Injectivity theorems and cubical descent for schemes, stacks, and analytic spaces

TL;DR

This paper extends key theorems like injectivity, torsion-freeness, and vanishing to a broad range of geometric spaces, using cubical hyperresolutions, thereby advancing the minimal model program and Deligne-Hodge theory.

Contribution

It provides a uniform proof of fundamental theorems across various categories of spaces and introduces a general extension criterion using cubical hyperresolutions.

Findings

Proved injectivity, torsion-freeness, and vanishing theorems in diverse geometric categories.

Developed a general extension criterion for functors using cubical hyperresolutions.

Established foundations for Deligne-Du Bois complexes and weight filtrations in new settings.

Abstract

We prove relative injectivity, torsion-freeness, and vanishing theorems for generalized normal crossing pairs on schemes, algebraic stacks, formal schemes, semianalytic germs of complex analytic spaces, rigid analytic spaces, Berkovich spaces, and adic spaces locally of weakly finite type over a field, all in equal characteristic zero. We give a uniform proof for all these theorems in all the categories of spaces mentioned above, which were previously only known for varieties and complex analytic spaces due to work of Ambro and Fujino. Ambro and Fujino's results are integral in the proofs of the fundamental theorems of the minimal model program for (semi-)log canonical pairs and the theory of quasi-log structures. Our results resolve a significant barrier to extending these results on (semi-)log canonical pairs and quasi-log structures beyond the setting of varieties and complex analytic spaces. In order to prove our most general injectivity theorems, we generalize to all these categories of spaces a criterion due to Guill\'en and Navarro Aznar characterizing when functors defined on smooth varieties extend to all varieties. This extension result uses cubical hyperresolutions, which we construct in all categories of spaces mentioned above. Our extension result is very general and is of independent interest. We use this extension result to prove our injectivity theorems for generalized normal crossing pairs. We also apply our extension result to develop the theoretical foundations for the Deligne-Du Bois complex in these categories of spaces and to construct a weight filtration on the (pro-)\'etale cohomology of schemes and rigid analytic spaces. These results establish some aspects of Deligne-Hodge theory in all categories of spaces mentioned above.