Abundance-type problems for generalised pairs

Abstract: In algebraic geometry, abundance-type problems are those that
relate numerical properties of a certain Cartier divisor with properties
of the associated line bundle (such as global generation or the existence
of nonzero sections). In its most basic form, the abundance conjecture
(which is one of the central open problems in higher dimensional algebraic
geometry) says that if the canonical divisor of a smooth projective
variety is nef (i.e. it intersects every curve non-negatively) then some
multiple of the canonical line bundle is generated by global sections. In
recent years, several abundance-type conjectures have been proposed for
generalised pairs. Instead of dealing just with the canonical divisor K_X
of a smooth projective variety X, (roughly speaking) these concern
divisors of the form K_X+L where L is a nef divisor on X. In this talk, I
will discuss my recent results on generalised abundance , where I extend
some classical results of Kawamata and Ambro to the setting of generalised
pairs.