Theta-Regularity and Log-Canonical Threshold

Abstract

The log-canonical threshold is an invariant that is widely used in modern birational geometry. It contains information regarding the singularities of sheaves of ideals. The theta-regularity index is a regularity condition for coherent sheaves on principally polarized abelian varities, that in many ways is an analogue to the Castelnuovo-Mumford regularity index for projective spaces. Amongst other properties, theta-regularity contains information on when a coherent sheaf is generated by its global sections. The main result of this thesis is an inequality relating the log-canonical threshold and theta-regularity of non-trivial ideal sheaves on principally polarized abelian varieties.