# Good lattices of algebraic connections

## TL;DR

This paper constructs a logarithmic model of algebraic connections on smooth varieties, utilizing good compactifications and lattices stabilized by log differential operators, and relates it to de Rham cohomology and characteristic classes.

## Contribution

It introduces a new method to build good logarithmic models of connections using good compactifications and lattices, extending the theory of Deligne-Malgrange lattices.

## Key findings

- Constructed a logarithmic model of connections on smooth varieties.
- Proved the model computes algebraic de Rham cohomology.
- Derived a formula for the characteristic variety class in K-theory.

## Abstract

We construct a logarithmic model of connections on smooth quasi-projective $n$-dimensional geometrically irreducible varieties defined over an algebraically closed field of characteristic $0$. It consists of a good compactification of the variety together with $(n+1)$ lattices on it which are stabilized by log differential operators, and compute algebraically de Rham cohomology. The construction is derived from the existence of good Deligne-Malgrange lattices, a theorem of Kedlaya and Mochizuki which consists first in eliminating the turning points. Moreover, we show that a logarithmic model obtained in this way, called a good model, yields a formula predicted by Michael Groechenig, computing the class of the characteristic variety of the underlying D-module in the $K$-theory group of the variety.