On the injectivity and non-injectivity of the \(l\)-adic cycle class maps
Friday, 26.4.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
We study the injectivity of the cycle class map with values in Jannsen's continuous étale cohomology, by using refinements that go through étale motivic cohomology and the ``tame'' version of Jannsen's cohomology. In particular, we use this to show that the Tate and the Beilinson conjectures imply that its kernel is torsion in positive characteristic, and to revisit recent counterexamples to injectivity.\n
Periods via Motives and Species
Friday, 31.5.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
We explain how the structure theory of finite dimensional algebras can be used to deduce dimension formulas for period spaces of motives. They are sharp and unconditional in the case of 1-motives, i.e., periods of curves. (Joint work with Martin Kalck, Graz)
On finite generation of fundamental groups in algebraic geometry
Friday, 28.6.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The étale fundamental group of a (quasicompact) variety over complex numbers is (topologically) finitely presented by comparison with the topological case.\nIn characteristic p, the situation is much more subtle, as affine varieties have very large fundamental groups.\nBuilding on a recent breakthrough result by Esnault, Shusterman and Srinivas, I will explain how to extend the finite presentation statement to arbitrary proper varieties (joint work with Srinivas and Stix) and then (at least the finite generation part) to log/tame fundamental groups of schemes and rigid analytic spaces (joint work with Achinger, Hübner and Stix).\nThis requires revisiting the tame topology of rigid spaces and working with a certain class of non-fs log schemes.
Global logarithmic deformation theory
Friday, 12.7.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Global logarithmic deformation theory
Friday, 12.7.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
A classical problem in algebraic geometry is the construction of smooth projective Calabi-Yau varieties, in particular of mirror pairs. In the approach via smoothings, the first step is to construct a reducible Gorenstein Calabi-Yau variety (or a pair thereof) by closed gluing of simple pieces. The second step is to find a family of Calabi-Yau varieties whose special fiber is the already constructed reducible Calabi-Yau variety, and whose general fiber is smooth. Logarithmic geometry, and especially logarithmic deformation theory, has given new impulses to the second step of this approach. In particular, the logarithmic version of the Bogomolov-Tian-Todorov theorem implies the existence of smoothings.\n\nIn this talk, we will see what logarithmic deformations are and by which types of Lie algebras they are controlled; we will discuss why logarithmic deformations are unobstructed in the Calabi-Yau case, and how their existence implies the existence of (non-logarithmic) smoothings.