Forschungsseminar Komplexe Analysis - Winter Semester 2014/2015

Motives and moduli of bundles

Wednesdays 4-6pm - Arnimallee 6, SR 025/026. ( 15th October 2014 - 11th February 2015 )

The main focus of our seminar is to understand various types of motives in algebraic geometry. The theory of motives is intended to be a universal cohomology theory for varieties. In the seminar, we will describe the conjectural expectations concerning mixed motives, study the category of pure motives and the construction of the triangulated category of mixed motives. We also plan to describe the relationship between these motives and virtual motives (that is, motivic classes in the Grothendieck ring of varieties).

A more detailed description of the talks including references can be found here.


Schedule 2014/2015

♠ 15.10 - 22.10.2014 // Algebraic cycles and Roitman's theorem - Alejandra Rincón We give some general results about algebraic cycles and Chow rings. We then describe a theorem of Roitman that, for a smooth complex projective variety, the Abel-Jacobi map is an isomorphism on torsion subgroups and give further details on this for the case of K3 surfaces.
♠ 29.10 - 05.11.2014 // An introduction to motives - Victoria Hoskins We define the notion of a (mixed) Weil cohomology theory and outline the main conjectures for mixed motives. In the remainder of the talk, we describe the construction and properties of the category of pure motives (these are the motives of smooth projective varieties).
♠ 05.11 - 12.11.2014 // Stacks and virtual motives - Nikolai Beck In this first part of this talk we give a brief introduction to the theory of stacks. Then we describe virtual motives of varieties (and certain stacks); that is, the motivic class in (the dimensional completion of) the Grothendieck ring of varieties.
♠ 12.11 - 19.11.2014 // The virtual motive of the stack of principal bundles - Anna Wißdorf We survey a paper of Behrend and Dhillon that gives a conjectural formula for the virtual motive of the stack of principal bundles on a smooth projective curve. We outline the ideas of the proof of this formula for i) SL_n-bundles and ii) principal bundles over the projective line.
♠ 26.11.2014 // Triangulated categories and Verdier localisation - Sina Rezazadeh In this talk we give an introduction to triangulated categoried and t-structures. We also describe Verdier localisation by a thick subcategory of a triangulated category and, as an example, describe the derived category of an abelian category.
♠ 03.12.2014 // Geometric motives via Verdier localisation - Joana Cirici We construct the triangulated category of effective geometric motives over k using Verdier localisation and enlarge this category, by inverting the tate twist, to produce a stablilised version, the triangulated category of geometric motives.
♠ 10.12.2014 // Properties of geometric motives - Victoria Hoskins We define the motivic cohomology groups and state some important properties of geometric motives (many of the proofs use a larger triangulated category of motives, which we will construct in a future talk). We also relate geometric motives with virtual motives and chow motives.
♠ 17.12.2014 // A review of pure, geometric and virtual motives - Alejandra Rincón, Anna Wißdorf and Ángel Muñoz Castañeda In these week's seminar, we will review the key ideas and constructions that we have discussed so far.
♠ 07.01.2015 // Larger categories of motives - Joana Cirici We will introduce Nisnevich sheaves with transfers and from this construct a larger category of motives DM, which is needed for many of the proofs of the fundamental properties of the triangulated category of effective geometric motives.
♠ 14.01.2015 // Mixed Tate motives - Simon Pepin Lehalleur We introduce the category DMT(k) of mixed Tate motives, a subcategory of DM(k) whose structure is much easier to study. We discuss examples of varieties whose motives are mixed Tate, and explain how for some fields k we can actually construct an abelian category of mixed Tate motives MT(k).
♠ 21.01.2015 // Construction of moduli spaces of slope semistable sheaves - Alejandra Rincón We describe the construction of compact 'moduli' spaces of slope semistable sheaves on higher-dimensional complex manifolds due to Greb and Toma, which uses a modified notion of slope semistability. This approach generalises the algebro-geometric construction of the Donaldson-Uhlenbeck compactification on surfaces given by J. Li and J. Le Potier.
♠ 28.01.2015 // Geometry of moduli spaces of slope semistable sheaves - Ángel Muñoz Castañeda We give a brief summary of the construction of the moduli space and its universal properties. We study its geometry via the separation properties of the 'quotient' map and show that it can be seen as a natural compactification of the moduli space of reflexive sheaves. Finally, we compare this moduli space with the classical (Gieseker-Maruyama) moduli space of sheaves.
♠ 04.02.2015 // Wall-crossing problems - Ángel Muñoz Castañeda This talk will be focused on the wall crossing problem for slope semistable sheaves. The problem is to describe the slope semistability when the polarization of the base manifold varies along the ample cone. Some examples will be described showing some strange phenomena and we will show how to explain these phenomena with our new moduli space.
♠ 11.02.2015 // On stability of syzygy bundles on projective spaces - Vincent Trageser We will consider cohomological stability and slope-stability of certain syzygy bundles on projective spaces, which are kernels of surjective maps between sums of line bundles.

