Trilinear forms and chern classes of calabi-yau threefolds

Let X be a Calabi-Yau threefold and μ the symmetric trilinear form on the second cohomology group H²(X, ℤ) defined by the cup product. We investigate the interplay between the Chern classes c₂(X), c₃(X) and the trilinear form μ, and demonstrate some numerical relations between them. When the cubic form μ (x, x, x) has a linear factor over ℝ, some properties of the linear form and the residual quadratic form are also obtained.