# Difference between revisions of "Algebraic Geometry Seminar Spring 2017"

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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.) | 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.) | ||

+ | |||

+ | ===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. |

## Revision as of 14:47, 18 April 2017

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

Here is the schedule for the previous semester.

## Contents

## Algebraic Geometry Mailing List

- Please join the AGS Mailing List to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).

## 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 | TBA | local |

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]\mathfrak{S}_n[/math] and [math]\mathbf{GL}(N)[/math] on [math]\mathbf{V}^{\otimes n}[/math] generate each others' commutants. Here [math]\mathfrak{S}_n[/math] is the symmetric group and [math]\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]\mathbf{V}^{\otimes n} \otimes \Delta[/math]. Here [math]\mathbf{V}[/math] is again the standard [math]N[/math]-dimensional complex representation of [math]{\rm Pin}(N)[/math] and [math]\Delta[/math] is the spin representation. We will give a general construction of the Spin-Brauer diagram algebra, discuss its connection to [math]{\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.)

### 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.