Birational automorphism groups and the movable cone theorem for Calabi-Yau complete intersections of products of projective spaces

Abstract

For a Calabi-Yau manifold $X$, the Kawamata - Morrison movable cone conjecture connects the convex geometry of the movable cone $\overline{\mathrm{Mov}}(X)$ to the birational automorphism group. Using the theory of Coxeter groups, Cantat and Oguiso proved that the conjecture is true for general varieties of Wehler type, and they described explicitly $\mathrm{Bir}(X)$. We generalize their argument to prove the conjecture and describe $\mathrm{Bir}(X)$ for general complete intersections of ample divisors in arbitrary products of projective spaces. Then, under a certain condition, we give a description of the boundary of $\overline{\mathrm{Mov}}(X)$ and an application connected to the numerical dimension of divisors.

Figures (2)

• Figure 1: Example of the movable cone for $X$ given by the intersection of divisors of multidegree $(2,2,1)$ and $(2,1,2)$ in ${\mathbb{P}}^3\times {\mathbb{P}}^2 \times {\mathbb{P}}^2$. In this case $J = \{2,3\}$. On the left, the Movable cone obtained as the $\mathop{\mathrm{Bir}}\nolimits(X)$-orbit of $\mathop{\mathrm{Nef}}\nolimits(X)$ (in gray). Notice that the top right and top left thicker lines correspond to accumulations of chambers. On the right, the boundary of the Movable cone computed using Theorem \ref{['mainThm2']}. In blue, the cones corresponding to Theorem \ref{['mainThm2']} (1), and in red, the cones corresponding to Theorem \ref{['mainThm2']} (2).
• Figure 2: An example of the situation in Lemma \ref{['lemmaChambers']}.

Theorems & Definitions (61)

• Conjecture 1.1: Morrison93,Kawamata97
• Theorem 1.2: CantatOguiso15
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• proof
• Theorem 2.2: Kawamata88, Theorem 5.7
• Theorem 2.3: Kawamata08, Theorem 1
• Theorem 2.4: CantatOguiso15, Theorem 3.1
• proof