This work was done during authors’ visit to Kavli IPMU and was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. The reported study was partially supported by RFBR, Research Projects 13-01-00234, 14-01-00416 and 15-51-50045. The article was prepared within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program.

For a morphism of smooth schemes over a regular affine base we define functors of derived direct image and extraordinary inverse image on coderived categories of DG-modules over deRham DG-algebras. Positselski proved that for a smooth algebraic variety X over a field k of characteristic zero the coderived category of DGmodules over •X/k is equivalent to the unbounded derived category of quasi-coherent right DX -modules.We prove that our functors correspond to the functors of the same name for DX -modules under Positselski equivalence.

We give evidence for a uniformization-type conjecture, that any algebraic variety can be altered into a variety endowed with a tower of smooth fibrations of relative dimension one.

In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.