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StabCondEn SIGNED

Stability Conditions, Moduli Spaces and Enhancements

Total Cost €

EC-Contrib. €

Partnership

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StabCondEn project word cloud

Explore the words cloud of the StabCondEn project. It provides you a very rough idea of what is the project "StabCondEn" about.

functors ones leaving semiorthogonal threefolds exist decompositions techniques space describing categories perfect canonicity interesting deep intricate yau category relations projective fourfolds admissible innovative rational introduce dg considering calabi geometry give corresponding showing enhancements complexes varieties hyperkaehler subcategories stable intend notion objects once uniqueness empty difficult exact algebraic prove theory curves prominently consisted bridgeland moduli big first fundamental stability spaces categorical questions triangulated parametrizing description foundational successful question normal conjectures completely provides solution cubic

Project "StabCondEn" data sheet

The following table provides information about the project.

Coordinator	UNIVERSITA DEGLI STUDI DI MILANO Organization address address: Via Festa Del Perdono 7 city: MILANO postcode: 20122 website: www.unimi.it contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.
Coordinator Country	Italy [IT]
Total cost	785˙866 €
EC max contribution	785˙866 € (100%)
Programme	1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
Code Call	ERC-2017-COG
Funding Scheme	ERC-COG
Starting year	2018
Duration (year-month-day)	from 2018-02-01 to 2023-01-31

Partnership

Take a look of project's partnership.

#	participants	country	role	EC contrib. [€]
1	UNIVERSITA DEGLI STUDI DI MILANO UNIVERSITA DEGLI STUDI DI MILANO Organization address address: Via Festa Del Perdono 7 city: MILANO postcode: 20122 website: www.unimi.it contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.	IT (MILANO)	coordinator	785˙866.00

Map

Project objective

I will introduce new techniques to address two big open questions in the theory of derived/triangulated categories and their many applications in algebraic geometry.

The first one concerns the theory of Bridgeland stability conditions, which provides a notion of stability for complexes in the derived category. The problem of showing that the space parametrizing stability conditions is non-empty is one of the most difficult and challenging ones. Once we know that such stability conditions exist, it remains to prove that the corresponding moduli spaces of stable objects have an interesting geometry (e.g. they are projective varieties). This is a deep and intricate problem.

On the more foundational side, the most successful approach to avoid the many problematic aspects of the theory of triangulated categories consisted in considering higher categorical enhancements of triangulated categories. On the one side, a big open question concerns the uniqueness and canonicity of these enhancements. On the other side, this approach does not give a solution to the problem of describing all exact functors, leaving this as a completely open question. We need a completely new and comprehensive approach to these fundamental questions.

I intend to address these two sets of problems in the following innovative long-term projects:

1. Develop a theory of stability conditions for semiorthogonal decompositions and its applications to moduli problems. The main applications concern cubic fourfolds, Calabi-Yau threefolds and Calabi-Yau categories.

2. Apply these new results to the study of moduli spaces of rational normal curves on cubic fourfolds and their deep relations to hyperkaehler geometry.

3. Investigate the uniqueness of dg enhancements for the category of perfect complexes and, most prominently, of admissible subcategories of derived categories.

4. Develop a new theory for an effective description of exact functors in order to prove some related conjectures.

Publications

List of publications.
year	authors and title	journal	last update
2018	Canonaco, Alberto; Ornaghi, Mattia; Stellari, Paolo Localizations of the category of $A_infty$ categories and internal Homs published pages: , ISSN: , DOI:	1	2019-10-01
2019	Bayer, Arend; Lahoz, MartÃ; MacrÃ¬, Emanuele; Nuer, Howard; Perry, Alexander; Stellari, Paolo Stability conditions in families published pages: , ISSN: , DOI:	1	2019-10-01
2019	MacrÃ¬, Emanuele; Stellari, Paolo Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces published pages: , ISSN: , DOI:	1	2019-10-01

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The information about "STABCONDEN" are provided by the European Opendata Portal: CORDIS opendata.