On the Hodge theory of toroidal embeddings and corresponding vanishings

TL;DR

This paper advances the understanding of Hodge theory in toroidal embeddings by proving key theorems and vanishing results, enhancing the mathematical framework for these complex geometric structures.

Contribution

It establishes Deligne's logarithmic comparison theorem and the $E_1$-degeneration for toroidal embeddings, along with proving Kawamata-Viehweg and Bott Vanishing theorems in this context.

Findings

Proved Deligne's logarithmic comparison theorem for toroidal embeddings.

Established $E_1$-degeneration of the Hodge-de Rham spectral sequence.

Proved Kawamata-Viehweg Vanishing for toroidal varieties and Bott Vanishing for toric varieties.

Abstract

In this paper, we establish Deligne's logarithmic comparison theorem and the $E_1$-degeneration of the corresponding Hodge-de Rham spectral sequence, in the setting of toroidal embeddings. Along the way, we prove Kawamata-Viehweg Vanishing and Bott Vanishing for toroidal varieties and toric varieties respectively.