Hyperkähler manifolds

Winter Semester 2014/2015

Meet Tuesdays at 10:45 in the seminar room 1.411 on our floor (Haus 1 floor 4 room 11). Starting week 3 on the 28th of October.


Here are the final handwritten notes from the course.

Last updated: 29/1/2015


Talk 1 (Frank 28/10/2014)

  • definitions

  • Mukai flop etc

  • deformation theory

  • Beauville-Bogomolov form

Talk 2 (Frank 4/11/2014)

  • Beauville-Bogomolov form

  • local Torelli

  • Fujiki's Lefschetz theorems and cohomology results

Talk 3 (Niels 11/11/2014)

  • birationally equivalent implies deformation equivalent

  • non-separability of points implies birational

  • examples of the above and why Verbitsky's global Torelli is stated as it is

Talk 4 (Mike 18/11/2014)

  • Kahler cone

  • Projectivity from section 9 in Huybrecht's Habilitation.

Talk 5 (Mike 25/11/2014)

  • More on projectivity

Talk 6 (Michael 9/12/2014)

  • Even more on projectivity

Talk 7 (Gregor 27/1/2015)

  • Hilbert schemes of points: generalities on construction

  • on a K3, holomorphic symplectic structure (see Beauville)

Talk 8 (Michael 3/2/2015)

  • intro to moduli of sheaves

  • From Huybrechts and Lehn's book and Mukai's papers

Talk 9 (?? 10/2/2015)

  • ??


From here on we can see what everyone is interested in. Some options are:

  • Bayer-Macri: get a feeling for the geometry here

  • Charles and Markman on the standard conjectures for IHSMs

  • Markman and Mehrotra density of Hilbert schemes of points

  • LMMP for ihsms

  • Markman's lagrangian fibrations paper

  • Results on cones: Amerik and Verbitsky's paper or Markman and Yoshioka

  • O'Grady's examples


Various Papers

Amerik and Verbitsky ``Rational curves on hyperkahler manifolds''

Bayer and Macri ``MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations'' Has nice summary of results on hyperkahler stuff

Beauville ``Varietes Kahleriennes dont la premiere classe de Chern est nulle''

Charles and Markman ``The Standard Conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces''

Charles and Pacienza ``Families of rational curves on holomorphic symplectic varieties''

Huybrechts ``Compact hyperkahler manifolds: Basic results''

Huybrechts ``A global Torelli theorem for hyperkaehler manifolds (after Verbitsky)''

Markman ``Lagrangian fibrations of holomorphic-symplectic varieties of K3^[n]-type''

Markman and Mehrotra ``Hilbert schemes of K3 surfaces are dense in moduli''

Markman and Yoshioka ``A proof of the Kawamata-Morrison Cone Conjecture for holomorphic symplectic varieties of K3^[n] or generalized Kummer deformation type''

Mukai ``Symplectic structure of the moduli space of sheaves on an abelian or K3 surface''

Mukai ``Moduli of vector bundles on K3 surfaces, and symplectic manifolds''

Mukai ``On the moduli space of bundles on K3 surfaces I''

Namikawa ``Deformation theory of singular symplectic n-folds''

O'Grady ``A new six-dimensional irreducible symplectic variety''

O'Grady ``Desingularized moduli spaces of sheaves on a K3''

Yoshioka ``Irreducibility of moduli spaces of vector bundles on K3 surfaces''


Various Notes

Huybrechts ``Compact hyperkahler manifolds notes'' (see also book with Gross and Joyce)

Huybrechts ``Hyperkahler manifolds and sheaves''

Huybrechts ``Compact Hyperkahler manifolds'' Contains more details than the paper.

O'Grady ``Compact Hyperkahler manifolds: an introduction''