Families of K3 surfaces. 
J. Algebraic Geom. 7 (1998), no. 1, 183-193. 

Richard E. Borcherds
Mathematics department, Evans Hall, 3840, UC Berkeley, CA 94720-3840 
D.P.M.M.S., 16 Mill Lane, Cambridge CB2 1SB, UK 

Ludmil Katzarkov 
M.S.R.I., 1000 Centennial Drive, Berkeley, CA 94720 
Department of Mathematics, UC Irvine, Irvine, CA 92697-3875 

Tony Pantev
Department of Mathematics, MIT, Cambridge, MA 02139 

N. I. Shepherd-Barron
D.P.M.M.S., 16 Mill Lane, Cambridge CB2 1SB, UK 

Introduction. We will prove the following theorem and give some examples to 
show that most of the conditions in it are necessary. Recall that a family 
X -> B of varieties over a base space B is called isotrivial if there is an 
\'etale covering [B\tilde] -> B such that X\times_B[B\tilde] is a trivial 
family over [B\tilde]. 

Theorem 1.1. Any complete family of minimal K\"ahler surfaces of Kodaira 
dimension 0 and constant Picard number is isotrivial. 

This is a generalization to surfaces of the well known fact that any complete 
family of complex elliptic curves is isotrivial, because any complex elliptic 
curve is automatically minimal, K\"ahler, of Kodaira dimension 0, and has 
Picard number 1. Roughly speaking, it gives some cases when moduli spaces of 
surfaces contain no complete subvarieties.