# Dualities and Correspondences in Algebraic Geometry via Derived Categories and Noncommutative Methods

Lead Research Organisation:
University of Edinburgh

Department Name: Sch of Mathematics

### Abstract

The study of curves and surfaces given by the common zeroes of a set of polynomials has been pursued by humanity for thousands of years. In modern algebraic geometry, we study such sets in any dimension: these are called algebraic varieties. There are a number of questions that one can ask about one such variety: how "nice" is it? If we were standing on it, would it look to us like a curvy hill or like a rough mountain? If we are given two such varieties, can one tell if they are the same? Or if they are similar, for example if standing in most places on them they would look the same, and they only look different when looking at them from certain precise spots?

Derived categories are a way to consider these geometric objects and translate much of the information about them into algebraic notions. While the derived category of a variety retains much of the information about the variety we started with, at the same time it allows us extra flexibility to work in an algebraic context. In the past two decades the field of derived categories has experienced an outpouring of activity as many classical algebraic geometry problems are solved passing through derived categories techniques.

One fundamental question about derived categories is about how the derived categories of two different geometric objects are related. Some of these relations might come from relations and symmetries between the two varieties, but there are also other kinds of relations between them, which are deeper and harder to understand:

1. First of all, it is important to understand what the maps (functors) between two derived categories are like. Many of these - but not all, as people used to think! - have a very pleasant and useful geometric description as "Fourier-Mukai functors". Part of my project will consist in analyzing and describing the "bad" maps that are not Fourier-Mukai functors, and how these arise naturally by deforming the "good" maps we know about.

2. Another relation between two derived categories, which will be investigated as part of my project, is given by a concept of "duality" at the categorical level. Describing this duality gives us a way to understand deeper relations between derived categories that haven't yet been discovered, and that will shed more light on the symmetries and behavior both at the level of derived categories and at the level of the geometric objects.

3. Finally, in some instances the relations between derived categories turn out to be equivalences and hence representable by Fourier-Mukai functors, and the analysis on the level of derived categories gives us back a big amount of geometric information. My project will tackle one such instance, namely the investigation of some quotient singularities that are a generalization of the Kleinian singularities, and their resolutions of singularities.

Derived categories are a way to consider these geometric objects and translate much of the information about them into algebraic notions. While the derived category of a variety retains much of the information about the variety we started with, at the same time it allows us extra flexibility to work in an algebraic context. In the past two decades the field of derived categories has experienced an outpouring of activity as many classical algebraic geometry problems are solved passing through derived categories techniques.

One fundamental question about derived categories is about how the derived categories of two different geometric objects are related. Some of these relations might come from relations and symmetries between the two varieties, but there are also other kinds of relations between them, which are deeper and harder to understand:

1. First of all, it is important to understand what the maps (functors) between two derived categories are like. Many of these - but not all, as people used to think! - have a very pleasant and useful geometric description as "Fourier-Mukai functors". Part of my project will consist in analyzing and describing the "bad" maps that are not Fourier-Mukai functors, and how these arise naturally by deforming the "good" maps we know about.

2. Another relation between two derived categories, which will be investigated as part of my project, is given by a concept of "duality" at the categorical level. Describing this duality gives us a way to understand deeper relations between derived categories that haven't yet been discovered, and that will shed more light on the symmetries and behavior both at the level of derived categories and at the level of the geometric objects.

3. Finally, in some instances the relations between derived categories turn out to be equivalences and hence representable by Fourier-Mukai functors, and the analysis on the level of derived categories gives us back a big amount of geometric information. My project will tackle one such instance, namely the investigation of some quotient singularities that are a generalization of the Kleinian singularities, and their resolutions of singularities.

### Planned Impact

The main impact of this research proposal will be to increase the body of knowledge in pure mathematics in the UK. This proposal will build on the existing strengths in UK mathematics in algebraic geometry and noncommutative geometry, in particular in the field of derived categories; it will moreover complement and strengthen the existing algebraic expertise with general DG and A-infinity techniques.

The potential for knowledge transfer is high due to the intradisciplinary nature of this proposal, which connects algebraic geometry, homological algebra, noncommutative algebraic geometry and representation theory. A great occasion to foster this knowledge transfer is the workshop that I will organize in Edinburgh in the second year of my fellowship, further increasing the impact of this proposal.

My track record, outlined in my case for support, demonstrates my ability to carry on research in collaboration and independently, my intradisciplinary approach to mathematics, and my commitment to interact with a wide section of the mathematical community. This fellowship would provide the ideal framework to further increase my impact in the mathematical community in the UK and worldwide.

The potential for knowledge transfer is high due to the intradisciplinary nature of this proposal, which connects algebraic geometry, homological algebra, noncommutative algebraic geometry and representation theory. A great occasion to foster this knowledge transfer is the workshop that I will organize in Edinburgh in the second year of my fellowship, further increasing the impact of this proposal.

My track record, outlined in my case for support, demonstrates my ability to carry on research in collaboration and independently, my intradisciplinary approach to mathematics, and my commitment to interact with a wide section of the mathematical community. This fellowship would provide the ideal framework to further increase my impact in the mathematical community in the UK and worldwide.

## People |
## ORCID iD |

Alice Rizzardo (Principal Investigator / Fellow) |

### Publications

Rizzardo A
(2018)

*A note on non-unique enhancements*in Proceedings of the American Mathematical Society
Rizzardo A
(2019)

*An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves*in Inventiones mathematicae
Rizzardo
(2020)

*A <em>k</em>-linear triangulated category without a model*in Annals of Mathematics### Related Projects

Project Reference | Relationship | Related To | Start | End | Award Value |
---|---|---|---|---|---|

EP/N021649/1 | 03/10/2016 | 31/08/2017 | £252,375 | ||

EP/N021649/2 | Transfer | EP/N021649/1 | 01/09/2017 | 30/09/2019 | £182,326 |

Description | Work funded through this award substantially deepened our understanding of the behavior of derived categories and triangulated categories. It gave us insight about behaviour that used to be considered pathological: several results in different papers show that in general triangulated categories and functors between them are not as well behaved as people used to believe. |

Exploitation Route | There is a lot more to be done. Many of the original objectives have been achieved, but some research directions that I indicated in the original proposal are still in progress. |

Sectors | Education,Other |

Description | LMS Research Grant (Scheme 1) |

Amount | £2,000 (GBP) |

Funding ID | 11853 |

Organisation | London Mathematical Society |

Sector | Academic/University |

Country | United Kingdom |

Start | 09/2019 |

End | 09/2019 |

Description | Non-enhanceable functors and applications |

Organisation | University of Hasselt |

Country | Belgium |

Sector | Academic/University |

PI Contribution | I have visited Prof. Michel Van den Bergh several times at the University of Hasselt. Prof. Van den Bergh has also visited me at the University of Edinburgh. This collaboration started in 2013 and has continued through the years, partially thanks to my current grant. Together we have made major progress in investigating the relationship between derived/triangulated categories and their enhancements. |

Collaborator Contribution | Prof. Van den Bergh has contributed his substantial and wide-reaching expertise and introduced me to new methods in non-commutative geometry. He has also partially funded my visits to the University of Hasselt. |

Impact | Since the beginning of the grant, we completed one paper which has been accepted for publication: "A note on non-unique enhancements" (accepted in Proceedings of the American Mathematical Society, not yet published). One more paper has been completed (not yet accepted for publication) and is currently posted on the Arxiv: "A k-linear triangulated category without a model". |

Start Year | 2013 |