Originating from the School on Birational Geometry of Hypersurfaces, this volume focuses on the notion of (stable) rationality of projective varieties and, more specifically, hypersurfaces in projective spaces, and provides a large number of open questions, techniques and spectacular results.

The aim of the school was to shed light on this vast area of research by concentrating on two main aspects: (1) Approaches focusing on (stable) rationality using deformation theory and Chow-theoretic tools like decomposition of the diagonal; (2) The connection between K3 surfaces, hyperkähler geometry and cubic fourfolds, which has both a Hodge-theoretic and a homological side.

Featuring the beautiful lectures given at the school by Jean-Louis Colliot-Thélène, Daniel Huybrechts, Emanuele Macrì, and Claire Voisin, the volume also includes additional notes by János Kollár and an appendix by Andreas Hochenegger.

Describes the most intriguing Hodge-theoretic aspects of cubic fourfolds Presents well-written surveys by leading experts on recent developments on rationality questions for hypersurfaces Provides a comprehensive and state-of-the-art introduction to the new and exciting subject of non-commutative K3 surfaces

Inhalt

• Part I Birational Invariants and (Stable) Rationality . - Birational Invariants and Decomposition of the Diagonal. - Non rationalité stable sur les corps quelconques. - Introduction to work of Hassett-Pirutka-Tschinkel and Schreieder. - Part II Hypersurfaces. - The Rigidity Theorem of FanoSegreIskovskikhManinPukhlikovCortiCheltsovdeFernexEinMustaZhuang. - Hodge Theory of Cubic Fourfolds, Their Fano Varieties, and Associated K3 Categories. - Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces. - Appendix: Introduction to Derived Categories of Coherent Sheaves.