  In this paper, we study boundedness questions for (simply-connected) smooth
Calabi-Yau threefolds. The diffeomorphism class of such a threefold is known to
be determined up to finitely many possibilities by the integral middle
cohomology and two integral forms on the integral second cohomology, namely the
cubic cup-product form and the linear form given by cup-product with the second
Chern class. The motivating question for this paper is whether knowledge of
these cubic and linear forms determines the threefold up to finitely many
families, that is the moduli of such threefolds is bounded. If this is true,
then in particular the middle integral cohomology would be bounded by knowledge
of these two forms on the second cohomology. Crucial to this question is the
study of rigid rational surfaces on the threefold, which are the irreducible
surfaces for which no multiple moves and which do not disappear under a general
deformation of the complex structure of the threefold. If for instance there
are no such surfaces, then the answer to the motivating question is yes
(Theorem 0.1). In particular, for given cubic and linear forms on the second
cohomology, there must exist such surfaces for large enough third Betti number
(Corollary 0.2). The paper starts by proving general results on these rigid
rational surfaces and boundedness of the family of threefolds. The basic
principle is that if the cohomology classes of these surfaces are also known,
then boundedness should hold (Theorem 4.5). The second half of the paper
restricts to the case of Picard number 2, where it is shown that knowledge of
the cubic and linear forms does indeed bound the family of Calabi-Yau
threefolds (Theorem 0.3). As far as the author is aware, this is the first
non-trivial case where a general boundedness result for Calabi-Yau threefolds
has been proved (without the assumption of a special structure).
