Algebraic Geometry Seminar Spring 2017

The seminar meets on Fridays at 2:25 pm in Van Vleck B113.

Here is the schedule for the previous semester, the next semester. and for this semester.

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Spring 2017 Schedule

date	speaker	title	host(s)
January 20	Sam Raskin (MIT)	W-algebras and Whittaker categories	Dima
January 27	Nick Salter (U Chicago)	Mapping class groups and the monodromy of some families of algebraic curves	Jordan
March 3	Robert Laudone (UW Madison)	The Spin-Brauer diagram algebra	local (Steven)
March 10	Nathan Clement (UW Madison)	Parabolic Higgs bundles and the Poincare line bundle	local
March 17	Amy Huang (UW Madison)	Equations of Kalman varieties	local (Steven)
March 31	Jie Zhou (Perimeter Institute)	Gromov-Witten invariants of elliptic curves and moments of Weierstrass P-function	Andrei
April 7	Vladimir Dokchitser (Warwick)	Arithmetic of hyperelliptic curves over local fields	Jordan
April 14	Laurentiu Maxim (UW-Madison)	Characteristic classes of complex hypersurfaces and multiplier ideals	local
April 21	Vladimir Sotirov	Cohomology of compactified Jacobians of singular curves	local
April 28	Lucia Mocz (Princeton)	A New Northcott Property for Faltings Height	Jordan
May 5	Qingyuan Jiang (Chinese University of Hong Kong)	Categorical Plücker formula and Homological Projective Duality	Andrei

Abstracts

Sam Raskin

W-algebras and Whittaker categories

Affine W-algebras are a somewhat complicated family of (topological) associative algebras associated with a semisimple Lie algebra, quantizing functions on the algebraic loop space of Kostant's slice. They have attracted a great deal of attention because of Feigin-Frenkel's duality theorem for them, which identifies W-algebras for a Lie algebra and for its Langlands dual through a subtle construction.

The purpose of this talk is threefold: 1) to introduce a ``stratification" of the category of modules for the affine W-algebra, 2) to prove an analogue of Skryabin's equivalence in this setting, realizing the categoryof (discrete) modules over the W-algebra in a more natural way, and 3) to explain how these constructions help understand Whittaker categories in the more general setting of local geometric Langlands. These three points all rest on the same geometric observation, which provides a family of affine analogues of Bezrukavnikov-Braverman-Mirkovic. These results lead to a new understanding of the exactness properties of the quantum Drinfeld-Sokolov functor.

Nick Salter

Mapping class groups and the monodromy of some families of algebraic curves

In this talk we will be concerned with some topological questions arising in the study of families of smooth complex algebraic curves. Associated to any such family is a monodromy representation valued in the mapping class group of the underlying topological surface. The induced action on the cohomology of the fiber has been studied for decades- the more refined topological monodromy is largely unexplored. In this talk, I will discuss some theorems concerning the topological monodromy groups of families of smooth plane curves, as well as families of curves in CP^1 x CP^1. This will involve a blend of algebraic geometry, singularity theory, and the mapping class group, particularly the Torelli subgroup.

Robert Laudone

The Spin-Brauer diagram algebra

Schur-Weyl duality is an important result in representation theory which states that the actions of [math]\displaystyle{ \mathfrak{S}_n }[/math] and [math]\displaystyle{ \mathbf{GL}(N) }[/math] on [math]\displaystyle{ \mathbf{V}^{\otimes n} }[/math] generate each others' commutants. Here [math]\displaystyle{ \mathfrak{S}_n }[/math] is the symmetric group and [math]\displaystyle{ \mathbf{V} }[/math] is the standard complex representation. In this talk, we investigate the Spin-Brauer diagram algebra, which arises from studying an analogous form of Schur-Weyl duality for the action of the spinor group on [math]\displaystyle{ \mathbf{V}^{\otimes n} \otimes \Delta }[/math]. Here [math]\displaystyle{ \mathbf{V} }[/math] is again the standard [math]\displaystyle{ N }[/math]-dimensional complex representation of [math]\displaystyle{ {\rm Pin}(N) }[/math] and [math]\displaystyle{ \Delta }[/math] is the spin representation. We will give a general construction of the Spin-Brauer diagram algebra, discuss its connection to [math]\displaystyle{ {\rm End}_{{\rm Pin}(N)}(V^{\otimes n} \otimes \Delta) }[/math] and time permitting we will mention some interesting properties of the algebra, in particular its cellularity.

