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

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|November 3 | |November 3 | ||

|Dario Beraldo (Oxford) | |Dario Beraldo (Oxford) | ||

− | |[[#Dario Beraldo| | + | |[[#Dario Beraldo|Quasi-quasi-smooth schemes and Hecke eigensheaves]] |

|Dima | |Dima | ||

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|December 1 | |December 1 | ||

|Jonathan Wang (IAS) | |Jonathan Wang (IAS) | ||

− | | | + | |[[#Jonathan Wang|The Drinfeld-Gaitsgory operator on automorphic functions]] |

|Dima | |Dima | ||

|- | |- | ||

|December 8 | |December 8 | ||

− | |[https://www.math.brown.edu/~mtchan/ Melody Chan (Brown | + | |[https://www.math.brown.edu/~mtchan/ Melody Chan (Brown)] |

− | |[[#Melody Chan| | + | |[[#Melody Chan|The Euler characteristic of Brill-Noether varieties]] |

|Jordan | |Jordan | ||

+ | |||

+ | |- | ||

+ | |December 15 | ||

+ | |[http://math.rice.edu/~jrc9/ John Calabrese (Rice)] | ||

+ | |[[# John Calabrese|Towards the crepant resolution conjecture for Donaldson-Thomas invariants]] | ||

+ | |Andrei | ||

|} | |} | ||

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'''LePotier's Strange Duality and Quot Schemes''' | '''LePotier's Strange Duality and Quot Schemes''' | ||

+ | |||

Finite schemes of quotients over a smooth curve are a vehicle | Finite schemes of quotients over a smooth curve are a vehicle | ||

for proving strange duality for determinant line bundles on | for proving strange duality for determinant line bundles on | ||

Line 75: | Line 82: | ||

'''Poincare sheaf on the stack of rank two Higgs bundles''' | '''Poincare sheaf on the stack of rank two Higgs bundles''' | ||

+ | |||

It is well known that for a smooth projective curve, there exists a Poincare line bundle on the product of the Jacobian of the curve which is the universal family of topologically trivial line bundles of the Jacobian. It is natural to ask whether similar results still hold for the compactified Jacobian of singular curves. There has been a lot of work in this problem. In this talk I will sketch the construction of the Poincare sheaf for spectral curves in Hitchin fibration. The new feature is that we are able to extend the construction of Poincare sheaf to nonreduced spectral curves. | It is well known that for a smooth projective curve, there exists a Poincare line bundle on the product of the Jacobian of the curve which is the universal family of topologically trivial line bundles of the Jacobian. It is natural to ask whether similar results still hold for the compactified Jacobian of singular curves. There has been a lot of work in this problem. In this talk I will sketch the construction of the Poincare sheaf for spectral curves in Hitchin fibration. The new feature is that we are able to extend the construction of Poincare sheaf to nonreduced spectral curves. | ||

+ | |||

+ | ===Dario Beraldo=== | ||

+ | |||

+ | '''Quasi-quasi-smooth schemes and Hecke eigensheaves''' | ||

+ | |||

+ | Quasi-smooth schemes (or stacks) arise in several geometric situations: an important example is the stack of G-local systems on a Riemann surface, but I will provide many other simpler examples. The degree of non-smoothness of a quasi-smooth scheme is captured nicely by the difference between quasi-coherent sheaves and ind-coherent sheaves on it. | ||

+ | |||

+ | I will explain how, while studying a quasi-smooth scheme, one is led to go to the next level: the level of "quasi-quasi-smooth scheme" (pun intended). By definition, if quasi-smooth means that the tangent complex lives in amplitude [0,1], quasi-quasi-smooth is the next best thing: amplitude [0,2]. | ||

+ | |||

+ | In this situation, the relation between QCoh and IndCoh gets unwieldy. This wildness has a positive consequence though: it helps define a new theory of D-modules on quasi-quasi-smooth schemes that is of use in geometric Langlands (for instance, in the description of Hecke eigensheaves for reducible local systems). | ||

+ | |||

+ | ===Jonathan Wang=== | ||

+ | |||

+ | '''The Drinfeld-Gaitsgory operator on automorphic functions''' | ||

+ | |||

+ | Let F be a function field and G a connected split reductive group over F. We define a "strange" operator between spaces of automorphic functions on G(A)/G(F), and show that this operator is natural from the viewpoint of the geometric Langlands program via the functions-sheaves dictionary. We discuss how to define this operator over a number field by relating it to pseudo-Eisenstein series and inversion of the standard intertwining operator. The Drinfeld-Gaitsgory operator is also connected to Deligne-Lusztig duality and cohomological duality of representations over a local field. |

## Revision as of 11:42, 7 December 2017

The seminar meets on Fridays at 2:25 pm in Van Vleck B321.

