Abstract:

In birational geometry, the (anti)canonical divisor is one of the most important ingredients of (projective) algebraic varieties. For example, if either the canonical divisor or anticanonical divisor of a given variety is "positive" in some sense, then the positivity of the (anti)canonical divisor will provide us with important information about the geometry structure of the variety. On the other hand, given a morphism  f:X→Y, it is also interesting to study the relation between the (anti)canonical divisor of X and Y. In this talk, we will give a brief survey about some conjectures and known results around the positivity about varieties with positive (anti)canonical divisor in the few decades.