Nathan Clement

Parabolic Higgs bundles and the Poincare line bundle

We work with some moduli spaces of (parabolic) Higgs bundles which come in infinite families indexed by rank. I'll give some motivation for the study of parabolic Higgs bundles, but the main problem will be to describe the moduli spaces. By applying some integral transforms, most importantly the Fourier-Mukai transform associated to the Poincare line bundle, we are able to reduce the rank of the problem and eventually get a good presentation of the moduli spaces. One fun technique involved in the argument deals with the spectrum of a one-parameter family of linear operators. When such an operator degenerates to one that is diagonalizable with repeated eigenvalues, the spectrum of the operator admits a scheme-theoretic refinement in a certain blowup which carries more information than simply the eigenvalues with multiplicity.

Amy Huang

Equations of Kalman Varieties

Given a subspace L of a vector space V, the Kalman variety consists of all matrices of V that have a nonzero eigenvector in L. We will discuss how to apply Kempf Vanishing technique with some more explicit constructions to get a long exact sequence involving coordinate ring of Kalman variety, its normalization and some other related varieties in characteristic zero. This long exact sequence is first conjectured by Sam in 2011. Time permitting we will also discuss how to extract more information from the long exact sequence including the minimal defining equations for Kalman varieties.

Jie Zhou

Gromov-Witten invariants of elliptic curves and moments of Weierstrass P-function

I will talk about a joint work with Si Li on the computation of higher genus Gromov-Witten invariants of elliptic curves using mirror symmetry.

The Gromov-Witten theory for elliptic curves is proved by Si Li, basing on the works of Bershadsky-Cecotti-Ooguri-Vafa and Costello-Li, to be equivalent to a quantum field theory on the mirror elliptic curve. Taking the Feynman graph integrals as the definition of the quantum field theory, I will explain the computations on the integrals (which are closely related to moments of the Weierstrass P-function). I will also discuss the quasi-modularity and the modular completion of the integrals. The Hodge-theoretic interpretations of all of these will also be explained.

Vladimir Dokchitser

Arithmetic of hyperelliptic curves over local fields

Let C:y^2 = f(x) be a hyperelliptic curve over a local field K of odd residue characteristic. We show how several arithmetic invariants of the curve and its Jacobian, including its potential stable reduction, Galois representation and (in the semistable case) Tamagawa numbers, can be simply extracted from combinatorial data coming from the roots of f(x).

Laurentiu Maxim

Characteristic classes of complex hypersurfaces and multiplier ideals

I will discuss two different ways to measure the complexity of singularities of a (globally-defined) complex hypersurface. The first is derived via (Hodge-theoretic) characteristic classes of singular complex algebraic varieties, while the second is provided by the multiplier ideals. I will also point out a natural connection between these two points of view. (Joint work with Morihiko Saito and Joerg Schuermann.)

Vladimir Sotirov

Cohomology of compactified Jacobians of singular curves

Qingyuan Jiang

Categorical Plücker formula and Homological Projective Duality

The talk will be based on the joint work with Prof. Conan Leung, and Mr. Ying Xie (arXiv:1704.01050). We will be mainly interested in the question of how derived categories of coherent sheaves of two varieties behave under intersections, and how they are related to that of the original varieties.

For the study of derive categories of linear sections of projective varieties, Kuznetsov introduced the concept of Homological Projective Duality (HPD). Since its introduction, the HPD theory becomes one of the most powerful frameworks in the homological study of algebraic geometry. The main result (HPD theorem) of the theory gives complete descriptions of bounded derived categories of coherent sheaves of (dual) linear sections of HP-dual varieties.

For general intersections beyond linear sections, we show same type results hold. More precisely, our results are twofold: i) Decomposition part. For any two varieties $X$, $T$ with maps to projective space $\mathbb{P}$ and Lefschetz decompositions, then there is a semiorthogonal decomposition of $D(X\times_{\mathbb{P}} T$ into ‘ambient’ part (contributions from ambient product $X \times T$) and ‘primitive’ part, as long as the fiber product $X\times_{\mathbb{P}} T$ has expected dimension; ii) Comparison part. If $Y$, $S$ are the respective HP-duals of $X$, $T$, then the ‘primitive’ parts of the derived categories of the two fiber products $D(X\times_{\mathbb{P}} T$ and $D(Y \times_{\mathbb{P} S)$ are equivalent, provided that the two pairs intersect properly. In the case when one pair of HP-dual varieties (say $(S,T)$) are given by dual linear subspaces, our method provides a more direct proof of the original HPD theorem.