A Perturbation of the de Rham Complex

We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings

Hodge theorem for the logarithmic de Rham complex via derived intersections

In a beautiful paper, Deligne and Illusie (Invent Math 89(2):247–270, 1987) proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. Kato (in: Igusa (ed) ALG analysis, geographic and numbers theory, Johns Hopkins University Press, Baltimore, 1989) generalized this result to logarithmic schemes. In this paper, we use the theory of twisted derived intersections developed in Arinkin et al. (Algebraic Geom 4:394–423, 2017) and the author of this paper to give a new, geometric interpretation of the Hodge theorem for the logarithmic de Rham complex.