  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 $X$
is simply connected with torsion free homology 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 irreducible. Under a further relatively mild
assumption that there are no non-movable surfaces $E$ on $X$ with $1 \le E^3
\le 8$, 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 compact
oriented 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. Moreover, except possibly if $c_2$ vanishes at a real inflexion
point of the elliptic curve, even when Calabi-Yau structures do occur under the
above conditions, there will be only a bounded family of them which are not
birationally elliptic.
