PRÄSENZ im SoSe22: Toric Geometry - Einzelansicht

Toric geometries are one of the best-understood examples in algebraic and differential geometry. They are explicit and not so complicated
to compute with, while their theory is rich enough to be interesting. This balance is what makes them play a prominent role in testing conjectures
in mathematics and theoretical physics. Despite their seeming simplicity several important problems remain unsolved.
Their systematic study, both in algebraic and differential geometry, started in '70s. As their name suggests
toric geometries are in particular varieties or smooth manifolds carrying an action of a torus. We start our lectures by explaining Lie group actions
and introducing symplectic geometry. By combining these we arrive at toric symplectic manifolds which, in particular, admit a momentum map.
By studying the momentum map we find a correspondence between compact connected toric symplectic manifolds and Delzant polytopes. This
striking fact, that the whole geometry is encoded in some kind of polytope and how to build the geometry out of such a polytope, is one of the
main goals of our lecture series. If time permits, we look at the algebraic geometry side of this correspondence, build a variety out of a Delzant
polytope and remark on their properties.

* I am almost certain that this course will not be thought again in the following semesters.

Books:

• Torus actions on symplectic manifolds, Audin
• Transformation groups (Symplectic torus actions and toric manifolds), Mukherjee
• Toric topology, Buchstaber and Panov
• Momentum maps and Hamiltonian reduction, Ortega and Ratiu
• Hamiltonian group action and equivariant cohomology, Dwivedi, Herman, Jeffrey and Hurk
• Introduction to toric varieties, Fulton
• Moment maps and combinatorial invariants of Hamiltonian T^n-spaces, Guillemin
• Introduction to symplectic topolgy, McDuff and Salamon
• Symplectic toric manifolds, Silva
• An introduction to symplectic geometry, Berndt

Papers:

• Convexity and commuting Hamiltonian, Atiyah
• Hamiltoniens periodiques et images convexes de l'application moment, Delzant
• Scalar curvature and stability of toric varieties, Donaldson
• Kähler structures on toric varieties, Guillemin
• Convexity properties of the moment mapping, Guillemin and Sternberg
• Hamiltonian torus action on symplectic orbifolds and toric varieties, Lerman and Tolman

This syllabus is work in progress and as such may include too much or too little material. Here, my intention is to provide you with an idea of what I would like to cover in our classes. This course aims to explain the Delzant correspondence in detail (all proofs included). To achieve this we need several concepts from topics 1,2 and 3 from the list below. The timespan of our course does not allow us to explore these in great detail, however, I try to present all proofs needed for full understanding of the Delzant correspondence and more. You may expect to see examples illustrating involved statements/theorems. I also have some extras for very interested students.

1. Lie group actions
2. symplectic and Hamiltonian geometry
3. Morse theory
4. Delzant construction
5. Fans and toric varieties

In a few days, I will expand each item in this list into subtopics. (24.02.22)

If you have questions, do not hesitate to contact me.

Basic differential geometry

Anyone who is familiar with the basics of differential geometry - no matter what you study.