Trilinear forms and Chern classes of Calabi--Yau threefolds

Abstract

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.