Birational Geometry of Hypersurfaces: Gargnano del Garda, Italy, 2018

Description

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. Front Matter ....Pages i-ix Front Matter ....Pages 1-1 Birational Invariants and Decomposition of the Diagonal (Claire Voisin)....Pages 3-71 Non rationalité stable sur les corps quelconques (Jean-Louis Colliot-Thélène)....Pages 73-110 Introduction to work of Hassett-Pirutka-Tschinkel and Schreieder (Jean-Louis Colliot-Thélène)....Pages 111-125 Front Matter ....Pages 127-127 The Rigidity Theorem of Fano–Segre–Iskovskikh–Manin–Pukhlikov–Corti–Cheltsov–de Fernex–Ein–Mustaţă–Zhuang (János Kollár)....Pages 129-164 Hodge Theory of Cubic Fourfolds, Their Fano Varieties, and Associated K3 Categories (Daniel Huybrechts)....Pages 165-198 Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces (Emanuele Macrì, Paolo Stellari)....Pages 199-265 Appendix: Introduction to Derived Categories of Coherent Sheaves (Andreas Hochenegger)....Pages 267-295 Back Matter ....Pages 297-297