Geodesic
Deligne-Beilinson cohomology of the universal K3 surface
Forum of Mathematics, Sigma, Tome 10 (2022)

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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. 
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     author = {Zhiyuan Li and Xun Zhang},
     title = {Deligne-Beilinson cohomology of the universal {K3} surface},
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Zhiyuan Li; Xun Zhang. Deligne-Beilinson cohomology of the universal K3 surface. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.60

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