UBC Algebraic Geometry Seminar

In recent work with Felix Thimm and Nick Kuhn, we proved a Joyce-style "universal" wall-crossing formula for certain equivariant moduli problems of 3-Calabi-Yau type, including Donaldson-Thomas theory. An immediate and productive question is how tautological classes like descendents transform under such wall-crossings. I will present an explicit descendent transformation formula for the Donaldson-Thomas/Pandharipande-Thomas wall-crossing of equivariant vertices, explain how the computation works, and speculate on how it may be generalized. This serves as a fairly down-to-earth example of how such wall-crossing formulas may be applied.

Let F be a number field (say, the field of rational numbers Q) or a p-adic field (say, the field of p-adic numbers Q_p), or a global function field (say, the field of rational functions of one variable over a finite field F_q). Let G be a connected reductive group over F (say, SO(n) ). One needs the first Galois cohomology set H^1(F,G) for classification problems in algebraic geometry and linear algebra over F. In the talk, I will give closed formulas for H^1(F,G) when F is as above, in terms of the algebraic fundamental group \pi_1(G) introduced by the speaker in 1998. All terms will be defined and examples will be given.

The talk is based on a joint work with Tasho Kaletha.

For a smooth and proper surface over a finite field, the formula of Artin and Tate relates the behavior of the zeta-function at 1 to other invariants of the surface. We give a version which equates invariants only depending on the Brauer group to invariants only depending on the Neron-Severi group. We estimate the terms appearing in the formula, and discuss the special case of abelian varieties and K3-surfaces. 

TBD