Positivity has long been a major theme in various branches of algebraic geometry, both as an object of study and as a technical tool. the subject has seen major developments in a large number of different directions particulary in Vanishing theorems and their applications.
Vanishing theorems has been initiated by The medal Fields Kodaira and developped after by the work of Nakano,-Akizuki, Griffith's, Kawamata-Viehweg, LE Potier, Schneider , Peterenell , Claire Voisin, Lazarsfeld, Kollar, Mori, Demailly, Manivel and Nahm-laytimi.
This subject was the crucial tool in solving some conjectures like Fujita conjecture, Hartshorne conjecture, in Quadratic Normality, Barth Lefshetz theorem.
