K(X> L) is de-fined by K(X, L) = I ta(d im ( ƒ,(*)) I-»oo 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 canoni-cal 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 sur-faces. 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 THEOREM. Given X0 a compact analytic space (or complete algebraic variety) and L0a line bundle on X0 satisfying (1) L®1 is spanned by its global sections for some i> 0, (2) K(X0, L0) = dim(X0), and (Xt, Lt) is any (flat) deformation of(X0, L0), then K(Xt, Lt) = K(X0i L0). When X0 is a smooth surface and L0 = ^ 0 it was shown by Mumford [3] that hypothesis (1) on L0 is implied by (2). For general L0 hypothesis AMS (MOS) subject classifications (1970). Primary 14D15, 32G05. Key words and phrases. Kodaira dimension, semicontinuity, deformation. lSloan Foundation Fellow, partially supported by NSF GP-28323A3