Deligne-Beilinson cohomology of the universal K3 surface

Abstract

O’Grady’s generalised Franchetta conjecture (GFC) is concerned with codimension 2 algebraic cycles on universal polarised K3 surfaces. In [4], 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 (smooth) Deligne-Mumford stacks. Using the automorphic cohomology group and Noether-Lefschetz theory, we compute the 4th DB-cohomology group of universal oriented polarised 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.