Logarithmic Kodaira-Akizuki-Nakano vanishing and Arakelov-Parshin boundedness for singular varieties

TL;DR

This paper extends vanishing theorems to singular varieties and applies them to establish boundedness results for families of canonically polarized varieties with certain singularities.

Contribution

It proves a singular version of the logarithmic Kodaira-Akizuki-Nakano vanishing theorem and uses it to derive Arakelov-Parshin boundedness for specific singular families.

Findings

Established a singular logarithmic vanishing theorem.

Proved boundedness for families of canonically polarized varieties with Gorenstein singularities.

Extended classical vanishing results to singular settings.

Abstract

The article has two parts. The first part is devoted to proving a singular version of the logarithmic Kodaira-Akizuki-Nakano vanishing theorem of Esnault and Viehweg. This is then used to prove other vanishing theorems. In the second part these vanishing theorems are used to prove an Arakelov-Parshin type boundedness result for families of canonically polarized varieties with rational Gorenstein singularities.