Bounding toric singularities with normalized volume

Abstract

We study the normalized volume of toric singularities. As it turns out, there is a close relation to the notion of (non-symmetric) Mahler volume from convex geometry. This observation allows us to use standard tools from convex geometry, such as the Blaschke-Santal\'o inequality and Radon's theorem to prove non-trivial facts about the normalized volume in the toric setting. For example, we prove that for every $\epsilon > 0$ there are only finitely many $\mathbb{Q}$-Gorenstein toric singularities with normalized volume at least $\epsilon$. From this result, it directly follows that there are also only finitely many toric Sasaki-Einstein manifolds of volume at least $\epsilon$ in each dimension. Additionally, we show that the normalized volume of every toric singularity is bounded from above by that of the rational double point of the same dimension. Finally, we discuss certain bounds of the normalized volume in terms of topological invariants of resolutions of the singularity. We establish two upper bounds in terms of the Euler characteristic and of the first Chern class, respectively. We show that a lower bound, which was conjectured earlier by He, Seong, and Yau, is closely related to the non-symmetric Mahler conjecture in convex geometry.