Seminar on further topics in GIT, Summer Semester 2016

Summer Semester 2016 (Erganzungsmodul: Forshungsseminar, Forschungsmodul: Algebra)
Tuesdays 18:00 - 19:30, SR 119 Arnimallee 3.

Organised by Dominic Bunnett and Victoria Hoskins.

In this seminar, we will build on the foundations studied in the Algebra III lectures on moduli problems and geometric invariant theory. We will study links with toric geometry and symplectic geometry, and variation of GIT, which studies how the quotient depends on the choice of linearisation. The main goals of this seminar are as follows.

To explore the relationship between toric geometry and GIT. More precisely, we will explain how any projective normal toric variety can be constructed as a GIT quotient of an affine space, where the action is linearised by a character.
To study how the GIT quotient depends on the choice of linearisation of the action, known as variation of GIT. There is a wall and chamber structure on the space of linearisations of a reductive group action such that the quotient only changes (often by an explicit birational transformation) as one crosses a wall.
To give an introduction to symplectic geometry and symplectic quotients and prove the Kempf-Ness Theorem which gives a homeomorphism between a projective GIT quotient over the complex numbers and a symplectic quotient.

A more detailed description of the talks is available here. For questions about the seminar, or if you would like to give a talk, please contact Dominic Bunnett ().


Schedule: Summer Semester 2016

♠ 19.04.2016 // An overview of projective normal toric varieties - Dominic Bunnett We start by giving the abstract definition of a toric variety, then explain how to construct affine toric varieties from cones and normal toric varieties from fans. We will then focus on projective normal toric varieties and their toric divisors and finally give an explicit description of their divisor class group. Notes for the first and second talk can be found here .
♠ 26.04.2016 // Toric geometric invariant theory - Dominic Bunnett We will prove that every projective normal toric variety is constructed as a GIT quotient of a diagonalisable group acting on an affine space which is linearised by a character. In the first part of this talk, we explain how to construct GIT quotients of affine spaces with respect to a character and, in the second part of the talk, we give the proof of the above result. Notes for the first and second talk can be found here .
♠ 03.05.2016 // An introduction to symplectic geometry - Maik Pickl We introduce symplectic manifolds and some examples (symplectic vector spaces, the cotangent bundle of a manifold and complex projective spaces). We introduce two types of morphisms in symplectic geometry: symplectomorphisms (which are diffeomorphisms which respect the symplectic form) and Lagrangian correspondences (which are defined using Lagrangian submanifolds in the product). Notes for the third talk can be found here .
♠ 10.05.2016 // Group quotients in symplectic geometry - Vicky Hoskins We consider the construction of symplectic quotients of actions of compact groups in symplectic geometry; this is known as symplectic reduction and is performed by taking a topological quotient of a level set of a moment map for the action. We prove the Marsden-Weinstein-Meyer Theorem, which states that if the group acts freely on this level set, then the reduction is a symplectic manifold. Notes for the fourth talk can be found here .
♠ 17.05.2016 // The Kempf-Ness theorem - Vicky Hoskins We prove the Kempf-Ness theorem which states that the GIT quotient of a reductive group acting linearly on a smooth projective variety X over the complex numbers is homeomorphic to the symplectic reduction of the action of the maximal compact subgroup. The main step is to prove that an orbit meets the zero level set of the moment map if and only if the orbit is GIT polystable. Notes for the fifth talk can be found here .
