A remark on an algebraic Riemann-Roch formula for flat bundles (Q1382118)

Let \(X\) be a smooth variety over an algebraically closed field, \({\mathcal K}_n\) the Zarisky sheaf image of Milnor \(K\)-theory in \(K^M_n(k(X))\), \((E,\nabla)\) a bundle over \(X\) with an integrable connection. Let further \(f:X\to S\) be a smooth projective morphism. In her former work, reviewed in section 1 of the paper, the author defined algebraic classes \(c_n(E,\nabla)\). The sheaves \(R^jf_* (\Omega^\bullet_{X/S} \otimes E,\nabla)\) carry the Gauss-Manin connection, therefore it is natural to ask for a formula relating the classes \(c_n(\sum(-1)^j (R^jf_* (\Omega^\bullet_{X/S}\otimes E,\nabla), \nabla))\) and \(c_n (E,\nabla)\). The author proves such a formula in case \(X=Y\times S\) and \(f\) is the projective map.