University of Chicago Algebraic Geometry Seminar

October 7
Timothy Logvinenko (University of Liverpool): Tyurin's isomorphism via spherical twists

Abstract: Tyurin's isomorphism is a birational isomorphism between a smooth projective K3 surface X and the moduli space M(+-H^2/2,H,+-1) of stable sheaves on X with the Mukai vector (+-H^2/2,H,+-1), where the signs are chosen so as to make the rank positive. Tyurin has constructed it explicitly for two classes of divisors H on X -- for H ample and for H with H^0(+-H) = 0. We show that both constructions are induced naturally by the same composition of a pair of spherical twists in the derived category D(X). We then use this to extend Tyurin's result to any H with H^2 \neq 0.

October 14 (Geometric Langlands Seminar)
Dima Arinkin (University of North Carolina): Autoduality for Jacobians of singular curves (Part 2)

Abstract: Let C be a (smooth projective algebraic) curve, in other words, a Riemann surface. It is well known that the Jacobian J of C is a self-dual complex torus, that is, J is identified with the space of topologically trivial line bundles on J. Suppose now that C is singular. The Jacobian J of C parametrizes topologically trivial line bundles on C; it is smooth, but no longer compact. By considering torsion-free sheaves instead of line bundles, one obtains a natural singular compactification J' of J. The subject of this talk is line bundles on J' and their cohomology. The main result is the following `autoduality': If C has planar singularities, J is identified with a space of line bundles on J'. I also plan to discuss the Fourier-Mukai transform arising from the autoduality. My next talk will cover the stronger `compactified' autoduality statement (identifying J' and with a space of torsion-free sheaves on itself), which requires additional restrictions on C. The compactified Jacobians play a role in the geometric Langlands correspondence (for GL(n)), where they appear as fibers of the Hitchin fibration. However, the talk relies on classical methods of algebraic geometry, and should be accessible to wide audience.

October 20 (Tuesday), 3pm, E206 (Notice special day, time and room)
Teruyoshi Yoshida (University of Cambridge): Converse of the weight-monodromy conjecture

Abstract: The long-standing weight-monodromy conjecture claims that all geometric local Galois representations (i.e. appearing in etale cohomology of varieties) are "pure" (have pure monodromy filtrations), but we can show that all pure local Galois representations are found in geometry. It seems that we can only do this via global argument (Honda-Tate theory etc), and we introduce Langlands correspondence for GL(n) in the course of going through several proofs. I'll also discuss the question of realizing local Galois representations as local components of _irreducible_ global Galoir representations (or motives; partly joint with Sug Woo Shin).

October 28
Vikraman Balaji (Chennai Mathematical Institute): Principal bundles on projective varieties and Donaldson-Uhlenbeck spaces for stable principal bundles on surfaces.

Abstract: In this talk we will first prove a semistable reduction theorem for semistable principal bundles with semisimple structure groups on smooth projective varieties over complex numbers. Then we construct the Donaldson-Uhlenbeck compactification for the moduli of stable principal bundles on algebraic surfaces and describe the boundary. Using the principle of holonomy, we then construct stable principal bundles for large Chern classes.

November 4
Henrik Russel : Generalized Albanese varieties and duality.

Abstract: We consider two generalizations of the classical Albanese variety: an Albanese variety for singular projective varieties (over a field k) of Esnault, Srinivas, Viehweg and a higher dimensional analogon of the generalized Jacobian with modulus of Rosenlicht-Serre. These are algebraic groups defined by universal mapping properties. We discuss the costruction of these objects using duality theory of generalized 1-motives. In particular, we introduce dual functors to the generalized Albanese varieties. (Work partly joint with Kazuya Kato.)

November 18
Takako Fukaya (University of Chicago): Title: Local units in non-commutative Galois extensions

Abstract: In local Iwasawa theory, for the cyclotomic extensions of the p-adic number field, Iwasawa, Coates-Wiles, and Coleman obtained the ``almost isomorphism'' of the two modules over the Iwasawa algebra: (the tower of local units with respect to the norm maps) to (Iwasawa algebra, regarded as a module over the Iwasawa algebra itself). I will discuss how to generalize this to a tower of local units in non-commutative Galois extensions and give a partial result.

December 2
Liang Xiao (University of Chicago): TBA