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 fundamental
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
Ng with g [ges ] 2 of algebraic K3 surfaces
of dimension 19 lie as a dense subset. More
explicitly, an element in Ng is (S, H),
where S is a K3 surface and H is a primitive
ample divisor on S with H2 = 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 = h1(S, TS) = 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 Mh,d to be
{(X, H)[mid ]H3
= h, c2(X) · H = d}, where H is
a primitive ample divisor on a smooth Calabi–Yau threefold X. There are two parameters
h, d for algebraic Calabi–Yau threefolds, while there is only one parameter
g for algebraic K3 surfaces. (Note that c2(S) = 24 for every K3 surface.) We know
that Ng is of dimension 19 for every g and is irreducible but we do not know the
dimension of Mh,d and whether or not
Mh,d is irreducible. In fact, the dimension of
Mh,d = h1(X,
TX), where
(X, H) ∈ Mh,d. Furthermore,
it is well known that χ(X) = 2 (rank of
Pic (X) − h1(X,
TX)), where χ(X) is the topological Euler
characteristic of X. Calabi–Yau threefolds with Picard rank one are primitive
[G] and play an important role in the moduli spaces
of all Calabi–Yau threefolds. In this paper we give
a bound on c3 of Calabi–Yau threefolds with Picard rank 1.
On the second Gaussian map for curves on a K3 surface