Arnaud Beauville (Université de Nice)
Slides: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5.
Lecture 1: Manifolds with c1=0
The decomposition theorem: idea of proof. Calabi-Yau manifolds; holomorphic symplectic manifolds.
Lecture 2: The classical examples
The Hilbert scheme S[r] of a K3 surface, the generalized Kummer variety Kr, their deformations.
Lecture 3: The period map
Definition. The standard quadratic form and its properties; the Fujiki constant. Discussion of the surjectivity and injectivity of the period map.
Lecture 4: Birational symplectic manifolds
Mukai's symplectic flops. Huybrechts' theorem.
Lecture 5: Further developments
Cohomology: the Bogomolov-Verbitzky theorem. Restrictions on symplectic
fourfolds. Lagrangian fibrations, etc.
Short list of references (more during the school):
• D. Huybrechts, Compact hyperkähler manifolds, in: Calabi-Yau manifolds and related geometries (Nordfjordeid, 2001), 161-225, Universitext, Springer, Berlin (2003).
• A. Beauville, Riemannian Holonomy and Algebraic Geometry . Enseign. Math 53 (2007), 97-126. It can be downloaded as [81] from here.
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Tom Bridgeland (University of Sheffield)
Slides: Lectures 1 and 2 , Lectures 3 and 4.
Lecture 1: Derived and triangulated categories I
Lecture 2: Derived and triangulated categories II
Lecture 3: t-structures and tilting
Lecture 4: Stability conditions
Lecture 5: Stability conditions on K3s
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Daniel Huybrechts (Universität Bonn)
Slides: go to the web page.
Lecture 1: Generalized K3s
Lecture 2: Fourier-Mukai functors, Seidel-Thomas twists
Lecture 3: Derived Torelli
Lecture 4: Autoequivalences of K3s
Lecture 5: Some comments on mirror symmetry for K3s
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Kieran O'Grady (Università di Roma “La Sapienza”)
Lecture notes: Lectures 1,2 and 3 , Lectures 4 and 5.
Lecture 1: A tour through K3 surfaces
(1) Definition of a K3. Projective K3's of low degree, Kummer surfaces, K3 surfaces with du Val singularities.
(2) Cohomology of a K3, local period map and uniqueness of deformation class.
(3) Global period map for K3 surfaces. The Torelli Theorem and surjectivity of the period map. Moduli of projective K3's.
Lecture 2: First examples of higher dimensional irreducible symplectic manifolds
(1) The Beauville-Bogomolov quadratic form and Fujiki's constant of irreducible symplectic manifolds of the standard series.
(2) The Fano variety of lines on a cubic 4-fold.
Lecture 3: Moduli of sheaves on a projective Calabi-Yau surface, 1
(1) Smoothness of the moduli space. The Mukai lattice. A symplectic form on the moduli space.
(2) The case when the polarization is suitable (Yoshioka et alia). Explicit examples.
Lecture 4: Moduli of sheaves on a projective Calabi-Yau surface, 2
(1) The case when the polarization is not suitable. New irreducible symplectic varieties in dimensions 6 and 10; computation of b2, the Beauville-Bogomolov quadratic form and Fujiki's constant.
(2) The case when the polarization is not suitable: negative results of Lehn-Sorger.
Lecture 5: Double EPW-sextics: moduli and periods
(1) Irreducible symplectic 4-folds numerically equivalent to (K3)[2]. A plan for proving uniqueness of deformation class and a birational global Torelli Theorem.
(2) Double EPW-sextics and their periods.
