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Deligne-Beilinson cohomology of the universal \(K3\) surface. (English) Zbl 1498.14101

On the moduli space \(\mathscr{F}_g^\circ\) of primitively polarised \(K3\) surfaces of genus \(g\) without automorphisms there is a universal family \(\pi\colon \mathscr{S}_g^\circ\to \mathscr{F}_g^\circ\). A famous result of Beauville and Voisin says that any \(K3\) surface \(S\) carries a canonical class \(c_S\in {\operatorname{CH}}_0(S)\) that generates both the intersection class of any two divisor classes and the Chern class \(c_2\). The generalised Franchetta conjecture, posited by O’Grady, is a generic relative version of this and states that if \(\mathcal{T}_\pi\) is the relative tangent bundle and \(\alpha\in {\operatorname{CH}}^2(\mathscr{S}_g^\circ)\) then there is an \(m\in \mathbb{Q}\) such that \(\alpha-mc_2(\mathcal{T}_\pi)\) is supported on a proper subset of \(\mathscr{F}_g^\circ\).
This is known for some small values of \(g\) and it is also known if one replaces the Chow groups by cohomology. In this paper, the authors move a bit closer to the full (Chow) version of the conjecture by replacing cohomology with Deligne-Beilinson cohomology \(H_{DB}\). This is an improvement because the cycle class map to cohomology factors through the Deligne-Beilinson cycle class map.
The precise statement is that \(\operatorname{cl}_{DB}(\alpha-mc_2(\mathcal{T}_\pi))\in H^4_{DB}(\mathscr{S}_g^\circ,\mathbb{Q}(2))\) restricts to zero on \(\pi^{-1}(V)\) for some open \(V\subset \mathscr{F}_g^\circ\). In view of the known cohomological version the proof reduces to a study of the kernel of \(H^4_{DB}(\mathscr{S}_g^\circ,\mathbb{Q}(2))\to H^4(\mathscr{S}_g^\circ,\mathbb{Q}(2))\). The technical difficulty is that while results of this kind are known over \(\mathscr{F}_g\) the cohomology of \(\mathscr{F}_g^\circ\) is hard to access. Nevertheless the authors are able to compute enough information for their purposes by passing to a suitable double cover of \(\mathscr{S}\) and restricting first to the complement of certain Noether-Lefschetz loci.

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14C25 Algebraic cycles

Citations:

Zbl 1498.14098
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References:

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