The topology of Calabi-Yau threefolds with Picard number three

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.