Sharp de Rham realization

Details Sharp de Rham realization Sharp de Rham realization

2006-07-05

arxiv.org/abs/math/0607115v1

Advances in Mathematics Vol. 222 Issue 4 (2009) 1308-1338

Mathematics

Algebraic Geometry

30 pages

Scientific paper

10.1016/j.aim.2009.06.003

We introduce the "sharp" (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the "sharp de Rham realization" by passing to the Lie-algebra. Over the complex numbers we then show a (sharp de Rham) comparison theorem in the category of formal Hodge structures. For a free 1-motive along with its Cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations. We thus provide "one-dimensional sharp de Rham cohomology" of algebraic varieties.

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