Mixed Hodge Structures of cluster varieties of dimension 3

TL;DR

This paper computes the mixed Hodge numbers of smooth 3-dimensional cluster varieties, demonstrating they are of mixed Tate type, and explores the Hodge structures of cohomology in some singular cases.

Contribution

It provides explicit calculations of mixed Hodge numbers for 3D cluster varieties and analyzes their Hodge structures, including singular cases, advancing understanding of their geometric properties.

Findings

Mixed Hodge numbers are explicitly calculated for smooth 3D cluster varieties.

These varieties are shown to be of mixed Tate type.

Hodge structures of cohomology and intersection cohomology are studied for singular cases.

Abstract

We calculate the mixed Hodge numbers of smooth 3-dimensional cluster varieties and show that they are of mixed Tate type. We also study the mixed Hodge structures of the cohomology and intersection cohomology groups of some singular cluster varieties.