Math. Proc. Camb. Phil. Soc. (2000), 128, 79 Printed in the United Kingdom c  2000 Cambridge Philosophical Society 79 The Euler number of certain primitive Calabi–Yau threefolds By MEI-CHU CHANG Department of Mathematics, University of California, Riverside, U.S.A. e-mail : mcc@math.ucr.edu and HOIL KIM† Topology and Geometry Research Center, Kyungpook University, Taegu, Korea e-mail : hikim@gauss.kyungpook.ac.kr (Received 24 November 1998; revised 8 June 1999) 1. Introduction Recently Calabi–Yau threefolds have been studied intensively by physicists and mathematicians. They are used as physical models of superstring theory [Y] and they are one of the building blocks in the classification of complex threefolds [KMM]. These are three dimensional analogues of K3 surfaces. However, there is a funda- mental difference as is to be expected. For K3 surfaces, the moduli space N of K3 surfaces is irreducible of dimension 20, inside which a countable number of families N g with g  2 of algebraic K3 surfaces of dimension 19 lie as a dense subset. More explicitly, an element in N g is (S, H ), where S is a K3 surface and H is a primitive ample divisor on S with H 2 =2g − 2. For a generic (S, H ), Pic (S) is generated by H , so that the rank of the Picard group of S is 1. A generic surface S in N is not algebraic and it has Pic (S) = 0, but dim N = h 1 (S, T S ) = 20 [BPV]. It is quite an interesting problem whether or not the moduli space M of all Calabi–Yau threefolds is irreducible in some sense [R]. A Calabi–Yau threefold is algebraic if and only if it is Kaehler, while every non-algebraic K3 surface is still Kaehler. Inspired by the K3 case, we define M h,d to be {(X, H )|H 3 = h, c 2 (X) · H = d}, where H is a primitive ample divisor on a smooth Calabi–Yau threefold X. There are two pa- rameters h, d for algebraic Calabi–Yau threefolds, while there is only one parameter g for algebraic K3 surfaces. (Note that c 2 (S) = 24 for every K3 surface.) We know that N g is of dimension 19 for every g and is irreducible but we do not know the dimension of M h,d and whether or not M h,d is irreducible. In fact, the dimension of M h,d = h 1 (X, T X ), where (X, H ) ∈ M h,d . Furthermore, it is well known that χ(X)=2 (rank of Pic (X) − h 1 (X, T X )), where χ(X) is the topological Euler characteristic of X. Calabi–Yau threefolds with Picard rank one are primitive [G] and play an im- portant role in the moduli spaces of all Calabi–Yau threefolds. In this paper we give a bound on c 3 of Calabi–Yau threefolds with Picard rank 1. As a corollary we find a bound, as a function of h and d, on the dimension of M h,d , whose generic member X has Pic (X)= 〈H 〉, where H is very ample. Hence, we also find a bound on the Euler characteristic of these Calabi–Yau threefolds. For bounds of the Euler characteristic † This work is partially supported by TGRC-KOSEF.