Positivity and Vanishing Theorems

Existence theorems are a central part of algebraic geometry. These results frequently involve linear problems where positivity assumptions are used to prove existence of solutions by establishing the vanishing of obstructions to that existence. Beginning with Riemann (algebraic curves), Picard (algebraic surfaces) and continuing into more recent times (Lefschetz, Hodge, Kodaira-Spencer and many others since this work) it has come to be understood that the vanishing theorems are intimatedly related to the topology of algebraic varieties. What remains is the case that although the results are about algebraic varieties, analytic tools are needed to establish them. Moreover the property of positivity also appears in other aspects where analytic methods are needed, an example being the proof of the Iitaka conjecture which is central in the classification of algebraic varieties. The purpose of these lectures is to present, sometimes from an historical perspective, some of the principal aspects of the theory.