Lecturer: Olivier Debarre
K3 surfaces are interesting for many reasons: they have interesting dynamics (up to bimeromorphism, they are the only compact complex surfaces which can have an automorphism with positive entropy and no fixed points) and interesting arithmetic and geometric properties. Hyperkähler manifolds are analogues of K3 surfaces in higher (even) dimensions. In both cases, the Torelli theorem, recently proved by Verbitksy, allows one to read the automorphism group of the variety on the Picard lattice.
I will review classical results on (complex projective) K3 surfaces and discuss recent results on hyperkähler manifolds, such as the Torelli theorem (results of Verbitsky and Markman) and the determination of their ample cone in some cases (results of Bayer and Macrì).
References:
Beauville, Arnaud, Complex algebraic surfaces, translated from the French by R. Barlow, N. I. Shepherd-Barron and M. Reid, London Mathematical Society Lecture Note Series 6, Cambridge University Press, Cambridge, 1983.
Huybrechts, Daniel, Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics 15, Cambridge University Press, Cambridge, 2016.
Voisin, Claire, Hodge theory and complex algebraic geometry.\ I and II, Translated from the French original by Leila Schneps, Cambridge Studies in Advanced Mathematics 76-77, Cambridge University Press, Cambridge, 2002.