A Riemann-Roch theorem for flat bundles, with values in the algebraic Chern-Simons theory

Let $f: X \to S$ be flat morphism over an algebraically closed field $k$ with a relative normal crossings divisor $Y\subset X$, $(E, \nabla)$ be a bundle with a connection with log poles along $Y$ and curvature with values in $f^*\Omega^2_{k(S)}$. Then the Gau\ss-Manin sheaf $R^if_*(\Omega^*_{X/S}({\rm log} Y)\otimes E)$ carries a Gau\ss-Manin connection $GM^i(\nabla)$. We establish a Riemann-Roch formula relating the algebraic Chern-Simons invariants of $\nabla$, $GM^i(\nabla)$ and the top Chern class of $\Omega^1_{X/S}({\rm log}Y)$.