# Degenerations of log Hodge de Rham spectral sequences, log Kodaira vanishing theorem in characteristic $p>0$ and log weak Lefschetz conjecture for log crystalline cohomologies

## TL;DR

This paper proves the degeneration of log Hodge de Rham spectral sequences and log Kodaira vanishing in characteristic p>0, and verifies the log weak Lefschetz conjecture for specific log crystalline cohomologies.

## Contribution

It establishes degeneration and vanishing results for log Hodge and Kodaira theories in positive characteristic, and confirms the log weak Lefschetz conjecture in particular cases.

## Key findings

- Log Hodge de Rham spectral sequences degenerate at E_1 for certain schemes.
- Log Kodaira vanishing holds for projective cases.
- The log weak Lefschetz conjecture is verified in specific instances.

## Abstract

In this article we prove that the log Hodge de Rham spectral sequences of certain proper log smooth schemes of Cartier type in characteristic $p>0$ degenerate at $E_1$. We also prove that the log Kodaira vanishings for them hold when they are projective. We formulate the log weak Lefschetz conjecture for log crystalline cohomologies and prove that it is true in certain cases.