Pushforward of Siegel flag varieties in the Chow ring

Abstract

Given a reductive group G over an algebraically closed field and subsets $I,J\subset \Delta$ of the simple roots Δ determined by a choice of maximal torus and Borel subgroup, there is a closed embedding of flag varieties $L_J/L_J\cap P_I\hookrightarrow G/P_I$. In this paper we compute the class of the sub flag variety $[L_J/L_J\cap P_I]\in A^{•}(G/P_I)$ in the Chow ring for the ‘Siegel’ case where G is a general symplectic group of semisimple rank g and $P_I$ is the parabolic stabilising a maximal isotropic subspace. This corresponds, under the isomorphism with the tautological ring of the moduli space of principally polarised abelian varieties $T^{•}\left(A_g^{\mathrm{tor}}\right)\cong A^{•}(G/P_I)$, to the generator of the classes in the tautological ring which are supported on the toroidal boundary. This provides basic evidence for a conjecture describing the tautological ring of a Hodge-type Shimura variety.