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

1998/04/30 by Spencer Bloch, Hélène Esnault
Mathematics · #math.AG

paper · pdf

published as Ann. of Math. (2) 151 (2000), no. 3, 1025-1070 · 46 pages, published version

arxiv created 2000/05/01 · arxiv updated 2009/11/30

Abstract

Let f: X → S be flat morphism over an algebraically closed field k with a relative normal crossings divisor Y⊂ X, (E, ∇) be a bundle with a connection with log poles along Y and curvature with values in f^*Ω²_k(S). Then the Gauß-Manin sheaf Rⁱf_*(Ω^*_X/S(\rm log Y)⊗ E) carries a Gauß-Manin connection GMⁱ(∇). We establish a Riemann-Roch formula relating the algebraic Chern-Simons invariants of ∇, GMⁱ(∇) and the top Chern class of Ω¹_X/S(\rm logY).

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