Variation of mixed Hodge structures and the positivity for algebraic fiber spaces

TL;DR

This paper discusses the variation of mixed Hodge structures and explores positivity properties in algebraic fiber spaces, providing new proofs and extending key theorems to broader contexts.

Contribution

It offers new, simplified proofs for fundamental theorems and extends decomposition results to differential forms and log cases in algebraic geometry.

Findings

New proofs for local freeness and semipositivity theorems

Extension of Kollár's decomposition theorem to differential forms

Log version results for boundary components

Abstract

These are the lecture notes based on earlier papers with some additional new results. New and simple proofs are given for local freeness theorem and the semipositivity theorem. A decomposition theorem for higher direct images of dualizing sheaves of Koll\'ar is extended to the sheaves of differential forms of arbitrary degrees in the case of a well prepared birational model. We will also prove the log versions of some of the results, i.e., the case where we allow horizontal boundary components.