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Note on tautological classes of moduli of \(K3\) surface. (English) Zbl 1124.14011
From the introduction: This note deals with a few properties of tautological classes on moduli spaces of \(K3\) surfaces. Let \({\mathcal M}_{2d}\) denote a moduli stack of \(K3\) surfaces over an algebraically closed field with a polarization of degree \(2d\) prime to the characteristic of the field. The Chern classes of the relative cotangent bundle \(\Omega^1_{{\mathcal X}/{\mathcal M}}\) of the universal \(K3\) surface \({\mathcal X}_{2d}={\mathcal X}\) define classes \(t_1\) and \(t_2\) in the Chow groups \(\text{CH}^i_{\mathbb{Q}}({\mathcal X}_{2d})\) of the universal \(K3\) surface over \({\mathcal M}_{2d}\). The class \(t_1\) is the pull-back from \({\mathcal M}_{2d}\) of the first Chern class \(v= c_1(V)\) of the Hodge line bundle \(V= \pi_*(\Omega^2_{{\mathcal X}/{\mathcal M}})\). We use Grothendieck-Riemann-Roch to determine the push-forwards of the powers of \(t_2\). These are powers of \(v\). We then prove that \(v^{18}= 0\) in the Chow group with rational coefficients of \({\mathcal M}_{2d}\). We show that this implies that a complete subvariety of \({\mathcal M}_{2d}\) has dimension at most 17 and that this bound is sharp. These results are in line with those for moduli of abelian varieties. There the top Chern class \(\lambda_g\) of the Hodge bundle vanishes in the Chow group with rational coefficients. The idea is that if the boundary of the Baily-Borel compactification has codimension \(r\), then some tautological class of codimension \(r\) vanishes. Our result means that \(v^{18}\) is a torsion class. It would be very interesting to determine the order of this class as well as explicit representations of this class as a cycle on the boundary [cf. T. Ekedahl and G. van der Geer, Acta Math. 192, No. 1, 95–109 (2004; Zbl 1061.14041) and Duke Math. J. 129, No. 1, 187–199 (2005; Zbl 1090.14002)].

MSC:
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14J28 \(K3\) surfaces and Enriques surfaces
14J10 Families, moduli, classification: algebraic theory
14D22 Fine and coarse moduli spaces
14K10 Algebraic moduli of abelian varieties, classification
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