  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 $\Omega^1_{X/S}({\rm
log}Y)$.
