The Grassmannian complex and Goncharov’s motivic complex in weight 4

For a field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a given integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n&gt;1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , Goncharov (1995) has given a motivic complex <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Subscript upper F Baseline left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma _F(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , generalising the Bloch-Suslin complex (Bloch, 2000; Suslin, 1990) for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 2"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which he expects to rationally compute the weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> motivic cohomology of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p e c upper F"> <mml:semantics> <mml:mrow> <mml:mi>Spec</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Spec}F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , hence its algebraic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -groups in Adams weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and he was also led to—conjecturally quasiisomorphic—‘thickened’ complexes thereof. These complexes involve tensor products of higher polylogarithm groups, the latter having been linked to the geometry of certain configurations in Goncharov’s proof of Zagier’s Polylogarithm Conjecture for weight 3, and an analogous picture has long been envisioned by Goncharov (1995) for higher weight. We provide a partial morphism in weight 4 by giving three out of four maps for configurations in general position. We moreover give partial results for the leftmost square.