We define a regulator map from the weight n polylogarithmic motivic complex to the weight n Deligne complex of an algebraic variety X. The regulator map is constructed explicitly via the classical polylogarithms with some funny combinations of Bernoulli numbers as coefficients. This leads to conjectures on special values of L-functions at s=n which reduce to Zagier's conjecture when X is of dimension zero over Q.
The cone of this map, shifted by one, should be called the Arakelov motivic complex of X. Its last cohomology group is identified with a group of codimension n Arakelov cycles on X.
We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti--de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. Th…
We construct an explicit regulator map from the weigh n Bloch Higher Chow group complexto the weight n Deligne complex of a regular complex projective algebraic variety X. We define the Arakelovweight n motivic complex as the cone of this map shifted by one. Its last cohomology group is (a version of) the Arakelov Chow group defined by H. Gillet. and C.Soule. We relate the Grassmannian n-logarithms (defined as in [G5]) to geometry of the symmetric space for GL_n(C). For n=2 we recover Lobachevsky's formula for the volume of an ideal geodesic tetrahedron via the dilogarithm. Using the relations…
The integral $t$-motivic cohomology and the class module of a (rigid analytically trivial) Anderson $t$-motive were introduced by the first author in [Gaz22b]. This paper is devoted to their determination in the particular case of tensor powers of the Carlitz $t$-motive, namely, the function field counterpart $\underline{A}(n)$ of Tate twists $\mathbb{Z}(n)$. We find out that these modules are in relation with fundamental objects of function field arithmetic: integral $t$-motivic cohomology governs linear relations among Carlitz polylogarithms, its torsion is expressed in terms of the denomina…
We show that the constructions done in part I generalize their classical counterparts: firstly, the classical Beilinson regulator is induced by the abstract Chern class map from $BGL$ to the Deligne cohomology spectrum. Secondly, Arakelov motivic cohomology is a generalization of arithmetic $K$-theory and arithmetic Chow groups. Finally, we give a conceptual explanation of the height pairing: it is given by the natural pairing of motivic homology and Arakelov motivic cohomology.
We prove that the Beilinson regulator, which is a map from $K$-theory to absolute Hodge cohomology of a smooth variety, admits a refinement to a map of $E_\infty$-ring spectra in the sense of algebraic topology. To this end we exhibit absolute Hodge cohomology as the cohomology of a commutative differential graded algebra over $\mathbb{R}$. The associated spectrum to this CDGA is the target of the refinement of the regulator and the usual $K$-theory spectrum is the source. To prove this result we compute the space of maps from the motivic $K$-theory spectrum to the motivic spectrum that repres…
In this paper we construct extensions of the Mixed Hodge structure on the fundamental group of a pointed algebraic curve. These extensions correspond to the regulator of certain explicit motivic cohomology cycles in the self product of the curve which were first constructed by Bloch and Beilinson. This leads to a new iterated integral expression for the regulator. Our result is a generalization of a result of Colombo's where she constructs the extension corresponding to a motivic cycle class in the Jacobian of a hyperelliptic curve constructed by Collino. This is to appear in the Mathematical …
Let $X$ be a variety defined over a local field $K$ of mixed characteristic $(0,p)$ with a totally degenerate reduction in the sense of Raskind and Xarles. Generalizing earlier work of Raskind and Xarles and relying on some conjectures we define a map, which we call the toric regulator, from the various motivic cohomology groups of $X$ to certain $p$-adically uniformized tori over $K$. This construction captures the part of the étale regulators on $X$ that land in the Galois cohomology of the submodules of cohomology which are extensions of $\mathbb{Z}_l$ by $\mathbb{Z}_l(1)$, simultaneously f…
We review and simplify A. Beilinson's construction of a basis for the motivic cohomology of a point over a cyclotomic field, then promote the basis elements to higher Chow cycles and evaluate the KLM regulator map on them.
A main theme of the paper is a conjecture of Bloch-Kato on the image of $p$-adic regulator maps for a proper smooth variety $X$ over an algebraic number field $k$. The conjecture for a regulator map of particular degree and weight is related to finiteness of two arithmetic objects: One is the $p$-primary torsion part of the Chow group in codimension 2 of $X$. Another is an unramified cohomology group of $X$. As an application, for a regular model ${\mathscr X}$ of $X$ over the integer ring of $k$, we show an injectivity result on torsion of a cycle class map from the Chow group in codimension …
Let X be a separated scheme of finite type over a field k and D a non-reduced effective Cartier divisor on it. We attach to the pair (X, D) a cycle complex with modulus, whose homotopy groups - called higher Chow groups with modulus - generalize additive higher Chow groups of Bloch-Esnault, Rülling, Park and Krishna-Levine, and that sheafified on $X_{Zar}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight 1. When X is smooth over k and D is such that $D_{red}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of re…
In this paper we construct extensions of mixed Hodge structures coming from the mixed Hodge structure on the graded quotients of the group ring of the fundamental group of a smooth, projective, pointed curve. These extensions correspond to the regulators of certain motivic cycles in the Jacobian of the curve which were constructed by Beilinson and Bloch. This leads to a new iterated integral expression for the regulator. This is a generalisation of a theorem of Colombo where she constructed the extension corresponding to Collino's cycles in the Jacobian of a hyperelliptic curve.
In this paper, by using the regulator map of Beilinson-Deligne on a curve, we show that the quantization condition posed by Gukov is true for the SL_2(C) character variety of the hyperbolic knot in S^3. Furthermore, we prove that the corresponding $\mathbb{C}^{*}$-valued closed 1-form is a secondary characteristic class (Chern-Simons) arising from the vanishing first Chern class of the flat line bundle over the smooth part of the character variety, where the flat line bundle is the pullback of the universal Heisenberg line bundle over $\mathbb{C}^{*}\times \mathbb{C}^{*}$. Based on this result…
We explicitly describe cycle-class maps c_H from motivic cohomology to absolute Hodge cohomology, for smooth quasi-projective and (some) proper singular varieties, and compute special cases of the latter. For smooth projective varieties, we also study Hodge-theoretically defined "higher Abel-Jacobi maps" on the kernel of c_H; this leads to new results on nontrivial indecomposable higher Chow cycles in the regulator kernel.
We introduce exponential complexes of sheaves on manifolds. They are resolutions of the (Tate twisted) constant sheaves of the rational numbers, generalising the short exact exponential sequence. There are canonical maps from the exponential complexes to the de Rham complex. Using this, we introduce new complexes calculating the rational Deligne cohomology. We call them exponential Deligne complexes. Their advantage is that, at least at the generic point of a complex variety, one can define Beilinson's regulator map to the rational Deligne cohomology on the level of complexes. Namely, we defin…
