Non-Archimedean Geometry, Birational Geometry and Resolution of Singularities

School in Heidelberg, 04.03.2024-08.03.2024

The school, which originally was planned to take place in Jerusalem, has been relocated to Heidelberg, Germany and will take place from 04.03.2024 to 08.03.2024. Due to the recent changes the registration is reopened for new participants until January 14.

For detailed information about the spring school and Heidelberg, you can check our booklet.

Organizers: Katharina Hübner (Frankfurt University), Michael Temkin (Hebrew University)

André Belotto: "Resolution of Singularties of Vector Fields and Foliations"

Hironaka’s proof of the existence of resolution of singularities for algebraic varieties over a field of characteristic zero stands as a landmark result in algebraic geometry with far reaching applications. Extending the technique to differential equations and foliations has intrigued mathematicians since at least the 19th century. However, the geometry of foliations involve transcendental phenomena and only low dimensional results are known.  

In this mini-course, I will present the main intended goals of resolution of singularities of foliations. We will cover the proof of the classical Bendixson-Seidenberg Theorem, which proves the existence of resolution of singularities when the ambient variety has dimension 2. We will follow it up by providing a brief panorama about the main results in the field, possibly including (depending on time constraints) the three dimensional case, nilpotent families of planar vector-fields, and local results on foliations generated by first integrals.

Katharina Hübner: "Non-archimedean Geometry and Applications to Birational Geometry"

We will study two approaches to nonarchimedean geometry: adic spaces and Berkovich spaces. They describe the same objects but with different flavor. We will see that adic spaces are connected to Berkovich spaces by taking the maximal Hausdorff quotient. Moreover we will explain how to construct formal models of adic spaces and Berkovich spaces. In fact we will see that formal models provide a third approach to nonarchimedean geometry.

Ming Hao Quek: "Resolution: Classical and New Tools"

This course is an invitation to resolution of singularities. I will sketch the classical methods (mostly following Kollár), as well as the complications that arise in the classical algorithm. I will then discuss the recent work of Abramovich-Temkin-Wlodarczyk and McQuillan, which demonstrated that complications in the classical algorithm can be better resolved by working more broadly with algebraic stacks instead of schemes. The output is an iterative procedure to embedded resolution of singularities in characteristic zero, where at every step, one blows up the ​"most singular locus", and immediately witnesses an improvement in singularities.

"Michael Temkin: "Non-archimedean Geometry and Semistable Reduction"

We will briefly discuss non-embedded approaches to resolution of singularities:  de Jong's method via alterations and the local uniformization approach of Zariski. Then we will concentrate on playing with these ideas in the framework of non-archimedean geometry. In particular, because a successful program  in the non-archimedean setting would also lead to classical resolution.  We will prove one-dimensional local uniformization of Berkovich spaces and deduce the analytic semi-stable reduction theorem. Applications to inseparable  local uniformization for algebraic varieties will be briefly discussed. Local uniformization of Berkovich spaces is still unknown even in dimension 2,  but in the last part of the minicourse we will discuss this problem, try to isolate the main difficulty and outline a way to attack it. 

For Michael's class, there are notes.