# HODGE REALIZATION Informe resumido

Project ID:
41076

Financiado con arreglo a:
FP6-MOBILITY

País:
Germany

## Final Activity Report Summary - HODGE REALIZATION (Mixed Hodge realization of triangulated mixed motives and comparison of triangulated categories of motives)

In the 90s, V. Voevodsky gave the construction of a triangulated category of motives. This work together with constructions by M. Levine and M. Hanamura is the first step towards establishing on a firm ground the universal framework for cohomology of algebraic varieties as conjectured by A. Grothendieck. It has been proved by those mathematicians that their categories have many expected properties of the derived category of Grothendieck's conjectural category of mixed motives. Therefore each of them should provide a different description of the same conjectural category and ought to be equivalent. In characteristic zero M. Levine has proved that his construction and the one due to Voevodsky are equivalent. However M. Levine's proof does not extend to positive characteristic. Indeed, his construction relies on a result of Voevodsky giving a concrete description of the dual in Voevodsky's triangulated category of motives by explicit complexes of equidimensionnal algebraic cycles, a result which uses Hironaka's resolution of singularity.

During this Marie Curie action the fellow, using the local decomposition of transfers constructed in his Ph.D., has extended to any perfect field the result proved by M. Levine in positive characteristic. The method to construct equivalence in positive characteristic is to use the outline of M. Levine's argument, choosing another complex of Nisnevich sheaves with transfers to represent the fake twisted dual motive of a smooth variety X that works in all characteristics without assuming resolution of singularities. This argument simplifies and clarifies also some of the rather technical aspects of M. Levine's original proof.

During this Marie Curie action the fellow, using the local decomposition of transfers constructed in his Ph.D., has extended to any perfect field the result proved by M. Levine in positive characteristic. The method to construct equivalence in positive characteristic is to use the outline of M. Levine's argument, choosing another complex of Nisnevich sheaves with transfers to represent the fake twisted dual motive of a smooth variety X that works in all characteristics without assuming resolution of singularities. This argument simplifies and clarifies also some of the rather technical aspects of M. Levine's original proof.