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

Abstract

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 $f^{\ast }{\Omega }_{k(S)}^2$. Then the Gauß-Manin sheaf $R^i{f}_{\ast }({\Omega }_{X/S}^{\ast }(\log Y)\otimes E)$ carries a Gauß-Manin connection $G{M}^i(\nabla )$. We establish a Riemann-Roch formula relating the algebraic Chern-Simons invariants of $\nabla$, $G{M}^i(\nabla )$ and the top Chern class of ${\Omega }_{X/S}^1(\log Y)$.

46 pages, published version