Math Ann 225, 77—88 (1977) © by Springer-Verlag 1977

Semicontinuity of L-Dimension

David Lieberman^ and Edoardo Sernesi^*

^ Department of Mathematics, Brandeis University, Waltham, Mass 02154 USA ^ Istituto Matematico Universita degh Studi, 1-44100 Ferrara Itaha

§ 1. Introduction

Let X be a compact complex space and L and invertible sheaf The notion of L-dimension of X, denoted /c(X, L), was introduced by Iitaka m [4J (for the definition see § 3) When X is a compact manifold and L-^œ^ the canonical mvertible sheaf, к (X, co^) is the canonical dimension of X, or the Kodaira dimension of X, sometimes denoted /cdim(X), it is the fundamental invariant in the Enriques- Kodaira classification of surfaces

An important open question is the behaviour of Kodaira dimension under deformation Not much is known about it One knows that the canonical sion IS a deformation invariant for curves and surfaces In the former case the result IS trivial, m the latter it has been proved by Iitaka [3] using the classification of surfaces It is also known that for higher dimensional manifolds the canonical dimension is not a deformation invariant Nakamura has produced an example of a family of threefolds {ïj^^j over a disc Л such that /cdim(3Eo) = 0 and /cdlm(Зe,)=-oolftфO(cf [8])

We investigate the behaviour of L-dimension under deformation Our mam results are the following

Theorem . Let £ be an mvertible sheaf on a complex space X and X —^ S a proper and flat morphism onto an irreducible complex space S There is a constant к and a set W^S, which is the complement of the union of a countable number of proper closed subvarieties, such that

/ c ( 3E „ £ , ) = /c if seW, /с(3е„£,)>1с if 5gS\^

Theorem . Let L be an mvertible sheaf on a compact, reduced, irreducible complex space X such that the following condition is satisfied for some n>Q

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