Steenbrink isomorphism and crystals on tubular neighbourhoods

TL;DR

This paper constructs a canonical isomorphism linking log de Rham cohomology on special fibers to higher direct images in a semistable family, demonstrating their crystalline invariance and algebraic properties.

Contribution

It introduces a canonical section in derived categories that establishes a fundamental isomorphism, revealing the crystalline nature of higher direct images in semistable families.

Findings

Proves the locally freeness of higher direct images algebraically.

Establishes a canonical isomorphism for base change of log de Rham cohomology.

Shows the invariance and crystalline nature of higher direct images.

Abstract

For a locally nilpotent integrable connection on a proper (strict) semistable family over a small polydisc with a relative horizontal simple normal crossing divisor, we construct a canonical section in derived categories inducing an isomorphism from the log de Rham cohomology of it on the log special fiber of this family to the stalk of the higher direct image of it at the origin modulo the maximal ideal of the localization of the structure sheaf at the origin. As an application of the existence of this section, we prove the following: (1): the locally freeness of the higher direct image by a purely algebraic method; (2): the existence of a canonical isomorphism between the base change of the log de Rham cohomology of the log special fiber to the small polydisc and the higher direct image above. The result (2) tells us that the higher direct image has the crystalline nature: the invariance of the higher direct image.