Project description

Moduli spaces play an important role in the classification of algebro geometric structures. Understanding the structure of a moduli space is an important step in understanding the underlying classification problem. There are two important techniques for investigating invariants attached to moduli spaces of bundles on algebraic varieties. Harder-Narasimhan stratifications: these enable computations of certain invariants, such as virtual motives, by an inductive procedure. Birational variations of moduli spaces: moduli spaces corresponding to different stability parameters are typically related by chains of geometric invariant theory flips. These modifications are reflected by wall-crossing formulae for the invariants under consideration. Harder-Narasimhan stratifications can be easily understood in terms of virtual motives in the Grothendieck ring of algebraic varieties. In this project, we seek to use stratifications to describe motives of moduli spaces in Voevodsky's triangulated category DM. Flips in geometric invariant theory are basic algebro geometric surgery operations. The second aim of this project is understanding them in the framework of algebraic cobordism.

Related publications

A. Schmitt, A remark on relative geometric invariant theory for quasi-projective varieties. Math. Nachr. 292 (2019), no. 2, 428–435.

Ángel Luis Muñoz Castañeda, A. Schmitt,, Singular principal bundles on reducible nodal curves. arxiv1911.01578 To appear: Trans. Amer. Math. Soc.

A.Schmitt, A general notion of coherent systems, hal-02391836

V. Hoskins, Victoria, S. Pepin Lehalleur, A formula for the Voevodsky motive of the moduli stack of vector bundles on a curve. Geom. Topol. 25 (2021), no. 7, 3555–3589.

V. Hoskins, Victoria, S. Pepin Lehalleur, On the Voevodsky motive of the moduli stack of vector bundles on a curve. Q. J. Math. 72 (2021), no. 1-2, 71–114.