UBC Algebraic Geometry Seminar

We study Quot schemes of rank 0 quotients on smooth projective curves. These Quot schemes exhibit a rich and highly structured geometry, with formal analogies to the Hilbert scheme of points on surfaces. In this talk, I will explain a formula for the twisted Hodge groups of the Quot scheme with values in tautological line bundles pulled back from the symmetric product.

We present the patching method, a machinery developed by Harbater–Hartmann–Krashen and various other authors, dedicated to the study of arithmetics of linear algebraic groups over function fields of curves over complete discretely valued field such as ℚₚ(T). Then, we present a new result in this direction, which gives a nine 9-term exact sequence for Galois cohomology of 2-term complexes of tori in the patching setting. This relies on the notion of (co-)flasque resolutions of such complexes, generalizing the previous work by Colliot-Thélène–Sansuc. As applications, we show that patching holds for nonabelian second Galois cohomology of reductive groups with a smooth center, as well as a weak local–global principle for this cohomology set. We also rediscover a local–global principle for indices of central simple algebras.

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.

We define the algebraic double loop space, and algebro-geometric analogue of the double loop space in topology. We discuss quiver descriptions of the algebraic loop space of various homogeneous spaces. We show how this leads to various new results both in topology and algebraic geometry.

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.