Algebraic Geometry Seminar Spring 2011

The seminar meets on Fridays at 2:25 pm in Van Vleck B305.

The schedule for the previous semester is here.

Spring 2011

date	speaker	title	host(s)
Jan. 21	Anton Geraschenko (UC Berkeley)	Toric Artin Stacks	David Brown
Jan. 28	Anatoly Libgober (UIC)	Mordell Weil groups of iso-trivial abelian varieties and Alexander polynomials	Laurentiu Maxim
Feb. 4	Valery Lunts (Indiana-Bloomington)	Lefschetz fixed point theorems for algebraic varieties and DG algebras	Andrei Caldararu
Feb. 18	Tony Várilly-Alvarado (Rice)	Transcendental obstructions to weak approximation on general K3 surfaces	David Brown
Feb. 25	Bhargav Bhatt (UMich)	TBA	Jordan Ellenberg
Mar. 4	Si Li (Harvard)	Higher Genus Mirror Symmetry	Junwu Tu
Mar. 25	Srikanth Iyengar (Nebraska)	TBA	Andrei Caldararu
April 8	Greg Pearlstein (Michigan State)	TBA	Laurentiu Maxim
May 6	Hal Schenck (UIUC)	TBA

Abstracts

Anton Geraschenko Toric Artin Stacks

Toric varieties provide a fantastic testing ground for ideas about varieties. Their main feature is that geometric properties of toric varieties correspond to combinatorial properties of their fans. In this talk, I'll briefly review some facts about toric varieties, then define a notion of a stacky fan and the toric Artin stack associated to a stacky fan. I'll state and prove some results relating the combinatorics of the fan to the geometry of the stack and show some illustrative examples. If time permits, I'll motivate this notion of toric Artin stacks a bit more by discussing how toric Artin stacks defined by fans compare to possible intrinsic definitions of toric Artin stacks. Some experience with toric varieties or stacks is a plus, but not essential.

Anatoly Libgober Mordell Weil groups of iso-trivial abelian varieties and Alexander polynomials

I will discuss a relation between the Mordell-Weil groups of isotrivial abelian varieties over function fields of cyclic coverings of projective plane and the Alexander polynomial of the complement to ramification locus of the latter. The results are based on joint work with J.I.Cogolludo on families of elliptic curves.

Valery Lunts Lefschetz fixed point theorems for algebraic varieties and DG algebras

I will report on my work in progress about a version of Lefschetz fixed point theorem for morphisms (more generally for Fourier-Mukai transforms) of smooth projective varieties. There is also a parallel version for smooth and proper DG algebras.

Si Li Higher genus mirror symmetry

I'll discuss my joint work with Kevin Costello on the geometric framework of constructing higher genus B-model from perturbative renormalization of BCOV theory on Calabi-Yau manifolds. This is conjectured to be the mirror of higher genus Gromov-Witten theory in the A-model. We carry out the construction in the one-dim cases, i.e., elliptic curves, and show that such constructed B-model correlation functions on the elliptic curve can be identified under the mirror map with the A-model descendant Gromov-Witten invariants on the mirror. This is the first compact example where mirror symmetry can be established at all genera.