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Mixed Hodge structures on character varieties of nilpotent groups
Carlos Florentino1, Sean Lawton2, Jaime Silva3
1Departamento de Matemática, Faculdade de Ciências, Univ. de Lisboa, Edf. C6, Campo Grande, 1749-016 Lisbon, Portugal.
Revista Matematica Complutense
|February 6, 2025
Summary
This study determines the mixed Hodge structure for representation and character varieties of nilpotent groups. Explicit formulas for mixed Hodge polynomials are derived for these complex algebraic structures.
Area of Science:
- Algebraic Geometry
- Representation Theory
- Hodge Theory
Background:
- The study focuses on the variety of representations of finitely generated nilpotent groups into complex reductive algebraic groups.
- Understanding the geometric and topological properties of these representation varieties is crucial in algebraic and geometric contexts.
Purpose of the Study:
- To determine the mixed Hodge structure on the representation variety and the character variety.
- To derive explicit closed and recursive formulas for the mixed Hodge polynomial of these varieties.
Main Methods:
- Utilizing concepts from algebraic geometry and representation theory.
- Applying techniques from Hodge theory to analyze the structure of the varieties.
Main Results:
- The mixed Hodge structure on the representation variety is explicitly determined.
- The mixed Hodge structure on the character variety is also determined.
- Explicit closed and recursive formulas for the mixed Hodge polynomial are obtained.
Conclusions:
- The research provides a comprehensive understanding of the mixed Hodge structure of representation and character varieties for nilpotent groups.
- The derived formulas offer powerful tools for further computations and theoretical investigations in the field.
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