♠ 24.05.2016 // The Bialynicki-Birula decomposition - Anna-Lena Winz Given an action of the multiplicative group on a smooth projective variety X, we prove that there are two decompositions of X into locally closed smooth subvarieties which are both indexed by the connected components of the fixed locus for the action. We explicitly describe the strata as locally trivial affine fibrations over the fixed loci.
♠ 31.05.2016 - 07.06.2016 // Variation of GIT for the multiplicative group - Koki Yoshimoto For a linear action of the multiplicative group on an affine variety, we describe the variation of GIT quotients as we twist by a character of the multiplicative group. There are three cases to consider: the trivial character, a positive character and a negative character. We use the Bialynicki-Birula decomposition to describe the morphisms from the positive (resp. negative) GIT quotient to the zero GIT quotient.
♠ 07.06.2016 -14.06.2016 // Toric quotients and flips - Marta Pieropan We study variation of GIT for the action of a subtorus of the torus for a toric variety associated to a polyhedron. The space of linearisations giving non-empty GIT quotients is closed related to the polyhedron defining this toric variety and on it we describe a polyhedral wall and chamber decomposition, and the birational transformations given by wall-crossings.
♠ 21.06.2016 // The space of linearisations - Markus Penner In this talk, for a reductive group G acting on a quasi-projective variety X, we define the G-Neron Severi group of X to be the group of G-linearisations on X up to G-algebraic equivalence. We define the G-effective cone inside the G-Neron Severi group and prove that there is a wall and chamber structure with finitely many chambers such that all linearisations in a given chamber give the same GIT quotient.
♠ 28.06.2016 // Wall-crossings in variation of GIT - Ignacio Barros We continue to study how the projective quotient depends on the G-linearised line bundle. In particular, we describe what happens when we move from one chamber to another. If time permits, we describe the whole construction on a concrete example and describe the VGIT wall-crossings.
♠ 5.07.2016 // The (Zariski local) description of the morphism to the wall - Dominic Bunnett We generalise the results discussed for actions of the multiplicative group in talk seven, under a certain smoothness hypothesis, to general reductive group actions. Suppose that G is a reductive group acting on a quasi-projective variety X over k, which is projective over an affine variety. We show that the maps the GIT quotient maps to the wall are locally trivial weighted projective space fibrations. If time permits, we describe local criteria which enable us to strengthen our results further.
♠ 12.07.2016 // The (étale local) description of the morphism to the wall - Marysia Donten-Bury For a reductive group G acting on a quasi-projective variety X, we show that under a weakened hypothesis, the morphism from the GIT quotient in a chamber to a wall is an etale locally trivial weighted projective space fibration. We will introduce Luna’s etale slice theorem, a key technical tool. We give a counter example to see why the hypothesis is necessary.
♠ 19.07.2016 // Mori dream spaces and GIT - Marta Pieropan We define an equivalence relation on the Neron-Severi group associated to a variety and define Mori chambers (as the closure of equivalence classes). Mori dream spaces are varieties with a 'good' Mori chamber decomposition. We then show that any Mori Dream space is a GIT quotient of an affine variety, generalising the results of the first talks.