Schedule 2014/2015

♠ 15.10 - 22.10.2014 // Algebraic cycles and Roitman's theorem - Alejandra Rincón
We give some general results about algebraic cycles and Chow rings. We then describe a theorem of Roitman that, for a smooth complex projective variety, the Abel-Jacobi map is an isomorphism on torsion subgroups and give further details on this for the case of K3 surfaces.

♠ 29.10 - 05.11.2014 // An introduction to motives - Victoria Hoskins
We define the notion of a (mixed) Weil cohomology theory and outline the main conjectures for mixed motives. In the remainder of the talk, we describe the construction and properties of the category of pure motives (these are the motives of smooth projective varieties).

♠ 05.11 - 12.11.2014 // Stacks and virtual motives - Nikolai Beck
In this first part of this talk we give a brief introduction to the theory of stacks. Then we describe virtual motives of varieties (and certain stacks); that is, the motivic class in (the dimensional completion of) the Grothendieck ring of varieties.

♠ 12.11 - 19.11.2014 // The virtual motive of the stack of principal bundles - Anna Wißdorf
We survey a paper of Behrend and Dhillon that gives a conjectural formula for the virtual motive of the stack of principal bundles on a smooth projective curve. We outline the ideas of the proof of this formula for i) SL_n-bundles and ii) principal bundles over the projective line.

♠ 26.11.2014 // Triangulated categories and Verdier localisation - Sina Rezazadeh
In this talk we give an introduction to triangulated categoried and t-structures. We also describe Verdier localisation by a thick subcategory of a triangulated category and, as an example, describe the derived category of an abelian category.

♠ 03.12.2014 // Geometric motives via Verdier localisation - Joana Cirici
We construct the triangulated category of effective geometric motives over k using Verdier localisation and enlarge this category, by inverting the tate twist, to produce a stablilised version, the triangulated category of geometric motives.

♠ 10.12.2014 // Properties of geometric motives - Victoria Hoskins
We define the motivic cohomology groups and state some important properties of geometric motives (many of the proofs use a larger triangulated category of motives, which we will construct in a future talk). We also relate geometric motives with virtual motives and chow motives.

♠ 17.12.2014 // A review of pure, geometric and virtual motives - Alejandra Rincón, Anna Wißdorf and Ángel Muñoz Castañeda
In these week's seminar, we will review the key ideas and constructions that we have discussed so far.

♠ 07.01.2015 // Larger categories of motives - Joana Cirici
We will introduce Nisnevich sheaves with transfers and from this construct a larger category of motives DM, which is needed for many of the proofs of the fundamental properties of the triangulated category of effective geometric motives.

♠ 14.01.2015 // Mixed Tate motives - Simon Pepin Lehalleur
We introduce the category DMT(k) of mixed Tate motives, a subcategory of DM(k) whose structure is much easier to study. We discuss examples of varieties whose motives are mixed Tate, and explain how for some fields k we can actually construct an abelian category of mixed Tate motives MT(k).

♠ 21.01.2015 // Construction of moduli spaces of slope semistable sheaves - Alejandra Rincón
We describe the construction of compact 'moduli' spaces of slope semistable sheaves on higher-dimensional complex manifolds due to Greb and Toma, which uses a modified notion of slope semistability. This approach generalises the algebro-geometric construction of the Donaldson-Uhlenbeck compactification on surfaces given by J. Li and J. Le Potier.

♠ 28.01.2015 // Geometry of moduli spaces of slope semistable sheaves - Ángel Muñoz Castañeda
We give a brief summary of the construction of the moduli space and its universal properties. We study its geometry via the separation properties of the 'quotient' map and show that it can be seen as a natural compactification of the moduli space of reflexive sheaves. Finally, we compare this moduli space with the classical (Gieseker-Maruyama) moduli space of sheaves.

♠ 04.02.2015 // Wall-crossing problems - Ángel Muñoz Castañeda
This talk will be focused on the wall crossing problem for slope semistable sheaves. The problem is to describe the slope semistability when the polarization of the base manifold varies along the ample cone. Some examples will be described showing some strange phenomena and we will show how to explain these phenomena with our new moduli space.

♠ 11.02.2015 // On stability of syzygy bundles on projective spaces - Vincent Trageser
We will consider cohomological stability and slope-stability of certain syzygy bundles on projective spaces, which are kernels of surjective maps between sums of line bundles.