We ask about the simply connected compact smooth 6-manifolds which can
support structures of Calabi-Yau threefolds. In particular, we study the
interesting case of Calabi-Yau threefolds $X$ with second betti number 3. We
have a cup-product cubic form on the second integral cohomology, a linear form
given by the second Chern class, and the integral middle cohomology, and if the
homology is torsion free this information determines precisely the
diffeomorphism class of the underlying 6-manifold by a result of Wall. For
simplicity, we assume that the cubic form defines a smooth real elliptic curve
whose Hessian is also smooth. Under a further relatively mild assumption that
there are no non-movable surfaces $E$ on $X$ with $1 \le E^3 \le 9$, we prove
that the real elliptic curve must have two connected components rather than
one, and that the K\"ahler cone is contained in the open positive cone on the
bounded component; we show moreover that the second Chern class is also
positive on this open cone. Using Wall's result, for any given third Betti
number we therefore have an abundance of examples of smooth 6-manifolds which
support no Calabi-Yau structures, both in the cases when the cubic defines a
real elliptic curve with one or two connected components.