Deligne-Beilinson cohomology of the universal K3 surface

TL;DR

This paper develops Deligne-Beilinson cohomology theory for stacks, computes the 4th cohomology of universal polarized K3 surfaces, and confirms O'Grady's generalized Franchetta conjecture within this cohomological framework.

Contribution

It introduces Deligne-Beilinson cohomology for stacks and verifies the GFC for certain K3 surface families in this setting, extending previous Betti cohomology results.

Findings

Confirmed GFC in DB cohomology for K3 surfaces with A1 singularities.

Computed the 4th DB-cohomology group of universal polarized K3 surfaces.

Developed the theory of DB cohomology on Deligne-Mumford stacks.

Abstract

O'Grady's generalized Franchetta conjecture (GFC) is concerned with codimension 2 algebraic cycles on universal polarized K3 surfaces. In \cite{BL17}, this conjecture has been studied in the Betti cohomology groups. Following a suggestion of Voisin, we investigate this problem in the Deligne-Beilinson (DB) cohomology groups. In this paper, we develop the theory of Deligne-Beilinson cohomology groups on separated (smooth) Deligne-Mumford stacks. Using the automorphic cohomology group and Noether-Lefschetz theory, we compute the 4-th DB-cohomology group of universal oriented polarized K3 surfaces with at worst an $A_1$-singularity and show that GFC for such family holds in DB-cohomology. In particular, this confirms O'Grady's original conjecture in DB cohomology.