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Let be a field of characteristic zero, and let . We construct an additive dilogarithm , where is the Bloch group which is crucial in studying weight two motivic cohomology. We use this construction to show that the Bloch complex of has cohomology groups expressed in terms of the K-groups as expected. Finally we compare this construction to the construction of the additive dilogarithm by Bloch and Esnault defined on the complex .
The class of standard braided vector spaces, introduced by Andruskiewitsch and the author in 2007 to understand the proof of a theorem of Heckenberger, is slightly more general than the class of braided vector spaces of Cartan type. In the present paper, we classify standard braided vector spaces with finite-dimensional Nichols algebra. For any such braided vector space, we give a PBW basis, a closed formula of the dimension and a presentation by generators and relations of the associated Nichols algebra.
We discuss a characteristic free version of Frobenius splittings for toric varieties and give a polyhedral criterion for a toric variety to be diagonally split. We apply this criterion to show that section rings of nef line bundles on diagonally split toric varieties are normally presented and Koszul, and that Schubert varieties are not diagonally split in general.