Birational Geometry, Iitaka Program, and Positivity of Canonical and Anticanonical Divisor

In birational geometry, one of the very interesting question is the Iitaka Program, that is, we want to "factorize" a given variety into "basic type" varieties. "Basic type" varieties are varieties of general type (canonincal divisor is ample), varieties of Calabi-Yau type (canonical divisor is "trivial"), and Fano type (anti-canonical divisor is ample).

The (anti)canonical divisor is one of the most important ingredients of (projective) algebraic varieties. Even if the canonical divisor or anticanonical divisor of a given variety is not ample, if it 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, it is also interesting to study the relation between the (anti)canonical divisor of the source space and the target space. In this talk, we will introduce some conjectures and known results around the positivity about varieties with positive (anti)canonical divisor in the few decades.