Higher direct images of ideal sheaves, correspondences in log Hodge cohomology and globally F-full varieties

Abstract

This document consists of three mathematically independent and more or less thematically independent parts. Chapter 1 concerns invariance of the cohomology groups of divisorial ideal sheaves under (a restricted class of) birational morphisms of pairs in arbitrary characteristic, and as an application extends some foundational results in the theory of rational pairs that were previously known only in characteristic 0. Chapter 2 discusses correspondences in logarithmic Hodge theory related to an as-of-yet-unsuccessful alternative strategy for proving the main theorems of Chapter 1. Chapter 3 introduces and studies a condition on a proper scheme over a field of positive characteristic defined in terms of the Frobenius action, which we call globally F-full.