On algorithms computing sheaf cohomology groups

Abstract

This paper is mostly a survey paper on algorithmic approachesto compute sheaf cohomology groups of a coherent sheaf on a projectivevariety. We review various computational methods together with an algorithmicpoint of view. We also review further generalizations to computehigher direct image sheaves of a coherent sheaf under a morphism betweentwo projective varieties. We also discuss a few open problems and workingideas on those computations and functorial properties.