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.
|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|
|Mar. 4||Si Li (Harvard)||Higher Genus Mirror Symmetry||Junwu Tu|
|Mar. 25||Srikanth Iyengar (Nebraska)||Detecting flatness over smooth bases||Andrei Caldararu|
|April 8||Greg Pearlstein (Michigan State)||TBA||Laurentiu Maxim|
|Apr. 15||Orit Davidovich (UT-Austin)||TBA||Andrei Caldararu|
|Apr. 22||Y. P. Lee (Utah)||Algebraic Cobordism of Bundles on Varieties||Andrei Caldararu|
|Apr. 29||Donu Arapura (Purdue)||TBA||Laurentiu Maxim|
|May 6||Hal Schenck (UIUC)||TBA||Andrei Caldararu|
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.
Srikanth Iyengar Detecting flatness over smooth bases
I will present a recent result, obtained in collaboration with Luchezar Avramov, that gives a criterion for detecting flatness of an essentially of finite type morphism of schemes, with the base a smooth scheme. Our preprint is available on the arxiv:1002.3652
Y. P. Lee Algebraic Cobordism of Bundles on Varieties
The double point relation defines a natural theory of algebraic cobordism for bundles on varieties. We construct a simple basis (over the rationals) of the corresponding cobordism groups over Spec(C) for all dimensions of varieties and ranks of bundles. The basis consists of split bundles over products of projective spaces. Moreover, we prove the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism.