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Description
An ICTP Virtual Meeting

Hyperkähler manifolds are a class of manifolds with vanishing first Chern class, constituting an active area of current research.  Another fundamental problem concerns moduli space of polarized abelian varieties, studied via Prym varieties and the Prym map to the moduli of abelian varieties.
 
Hyperkähler manifolds, an overview and some open problems
 
Hyperkähler manifolds are mainly characterized by their second cohomology. The period maps from the moduli spaces of hyperkähler manifolds to the period domains of their second cohomology are surjective, which is a rare phenomenon happening almost exclusively in weight 1 and weight 2.
The course will give an introduction to hyperkähler manifolds and their known examples, survey some of the known results, and present some open problems. In the examples, interesting connections between hyperkähler manifolds, Fano manifolds and abelian varieties will be shown.
 
Prym varieties
 
The course we will review the classical theory and recent advances on Prym varieties and the Prym map, with special focus on the low genera cases which display beautiful geometry. The moduli aspect and the appearances of Prym varieties in other mathematical contexts will also be discussed.
 
Topics:
  • Characterization of hyperkähler manifolds via their cohomology
  • Period maps from the moduli spaces of hyperkähler manifolds to the period domains of their second cohomology
  • Connections between hyperkähler manifolds, Fano manifolds and abelian varieties
  • Fano manifolds
  • Abelian varieties and Polarized abelian varieties
  • Prym varieties and the Prym map: classical theory and applications

Speakers:
E. IZADI, University of California, San Diego, USA
A. ORTEGA, Humboldt-Universität zu Berlin, Germany

Additional:
Contributed talks: Participants interested in giving a short communication are invited to submit an abstract.

Registration: There is no registration fee.

Material

Trieste Algebraic Geometry Summer School (TAGSS) 2021 - Hyperkähler and Prym varieties: classical and new results | (smr 3609)
Go to day
  • Monday, 19 July 2021
    • 13:45 - 14:00 Opening remarks 
      Time zone: You can choose your time zone (in place of of Europe/Rome) on the top right of the page and the programme time will be modified accordingly.
      Convener: Lothar Goettsche (ICTP, Italy)
    • 14:00 - 19:10
      • 14:00 Generalities about abelian varieties 45'
        Speaker: Angela ORTEGA (Berlin Humboldt University, Germany)
        Material: Papers Slides Video
      • 14:45 Break 5'
      • 14:50 Generalities about abelian varieties 45'
        Speaker: Angela ORTEGA (Berlin Humboldt University, Germany)
        Material: Video
      • 15:35 Gather Town Break 30'
      • 16:05 Hyperkähler manifolds, an introduction I 45'
        Speaker: Elham IZADI (UCSD, USA)
        Material: References Slides Video
      • 16:50 Break 5'
      • 16:55 Hyperkähler manifolds, an introduction I 45'
        Speaker: Elham IZADI (UCSD, USA)
        Material: Video
      • 17:40 Gather Town Break 30'
      • 18:10 Prof. Izadi's exercise session 1h0' 
        Tutor: Samir CANNING, University of California, USA
        Material: Video
  • Tuesday, 20 July 2021
    • 10:00 - 14:00
      • 10:00 On Ramified Prym Maps and Their Fibers 30'
        Speaker: Irene SPELTA (Università degli Studi di Pavia, Italy)
        Material: Video
      • 10:30 Hitchin fibration and Prym varieties 30'
        Speaker: Sumit ROY (Tata Institute of Fundamental Research, India)
        Material: Video
      • 11:00 Gather Town Break 30'
      • 11:30 Prof. Ortega's exercise session 1h0' 
        Tutor: Pawel Stefan BOROWKA, Jagiellonian University in Kraków, Poland
        Material: Exercises Video
      • 12:30 Gather Town Lunch Break 1h30'
    • 14:00 - 19:10
      • 14:00 Prym varieties and Prym map 45'
        Speaker: Angela ORTEGA (Berlin Humboldt University, Germany)
        Material: Slides Video
      • 14:45 Break 5'
      • 14:50 Prym varieties and Prym map 45'
        Speaker: Angela ORTEGA (Berlin Humboldt University, Germany)
        Material: Slides Video
      • 15:35 Gather Town Break 30'
      • 16:05 Hyperkähler manifolds, an introduction II 45'
        Speaker: Elham IZADI (UCSD, USA)
        Material: Video
      • 16:50 Group photo 5'
      • 16:55 Hyperkähler manifolds, an introduction II 45'
        Speaker: Elham IZADI (UCSD, USA)
        Material: Video
      • 17:40 Gather Town Break 30'
      • 18:10 Tropical Prym varieties 30'
        Speaker: Yoav LEN (University of St Andrews, UK)
        Material: Video
      • 18:40 Geometry of Prym semicanonical pencils 30'
        Speaker: Andres ROJAS (University of Barcelona, Spain)
        Material: Video
  • Wednesday, 21 July 2021
    • 10:00 - 14:00
      • 10:00 Kirchhoff’s theorem and semi-canonical representatives for the tropical Prym variety 30'
        Speaker: Dmitry ZAKHAROV (Central Michigan University, USA)
        Material: Video
      • 10:30 Involutions of a Cubic Fourfold 30'
        Speaker: Lisa MARQUAND (Stony Brook University, USA)
        Material: Video
      • 11:00 Gather Town Break 30'
      • 11:30 Prof. Ortega's exercise session 1h0' 
        Tutor: Pawel Stefan BOROWKA, Jagiellonian University in Kraków, Poland
        Material: Exercises Video
      • 12:30 Gather Town Lunch Break 1h30'
    • 14:00 - 19:10
      • 14:00 Degree computation of the Prym map of étale double coverings over genus 6 curves 45'
        Speaker: Angela ORTEGA (Berlin Humboldt University, Germany)
        Material: Slides Video
      • 14:45 Break 5'
      • 14:50 Degree computation of the Prym map of étale double coverings over genus 6 curves 45'
        Speaker: Angela ORTEGA (Berlin Humboldt University, Germany)
        Material: Slides Video
      • 15:35 Gather Town Break 30'
      • 16:05 Hyperkähler manifolds, an introduction III 45'
        Speaker: Elham IZADI (UCSD, USA)
        Material: Video
      • 16:50 Break 5'
      • 16:55 Hyperkähler manifolds, an introduction III 45'
        Speaker: Elham IZADI (UCSD, USA)
        Material: Video
      • 17:40 Gather Town Break 30'
      • 18:10 Prof. Izadi's exercise session 1h0' 
        Tutor: Yajnaseni DUTTA, Hausdorff Center for Mathematics, Germany
        Material: Exercise
  • Thursday, 22 July 2021
    • 14:00 - 19:10
      • 14:00 Degree computation of the Prym map of triple cyclic coverings over genus 4 curves 45'
        Speaker: Angela ORTEGA (Berlin Humboldt University, Germany)
        Material: Slides Video
      • 14:45 Break 5'
      • 14:50 Degree computation of the Prym map of triple cyclic coverings over genus 4 curves 45'
        Speaker: Angela ORTEGA (Berlin Humboldt University, Germany)
        Material: Slides Video
      • 15:35 Gather Town Break 30'
      • 16:05 Hyperkähler manifolds, an introduction IV 45'
        Speaker: Elham IZADI (UCSD, USA)
        Material: Video
      • 16:50 Break 5'
      • 16:55 Hyperkähler manifolds, an introduction IV 45'
        Speaker: Elham IZADI (UCSD, USA)
        Material: Video
      • 17:40 Gather Town Break 30'
      • 18:10 Prof. Izadi's exercise session 1h0' 
        Tutor: David STAPLETON, University of California, USA
        Material: Exercise
  • Friday, 23 July 2021
    • 11:30 - 14:00
      • 11:30 Prof. Ortega's exercise session 1h0' 
        Tutor: Pawel Stefan BOROWKA, Jagiellonian University in Kraków, Poland
        Material: Exercises
      • 12:30 Gather Town Lunch Break 1h30'
    • 14:00 - 18:10
      • 14:00 Overview of finite Prym maps 45'
        Speaker: Angela ORTEGA (Berlin Humboldt University, Germany)
        Material: Slides Video
      • 14:45 Break 5'
      • 14:50 Overview of finite Prym maps 45'
        Speaker: Angela ORTEGA (Berlin Humboldt University, Germany)
        Material: Slides Video
      • 15:35 Gather Town Break 30'
      • 16:05 Hyperkähler manifolds, an introduction V 45'
        Speaker: Elham IZADI (UCSD, USA)
        Material: Video
      • 16:50 Break 5'
      • 16:55 Hyperkähler manifolds, an introduction V 45'
        Speaker: Elham IZADI (UCSD, USA)
        Material: Video
      • 17:40 Gather Town final greeting 30'
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