Cohomology of torus bundles over Kuga fiber varieties
In this paper the author constructs a torus bundle over a Kuga fiber variety and computes its cohomology groups. Let \(G\) be a semi-simple Lie group over \(\mathbb Q\) of Hermitian type and \(D\) its associated Hermitian symmetric domain. Assume that there is a homomorphism \(\rho: G\to \text{Sp}(V,\alpha)\) defined over \(\mathbb Q\) and a holomorphic map \(\tau\) from \(D\) to a Siegel upper space \(\mathcal H_n\) which is equivariant with respect to \(\rho\). Let \(\Gamma\subset G\) be a co-compact arithmetic subgroup so that there is a torsion free arithmetic subgroup \(\Gamma_0\subset \text{Sp}(V,\alpha)\) with \(\rho(\Gamma)\subset \Gamma_0\). Put \(X:=\Gamma\backslash D\) and let \(Y:=\tau^* \tilde {\mathcal A}\), the pull-back of the universal family \( \tilde {\mathcal A}\to \Gamma_0\backslash \mathcal H_n\). Fix a real vector space \(U\) over \(\mathbb Q\) and an alternating pairing \(A:V\times V\to U\) defined over \(\mathbb Q\) so that \(\rho(G)\) perserves the pairing \(A\). One defines a group structure on \(G\times V\times U\) which is an extension of the Heisenberg group \(\mathbb H=V\times U\) assocaited to \(A\) by the group \(G\). The group \(G\times V\times U\) acts on \(D\times V\times U\) through the left translation. One extends the action to \(D\times V\times U_{\mathbb C}\), where \( U_{\mathbb C}:=U\otimes {\mathbb C}\). Let \(L_{\mathbb H}\subset \mathbb H\) be an arithmetic subgroup and let \(L:=p_V(L_{\mathbb H})\). Put \(\mathcal T:=\Gamma \cdot L_{\mathbb H} \backslash D\times V\times U_{\mathbb C}\) and \(Y:=\Gamma \ltimes L \backslash D\times V\). The main result (Theorem 5.2) states that \[ H^r(\mathcal T, \mathbb C)=\bigoplus_{p+q=r} H^p(Y,\mathbb C)\otimes \wedge ^q (U_{\mathbb C}). \] In the paper the author defines everything needed and carries all even elementary comuptations. It is well-written and is very friendly to read.
scientific article; zbMATH DE number 5345498