On direct images of pluricanonical bundles

TL;DR

This paper develops new theoretical results on the properties of direct images of pluricanonical bundles, using advanced techniques from algebraic geometry, with implications for vanishing theorems and positivity properties.

Contribution

It introduces new vanishing and positivity results for direct images of pluricanonical bundles based on Kollár and Viehweg's methods, formulating them as Fujita conjecture-type statements.

Findings

Effective vanishing theorems established

Weak positivity of direct images demonstrated

Generic vanishing properties confirmed

Abstract

We show that techniques inspired by Koll\'ar and Viehweg's study of weak positivity, combined with vanishing theorems for log-canonical pairs, lead to new consequences regarding generation and vanishing properties for direct images of pluricanonical bundles. We formulate the strongest such results as Fujita conjecture-type statements, which are then shown to govern a range of fundamental properties of direct images of pluricanonical and pluriadjoint line bundles, like effective vanishing theorems, weak positivity, or generic vanishing.