The additive polylogarithm cycle boundary conjecture

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Let TM(n)T\mathcal M(n) denote the additive part of the degree-nn component of the indecomposable space, and let Zn(a)Z_n(a) be a codimension-nn cycle on

A1×(P1−{1})2n−2.\mathbb A^1\times (\mathbb P^1-\{1\})^{2n-2}.

For a∈A1−{0}a\in\mathbb A^1-\{0\}, write ⟨a⟩n\langle a\rangle_n for the corresponding element of TM(n)T\mathcal M(n), with ⟨a⟩1=a\langle a\rangle_1=a, and let {a}n−1\{a\}_{n-1} and {1−a}1\{1-a\}_1 denote the polylogarithm cycles introduced above. Additive polylogarithm cycle boundary conjecture. There exist elements ⟨a⟩n∈TM(n)\langle a\rangle_n\in T\mathcal M(n) represented by such cycles Zn(a)Z_n(a), satisfying

∂⟨a⟩n=⟨a⟩n−1⋅{1−a}1+⟨1−a⟩1⋅{a}n−1∈(⋀2M~)(n).\partial\langle a\rangle_n=\langle a\rangle_{n-1}\cdot\{1-a\}_1+\langle 1-a\rangle_1\cdot\{a\}_{n-1}\in\Big(\bigwedge^2\widetilde{\mathcal M}\Big)(n).

This proposes additive analogues of the polylogarithm cycles whose boundaries encode the expected coproduct relation. The supplied text gives no evidence that the construction or boundary identity has been proved, so its status remains open.

References

Primary source

Spencer Bloch and Hélène Esnault, “The additive dilogarithm”, arXiv:math/0210138 (2002).

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