Publication

Motivic invariants of moduli spaces of rank 2 Bradlow-Higgs triples

Riccardo Grandi
2016
EPFL thesis
Abstract

In the present thesis we study the geometry of the moduli spaces of Bradlow-Higgs triples on a smooth projective curve C. There is a family of stability conditions for triples that depends on a positive real parameter σ. The moduli spaces of σ-semistable triples of rank r and degree d vary with σ. The phenomenon arising σ from this is known as wall-crossing. In the first half of the thesis we will examine how the moduli spaces and their universal additive invariants change as σ varies, for the case r = 2. In particular we will study the case of σ very close to 0, for which the moduli space relates to the moduli space of stable Higgs bundles, and σ very large, for which the moduli space is a relative Hilbert scheme of points for the family of spectral curves. Some of these results will be generalized to Bradlow-Higgs triples with poles. In the second half we will prove a formula relating the cohomology of the moduli spaces for small and odd degree and the perverse filtration on the cohomology of the moduli space of stable Higgs bundles. We will also partially generalize this result to the case of rank greater than 2.

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Moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space.
Projective variety
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .
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In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.
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