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).

## Fall 2017 Schedule

date | speaker | title | host(s) |
---|---|---|---|

October 6 | Michael Brown (UW-Madison) | Topological K-theory of equivariant singularity categories | local |

October 9 (Monday!!, 3:30-4:30, B119) | Aaron Bertram (Utah) | LePotier's Strange Duality and Quot Schemes | Andrei |

October 27 | Mao Li (UW-Madison) | Poincare sheaf on the stack of rank two Higgs bundles | local |

November 3 | Dario Beraldo (Oxford) | Quasi-quasi-smooth schemes and Hecke eigensheaves | Dima |

December 1 | Jonathan Wang (IAS) | The Drinfeld-Gaitsgory operator on automorphic functions | Dima |

December 8 | Melody Chan (Brown) | The Euler characteristic of Brill-Noether varieties | Jordan |

December 15 | John Calabrese (Rice) | Towards the crepant resolution conjecture for Donaldson-Thomas invariants | Andrei |

## Abstracts

### Michael Brown

**Topological K-theory of equivariant singularity categories**

This is joint work with Tobias Dyckerhoff. Topological K-theory of complex-linear dg categories is a notion introduced by Blanc in his recent article "Topological K-theory of complex noncommutative spaces". In this talk, I will discuss a calculation of the topological K-theory of the dg category of graded matrix factorizations associated to a quasi-homogeneous polynomial with complex coefficients in terms of a classical topological invariant of a complex hypersurface singularity: the Milnor fiber and its monodromy.

### Aaron Bertram

**LePotier's Strange Duality and Quot Schemes**

Finite schemes of quotients over a smooth curve are a vehicle for proving strange duality for determinant line bundles on moduli spaces of vector bundles on Riemann surfaces. This was observed by Marian and Oprea. In work with Drew Johnson and Thomas Goller, we extend this idea to del Pezzo surfaces, where we are able to use it to prove cases of Le Potier's strange duality conjecture.

### Mao Li

**Poincare sheaf on the stack of rank two Higgs bundles**

It is well known that for a smooth projective curve, there exists a Poincare line bundle on the product of the Jacobian of the curve which is the universal family of topologically trivial line bundles of the Jacobian. It is natural to ask whether similar results still hold for the compactified Jacobian of singular curves. There has been a lot of work in this problem. In this talk I will sketch the construction of the Poincare sheaf for spectral curves in Hitchin fibration. The new feature is that we are able to extend the construction of Poincare sheaf to nonreduced spectral curves.

### Dario Beraldo

**Quasi-quasi-smooth schemes and Hecke eigensheaves**

Quasi-smooth schemes (or stacks) arise in several geometric situations: an important example is the stack of G-local systems on a Riemann surface, but I will provide many other simpler examples. The degree of non-smoothness of a quasi-smooth scheme is captured nicely by the difference between quasi-coherent sheaves and ind-coherent sheaves on it.

I will explain how, while studying a quasi-smooth scheme, one is led to go to the next level: the level of "quasi-quasi-smooth scheme" (pun intended). By definition, if quasi-smooth means that the tangent complex lives in amplitude [0,1], quasi-quasi-smooth is the next best thing: amplitude [0,2].

In this situation, the relation between QCoh and IndCoh gets unwieldy. This wildness has a positive consequence though: it helps define a new theory of D-modules on quasi-quasi-smooth schemes that is of use in geometric Langlands (for instance, in the description of Hecke eigensheaves for reducible local systems).

### Jonathan Wang

**The Drinfeld-Gaitsgory operator on automorphic functions**

Let F be a function field and G a connected split reductive group over F. We define a "strange" operator between spaces of automorphic functions on G(A)/G(F), and show that this operator is natural from the viewpoint of the geometric Langlands program via the functions-sheaves dictionary. We discuss how to define this operator over a number field by relating it to pseudo-Eisenstein series and inversion of the standard intertwining operator. The Drinfeld-Gaitsgory operator is also connected to Deligne-Lusztig duality and cohomological duality of representations over a local field.