A criterion for toric varieties

We consider the pair of a smooth complex projective variety together with an anti-canonical simple normal crossing divisor (we call it "log Calabi- Yau"). Standard examples are toric varieties together with their toric boundaries (we call them "toric pairs"). We provide a numerical criterion for a general log Calabi-Yau to be toric by an inequality between its dimension, Picard number and the number of boundary components. The problem originates in birational geometry and our proof is constructive, motivated by mirror symmetry.