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Stability and applications to birational and hyperkaehler geometry - Lecture 1
Post-edited
Authors : Bayer, Arend (Author of the conference)
CIRM (Publisher )
14D20 - Algebraic moduli problems, moduli of vector bundles
14E30 - Minimal model program (Mori theory, extremal rays)
14J28 - $K3$ surfaces and Enriques surfaces
18E30 - Derived categories, triangulated categories
CIRM (Publisher )
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| definition of stability conditions support property stability conditions on surfaces |
Abstract :
This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.
- Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland's notion of stability conditions on derived categories [2, 5, 6]. I will then proceed to explain the concept of wall-crossing, both in theory, and in examples [1, 2, 4, 6].
- Wall-crossing and birational geometry. Every moduli space of Bridgeland-stable objects comes equipped with a canonically defined nef line bundle. This systematically explains the connection between wall-crossing and birational geometry of moduli spaces. I will explain and illustrate the underlying construction [7].
- Applications : Moduli spaces of sheaves on $K3$ surfaces. I will explain how one can use the theory explained in the previous talk in order to systematically study the birational geometry of moduli spaces of sheaves, focussing on $K3$ surfaces [1, 8].
MSC Codes : - Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland's notion of stability conditions on derived categories [2, 5, 6]. I will then proceed to explain the concept of wall-crossing, both in theory, and in examples [1, 2, 4, 6].
- Wall-crossing and birational geometry. Every moduli space of Bridgeland-stable objects comes equipped with a canonically defined nef line bundle. This systematically explains the connection between wall-crossing and birational geometry of moduli spaces. I will explain and illustrate the underlying construction [7].
- Applications : Moduli spaces of sheaves on $K3$ surfaces. I will explain how one can use the theory explained in the previous talk in order to systematically study the birational geometry of moduli spaces of sheaves, focussing on $K3$ surfaces [1, 8].
14D20 - Algebraic moduli problems, moduli of vector bundles
14E30 - Minimal model program (Mori theory, extremal rays)
14J28 - $K3$ surfaces and Enriques surfaces
18E30 - Derived categories, triangulated categories
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 05/01/16 Conference Date : 24/11/15 Subseries : Research talks arXiv category : Algebraic Geometry Mathematical Area(s) : Algebraic & Complex Geometry ; Algebra Format : MP4 (.mp4) - HD Video Time : 00:44:48 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2015-11-24_Bayer_part1.mp4 
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Information on the Event
Event Title : Algebraic geometry and complex geometry / Géométrie algébrique et géométrie complexeEvent Organizers : Broustet, Amaël ; Pasquier, Boris Dates : 23/11/15 - 27/11/15 Event Year : 2015 Event URL : http://conferences.cirm-math.fr/1393.html
Citation Data
DOI : 10.24350/CIRM.V.18897603Cite this video as: Bayer, Arend (2015). Stability and applications to birational and hyperkaehler geometry - Lecture 1. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18897603 URI : http://dx.doi.org/10.24350/CIRM.V.18897603 |
See Also
- [Multi angle] Stability and applications to birational and hyperkaehler geometry - Lecture 3 / Author of the conference Bayer, Arend.
- [Multi angle] Stability and applications to birational and hyperkaehler geometry - Lecture 2 / Author of the conference Bayer, Arend.
Bibliography
- [1] Arcara, D., Bertram, A., Coskun, I., Huizenga, J. (2013). The minimal model program for the Hilbert scheme of points on $\mathbb{P}2$ and Bridgeland stability. Advances in Mathematics, 235, 580-626. < arXiv:1203.0316> - http://arxiv.org/abs/1203.0316
- [2] Bridgeland, T. (2007). Stability condition on triangulated categories. Annals of Mathematics. Second Series, 166(2), 317-345.
- [3] Bridgeland, T. (2009). Spaces of stability conditions. In D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande, & M. Thaddeus (Eds.), Algebraic geometry: Seattle 2005 (pp. 1-21). Providence, RI: American Mathematical Society. (Proceedings of Symposia in Pure Mathematics 80.1).
- [4] Bridgeland, T. (2008). Stability conditions on $K3$ surfaces. Duke Mathematical Journal, 141(2), 241-291.
- [5] Caldararu, A. (2005). Derived categories of sheaves : a skimming. In R. Vakil (Ed.), Snowbird lectures in algebraic geometry (pp. 43-75). Providence, RI: American Mathematical Society. (Contemporary Mathematics, 388).
- [6] Bayer, A. ”A tour to stability conditions on derived categories”, Informal notes available on my homepage - http://www.maths.ed.ac.uk/~abayer/dc-lecture-notes.pdf
- [7] Bayer, A., & Macri, E. (2014). Projectivity and birational geometry of Bridgeland moduli spaces. Journal of the American Mathematical Society, 27(3), 707-752.
- [8] Bayer, A., & Macri, E. (2014). MMP for moduli of sheaves on $K3$s via wall-crossing : nef and movable cones, Lagrangian fibrations. Inventiones Mathematicae, 198(3), 505-590.
