Algebraic Geometry Seminar Spring 2011

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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
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 (at 1:20 in geometry/topology time/place) Ishai Dan-Cohen (Hannover) Moduli of unipotent representations Jordan Ellenberg
April 8 Greg Pearlstein (Michigan State) On the locus of Hodge classes Laurentiu Maxim
Apr. 15 Orit Davidovich (UT-Austin) State Sums in 2-dimensional Extended Topological Field Theories Andrei Caldararu
Apr. 22 Y. P. Lee (Utah) Algebraic Cobordism of Bundles on Varieties Andrei Caldararu
Apr. 29 Donu Arapura (Purdue) Nori's Hodge conjecture Laurentiu Maxim
May 6 Hal Schenck (UIUC) Equivariant Chow cohomology of nonsimplicial toric varieties Andrei Caldararu

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.

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

Ishai Dan-Cohen Moduli of Unipotent Representations

Let $G$ be a unipotent group over a field of characteristic zero. The moduli problem posed by all representations of a fixed dimension $n$ is badly behaved. We set out to define an appropriate nondegenracy condition, and to construct a quasi-projective variety parametrinzing isomorphism classes of nondegenerate representations. In my thesis I defined an invariant $w$ of $G$, its \textit{width}, and a nondegeneracy condition appropriate for representations of dimension $n \le w+1$. Unfortunately, the width is bounded by the depth. But for groups $G$, unipotent of depth $\le 2$, a different nondegeneracy condition gives rise to a quasi projective moduli space for \textit{all} $n$.

This talk is based in part on my thesis, and in part on joint work with Anton Geraschenko, part of which was covered by his recent talk in the number theory seminar here in Madison.

Greg Pearlstein On the locus of Hodge classes

I will discuss my work of over the past several years regarding the zero loci of normal functions and use this to show that the locus where a rational class in a variation of mixed Hodge structure is a Hodge class is a countable union of algebraic subvarieties.

Orit Davidovich State Sums in 2-dimensional Extended Topological Field Theories

A state sum is an expression approximating the partition function of a d-dimensional field theory on a closed d-manifold from a triangulation of that manifold. To consider state sums in completely local 2-dimensional topological field theories (TFTs), we introduce a mechanism for incorporating triangulations of surfaces into the cobordism (infinity,2)-category. This serves to produce a state sum formula for any fully extended 2-dimensional TFT possibly with extra structure. We then follow the Cobordism Hypothesis in classifying fully extended 2-dimensional G-equivariant TFTs for a finite group G. These are oriented theories in which bordisms are equipped with principal G-bundles. Combining the mechanism mentioned above with our classification result, we derive Turaev's state sum formula for such theories.

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.

Donu Arapura Nori's Hodge conjecture

Nori's conjecture, which is not so well known, says that his category of motives embeds fully and faithfully into the category of mixed Hodge structures. This should be viewed as a refinement of Deligne's absoluteness conjecture. I want to explain the conjecture, and then explain how to prove special case for the tensor subcategory generated by smooth affine curves, which contains things like semiabelian varieties.

Hal Schenck Equivariant Chow cohomology of nonsimplicial toric varieties

For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf on Proj(N), showing that the Chern classes depend on subtle geometry of P and giving criteria for the splitting of the sheaf as a sum of line bundles. For certain fans associated to the reflection arrangement A_n, we describe a connection between C^0(P) and logarithmic vector fields tangent to A_n.