Schedule: Summer Semester 2016

♠ 19.04.2016 // An overview of projective normal toric varieties - Dominic Bunnett
We start by giving the abstract definition of a toric variety, then explain how to construct affine toric varieties from cones and normal toric varieties from fans. We will then focus on projective normal toric varieties and their toric divisors and finally give an explicit description of their divisor class group.
Notes for the first and second talk can be found here .

♠ 26.04.2016 // Toric geometric invariant theory - Dominic Bunnett
We will prove that every projective normal toric variety is constructed as a GIT quotient of a diagonalisable group acting on an affine space which is linearised by a character. In the first part of this talk, we explain how to construct GIT quotients of affine spaces with respect to a character and, in the second part of the talk, we give the proof of the above result.
Notes for the first and second talk can be found here .

♠ 03.05.2016 // An introduction to symplectic geometry - Maik Pickl
We introduce symplectic manifolds and some examples (symplectic vector spaces, the cotangent bundle of a manifold and complex projective spaces). We introduce two types of morphisms in symplectic geometry: symplectomorphisms (which are diffeomorphisms which respect the symplectic form) and Lagrangian correspondences (which are defined using Lagrangian submanifolds in the product).
Notes for the third talk can be found here .

♠ 10.05.2016 // Group quotients in symplectic geometry - Vicky Hoskins
We consider the construction of symplectic quotients of actions of compact groups in symplectic geometry; this is known as symplectic reduction and is performed by taking a topological quotient of a level set of a moment map for the action. We prove the Marsden-Weinstein-Meyer Theorem, which states that if the group acts freely on this level set, then the reduction is a symplectic manifold.
Notes for the fourth talk can be found here .

♠ 17.05.2016 // The Kempf-Ness theorem - Vicky Hoskins
We prove the Kempf-Ness theorem which states that the GIT quotient of a reductive group acting linearly on a smooth projective variety X over the complex numbers is homeomorphic to the symplectic reduction of the action of the maximal compact subgroup. The main step is to prove that an orbit meets the zero level set of the moment map if and only if the orbit is GIT polystable.
Notes for the fifth talk can be found here .

♠ 24.05.2016 // The Bialynicki-Birula decomposition - Anna-Lena Winz
Given an action of the multiplicative group on a smooth projective variety X, we prove that there are two decompositions of X into locally closed smooth subvarieties which are both indexed by the connected components of the fixed locus for the action. We explicitly describe the strata as locally trivial affine fibrations over the fixed loci.

♠ 31.05.2016 - 07.06.2016 // Variation of GIT for the multiplicative group - Koki Yoshimoto
For a linear action of the multiplicative group on an affine variety, we describe the variation of GIT quotients as we twist by a character of the multiplicative group. There are three cases to consider: the trivial character, a positive character and a negative character. We use the Bialynicki-Birula decomposition to describe the morphisms from the positive (resp. negative) GIT quotient to the zero GIT quotient.

♠ 07.06.2016 -14.06.2016 // Toric quotients and flips - Marta Pieropan
We study variation of GIT for the action of a subtorus of the torus for a toric variety associated to a polyhedron. The space of linearisations giving non-empty GIT quotients is closed related to the polyhedron defining this toric variety and on it we describe a polyhedral wall and chamber decomposition, and the birational transformations given by wall-crossings.

♠ 21.06.2016 // The space of linearisations - Markus Penner
In this talk, for a reductive group G acting on a quasi-projective variety X, we define the G-Neron Severi group of X to be the group of G-linearisations on X up to G-algebraic equivalence. We define the G-effective cone inside the G-Neron Severi group and prove that there is a wall and chamber structure with finitely many chambers such that all linearisations in a given chamber give the same GIT quotient.

♠ 28.06.2016 // Wall-crossings in variation of GIT - Ignacio Barros
We continue to study how the projective quotient depends on the G-linearised line bundle. In particular, we describe what happens when we move from one chamber to another. If time permits, we describe the whole construction on a concrete example and describe the VGIT wall-crossings.

♠ 5.07.2016 // The (Zariski local) description of the morphism to the wall - Dominic Bunnett
We generalise the results discussed for actions of the multiplicative group in talk seven, under a certain smoothness hypothesis, to general reductive group actions. Suppose that G is a reductive group acting on a quasi-projective variety X over k, which is projective over an affine variety. We show that the maps the GIT quotient maps to the wall are locally trivial weighted projective space fibrations. If time permits, we describe local criteria which enable us to strengthen our results further.

♠ 12.07.2016 // The (étale local) description of the morphism to the wall - Marysia Donten-Bury
For a reductive group G acting on a quasi-projective variety X, we show that under a weakened hypothesis, the morphism from the GIT quotient in a chamber to a wall is an etale locally trivial weighted projective space fibration. We will introduce Luna’s etale slice theorem, a key technical tool. We give a counter example to see why the hypothesis is necessary.

♠ 19.07.2016 // Mori dream spaces and GIT - Marta Pieropan
We define an equivalence relation on the Neron-Severi group associated to a variety and define Mori chambers (as the closure of equivalence classes). Mori dream spaces are varieties with a 'good' Mori chamber decomposition. We then show that any Mori Dream space is a GIT quotient of an affine variety, generalising the results of the first talks.