Trilinear forms and Chern classes of Calabi--Yau threefolds

Let X be a Calabi--Yau threefold and \mu the symmetric trilinear form on the second cohomology group H^{2}(X,\mathbb{Z}) defined by the cup product. We investigate the interplay between the Chern classes c_{2}(X), c_{3}(X) and the trilinear form \mu, and demonstrate some numerical relations between them. When the cubic form \mu(x,x,x) has a linear factor over \mathbb{R}, some properties of the linear form and the residual quadratic form are also obtained.