Volumes in Calabi-Yau Complete Intersection of Products of Projective Space

Abstract

We prove that the birational automorphism group of a general Calabi-yau complete intersection $X$ given by ample divisors in $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_l}$ is always Lorentzain. Applying the Kawamata-Morrison cone theorem on such $X$, we compute $\operatorname{vol}_X(D+sA)$ for any divisor $D\in \partial\overline{\operatorname{Eff}}(X)$ and ample divisor $A$ when $s$ is small. We also provide examples of volumes of certain Cartier divisors that involve the digamma function.