As a PhD student at the Graduiertenkolleg "Cohomological Methods in Geometry", I was working on the application of homotopy theory to algebraic geometry. My advisor is Matthias Wendt. The project started in July 2012, took several different directions, settled around March 2015 and the thesis was submitted March 2016. The thesis defense was in June 2016. It is published electronically in the institutional archive FreiDok:
More concretely, the project is about motivic cell structures of certain spherical varieties. This means I'm studying ways to find cell structures in the stable -homotopy theory of schemes of certain schemes built out of affine algebraic groups.
The easiest examples of spherical varieties are smooth projective homogeneous spaces, where one can easily get a motivic cell structure by means of the Bialynicki-Birula filtration coming from an action of an algebraic torus, given by the group acting on the homogeneous space.
The next step are certain affine homogeneous spaces who have a wonderful completion by a single extra orbit. These were classified by Akhiezer and the completion turns out to be homogeneous (under a different group). This is already a list of quite interesting spaces, among them affine quadrics (for which a cell structure was recently found by Asok, Doran and Fasel: they are motivic spheres), quaternionic projective spaces (which are also rightly called symplectic Grassmannians) and the Cayley plane (the projective plane over the octonions). These projective spaces are related to elements in the (motivic) stable homotopy groups of spheres of Hopf invariant one.
In general, one can look at the wonderful completion of a group of adjoint type and analyse the cell structure of the completion with respect to the embedding. A localization sequence argument immediately gives you the motive (as was recently done by Karpenko and Merkurjev).
Another class of easy spherical varieties are toric varieties. Here one can also give motivic cell structures, and the Bialynicki-Birula filtration induced by a suitable cocharacter of the torus gives a smaller (but less symmetric) presentation than the usual one using all possible cells.
To understand the homotopy type of homogeneous spaces is important for the homotopy theory of the groups which act on them. For example, it is still not entirely known (or proven) how the rational motivic homotopy type of a (isotropic) reductive group looks like. Neither is the multiplicative structure on the motive understood. Not just the reductive groups themselves, also their topological and their algebraic loop spaces are interesting (for representation theory, arithmetic geometry and ultimately mathematical physics). To understand these, certain homogeneous spaces also help tremendously, if one looks at the world of compact Lie groups.
Parts of the thesis will be rewritten, improved and posted as individual preprints on the arXiv. You may contact me if you want to glimpse on a preliminary version of this.
This is a list of blog posts I wrote that have some (minor) connection to my PhD project, from old to new:
- Notes to Morel-Voevodsky's A¹-homotopy theory of schemes
- Thom Spaces
- What is ... a reductive group?
- Homotopy limits
- Model categories
- Cellular objects: CW complexes
- Cellular objects in the motivic model category
- Model structures on simplicial presheaves
- Motivic decomposition of projective space via a motivic cell structure
- Bialynicki-Birula decomposition of nonsingular complete varieties with a torus action
- Motivic Cell Structure of Toric Surfaces
- Classification of Division Algebras