Non-reductive geometric invariant theory and its applications to Higgs bundles

Thesis

Geometric Invariant Theory (GIT) is a powerful theory for constructing and studying the geometry of moduli spaces in algebraic geometry, as quotients of a parameter space by a linear group action. However GIT only applies to classification problems involving reductive group actions, and only produces a moduli space for the subclass of "stable" objects. Non-Reductive GIT was developed over the last ten years to overcome these limitations.

The aim of this thesis is firstly to describe a systematic way of constructing and studying the geometry of quotients using both classical and Non-Reductive GIT, and secondly to show how these methods can be applied to the classification problem for Higgs bundles.

In the first part we describe an algorithm which, given the linear action of a linear algebraic group with so-called "internally graded unipotent radical" on a projective variety, uses classical and Non-Reductive GIT to produce a non-empty open subset of the variety admitting a quasi-projective geometric quotient, together with an explicit projective completion of this quotient. Assuming the initial variety is smooth, we describe a procedure for calculating the Poincaré series of the resulting projective completion, which first requires showing that it can have at worst finite quotient singularities. We then obtain as a result a formula which can be viewed as the non-reductive analogue of the formula for the Poincaré series of the partial desingularisation of classical GIT quotients.

In the second part we show how Non-Reductive GIT can be used to refine two instability stratifications of the stack of Higgs bundles: the Higgs Harder-Narasimhan and the Harder-Narasimhan stratifications. These refined stratifications satisfy the property that each stratum admits a quasi-projective coarse moduli space with an explicit projective completion. In the rank 2 case we provide a complete moduli-theoretic interpretation of these refined stratifications and study the geometry of the corresponding moduli spaces.

Authors: Hamilton, E

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Moduli spaces; Non-Reductive Geometric Invariant Theory; Quotients by group actions; Cohomology of quotients; Higgs bundles
• Subjects: Algebraic Geometry