# Algebraic Geometry Seminar Spring 2014: Difference between revisions

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===Marci Hablcisek=== | ===Marci Hablcisek=== | ||

Given a smooth variety and two smooth closed subvarieties, derived algebraic geometry assigns to this data a differential graded scheme, the derived intersection. Equipping the ambient space with an Azumaya algebra, we obtain the notion of twisted derived intersections. In order to compare the twisted and the "untwisted" derived intersections, we assume that the Azumaya algebra is split along the two subvarieties. For such twisted intersection problem, we associate a natural line bundle on the derived intersection, which measures the difference between the two derived intersections. We give a criterion for the triviality of this line bundle. As an application, we prove a special case of the Barannikov-Kontsevich theorem, and we give a decomposition theorem for the hypercohomology spaces of the twisted de Rham complexes. The work is joint with Dima Arinkin and Andrei Caldararu. | Given a smooth variety and two smooth closed subvarieties, derived algebraic geometry assigns to this data a differential graded scheme, the derived intersection. Equipping the ambient space with an Azumaya algebra, we obtain the notion of twisted derived intersections. In order to compare the twisted and the "untwisted" derived intersections, we assume that the Azumaya algebra is split along the two subvarieties. For such twisted intersection problem, we associate a natural line bundle on the derived intersection, which measures the difference between the two derived intersections. We give a criterion for the triviality of this line bundle. As an application, we prove a special case of the Barannikov-Kontsevich theorem, and we give a decomposition theorem for the hypercohomology spaces of the twisted de Rham complexes. The work is joint with Dima Arinkin and Andrei Caldararu. | ||

===Rares Rasdeaconu=== | |||

Real enumerative invariants of real algebraic manifolds are derived from counting curves with suitable signs. | |||

I will discuss the case of counting real rational curves on K3 surfaces equipped with an anti-holomorphic involution. | |||

An adaptation to the real setting of a formula due to Yau and Zaslow will be presented. | |||

The proof passes through results of independent interest: a new insight into the signed counting, and | |||

a formula computing the Euler characteristic of the real Hilbert scheme of points on a K3 surface, the real | |||

version of a result due to Gottsche. | |||

The talk is based on a joint work with V. Kharlamov. |

## Revision as of 16:02, 19 February 2014

The seminar meets on Fridays at 2:25 pm in Van Vleck B231.

The schedule for the previous semester is here.

## Algebraic Geometry Mailing List

- Please join the Algebraic Geometry 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 2014 Schedule

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

January 31 | Marci Hablicsek (local) | Twisted derived intersections and twisted de Rham complexes | |

February 21 | Rares Rasdeaconu (Vanderbilt) | Counting real rational curves on K3 surfaces | Laurentiu |

March 28 | Kevin Tucker (University of Illinois at Chicago) | TBA | Daniel |

April 25 | Charles Doran (University of Alberta) | TBA | Song |

## Abstracts

### Marci Hablcisek

Given a smooth variety and two smooth closed subvarieties, derived algebraic geometry assigns to this data a differential graded scheme, the derived intersection. Equipping the ambient space with an Azumaya algebra, we obtain the notion of twisted derived intersections. In order to compare the twisted and the "untwisted" derived intersections, we assume that the Azumaya algebra is split along the two subvarieties. For such twisted intersection problem, we associate a natural line bundle on the derived intersection, which measures the difference between the two derived intersections. We give a criterion for the triviality of this line bundle. As an application, we prove a special case of the Barannikov-Kontsevich theorem, and we give a decomposition theorem for the hypercohomology spaces of the twisted de Rham complexes. The work is joint with Dima Arinkin and Andrei Caldararu.

### Rares Rasdeaconu

Real enumerative invariants of real algebraic manifolds are derived from counting curves with suitable signs. I will discuss the case of counting real rational curves on K3 surfaces equipped with an anti-holomorphic involution. An adaptation to the real setting of a formula due to Yau and Zaslow will be presented. The proof passes through results of independent interest: a new insight into the signed counting, and a formula computing the Euler characteristic of the real Hilbert scheme of points on a K3 surface, the real version of a result due to Gottsche.

The talk is based on a joint work with V. Kharlamov.