Contraction images of toric varieties

TL;DR

This paper proves that proper morphisms from toric varieties to normal varieties preserve toric structure, ensuring the target is also toric and the morphism is toric under compatible structures.

Contribution

It establishes that any proper morphism from a toric variety to a normal variety with trivial pushforward of structure sheaf is necessarily toric, confirming the stability of toric structures under such morphisms.

Findings

Y is toric if X is toric and f is proper with f_*O_X = O_Y

f is a toric morphism for some toric structures on X and Y

Y inherits toric structure from X under the given conditions

Abstract

Let $f:X \to Y$ be a proper morphism of normal varieties with $f_*\mathcal{O}_X = \mathcal{O}_Y$. If $X$ is toric, then $Y$ is toric and $f$ is a toric morphism for some toric structures on $X$ and $Y$.