A Vanishing Result for Toric Varieties Associated with Root Systems

TL;DR

This paper proves a combinatorial vanishing result for higher cohomology groups of line bundles on toric varieties associated with root systems, linking algebraic geometry with Lie theory and combinatorics.

Contribution

It establishes a new combinatorial proof of cohomology vanishing for line bundles on root system toric varieties, connecting to Mazur's Inequality.

Findings

Higher cohomology groups vanish for certain line bundles on $V_R$

The proof uses combinatorial properties of root systems

Results relate to a converse of Mazur's Inequality

Abstract

Consider a root system $R$ and the corresponding toric variety $V_R$ whose fan is the Weyl fan and whose lattice of characters is given by the root lattice for $R$. We prove the vanishing of the higher cohomology groups for certain line bundles on $V_R$ by proving a purely combinatorial result for root systems. These results are related to a converse to Mazur's Inequality for (simply-connected) split reductive groups.