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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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