In the study of the geometry of projective varieties, the theory of big and pseudo-effective line bundles plays a prominent role. It has long been known that there is a satisfactory theory of volumes of such line bundles. When the line bundles are equipped with singular pluri-subharmonic metrics, Tsuji initiated the study of (partial) volumes. In this talk, I will explain the recent progress in this field. As applications, I will explain the construction of partial Okounkov bodies associated to Hermitian pseudo-effective line bundles. The first part is a joint work with Tamás Darvas.

Mingchen Xia (Department of Mathematics, Chalmers Tekniska Högskola, Göteborg)