Exponential complexes, period morphisms, and characteristic classes
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 2-3, pp. 619-681.

We introduce a weight $n$ exponential complex of sheaves ${ℚ}_{\mathbf{E}}^{•}\left(n\right)$ on a manifold $X$:

 $𝒪\left(n-1\right)⟶{𝒪}^{*}\otimes 𝒪\left(n-2\right)⟶...⟶{\otimes }^{n-1}{𝒪}^{*}\otimes 𝒪⟶{\otimes }^{n}{𝒪}^{*}.\phantom{\rule{1em}{0ex}}\left(1\right)$

It is a resolution of the constant sheaf $ℚ\left(n\right)$, generalising the classical exponential sequence:

 $ℤ\left(1\right)⟶𝒪\stackrel{\mathrm{exp}}{⟶}{𝒪}^{*},\phantom{\rule{1em}{0ex}}ℤ\left(1\right):=2\pi iℤ.$

There is a canonical map from the complex ${ℚ}_{\mathbf{E}}^{•}\left(n\right)$ to the de Rham complex ${\Omega }^{•}$ of $X$. Using it, we define a weight $n$ exponential Deligne complex, calculating rational Deligne cohomology:

 ${\Gamma }_{𝒟}\left(X;n\right):=\mathrm{Cone}\left({ℚ}_{\mathbf{E}}^{•}\left(n\right)\oplus {F}^{\ge n}{\Omega }^{•}⟶{\Omega }^{•}\right)\left[-1\right].$

Its main advantage is that, at least at the generic point $𝒳$ of a complex variety $X$, it allows to define Beilinson’s regulator map to the rational Deligne cohomology on the level of complexes. (A regulator map to real Deligne complexes for any regular complex variety is known [18]).

Namely, we define a weight $n$ period morphism. We use it to define a map of complexes

 $\begin{array}{c}\text{a}\phantom{\rule{4pt}{0ex}}\text{weight}\phantom{\rule{4pt}{0ex}}n\phantom{\rule{4pt}{0ex}}\text{motivic}\phantom{\rule{4pt}{0ex}}\text{complex}\phantom{\rule{4pt}{0ex}}\text{of}\phantom{\rule{4pt}{0ex}}𝒳\phantom{\rule{4pt}{0ex}}⟶\hfill \\ \hfill \text{the}\phantom{\rule{4pt}{0ex}}\text{weight}\phantom{\rule{4pt}{0ex}}n\phantom{\rule{4pt}{0ex}}\text{exponential}\phantom{\rule{4pt}{0ex}}\text{complex}\phantom{\rule{4pt}{0ex}}\text{of}\phantom{\rule{4pt}{0ex}}𝒳.\left(2\right)\end{array}$

We show that it gives rise to a map of complexes

 $\begin{array}{c}\text{a}\phantom{\rule{4pt}{0ex}}\text{weight}\phantom{\rule{4pt}{0ex}}n\phantom{\rule{4pt}{0ex}}\text{motivic}\phantom{\rule{4pt}{0ex}}\text{complex}\phantom{\rule{4pt}{0ex}}\text{of}\phantom{\rule{4pt}{0ex}}𝒳\phantom{\rule{4pt}{0ex}}⟶\hfill \\ \hfill \text{the}\phantom{\rule{4pt}{0ex}}\text{weight}\phantom{\rule{4pt}{0ex}}n\phantom{\rule{4pt}{0ex}}\text{exponential}\phantom{\rule{4pt}{0ex}}\text{Deligne}\phantom{\rule{4pt}{0ex}}\text{complex}\phantom{\rule{4pt}{0ex}}\text{of}\phantom{\rule{4pt}{0ex}}𝒳.\left(3\right)\end{array}$

It induces Beilinson’s regulator map on the cohomology.

Combining the map (3) with the construction of Chern classes with coefficients in the bigrassmannian complexes [17], we get a local explicit formula for the $n$-th Chern class in the rational Deligne cohomology via polylogarithms, at least for $n\le 4$. Equivalently, we get an explicit construction for the universal Chern class in the rational Deligne cohomology

 ${c}_{n}^{𝒟}\in {H}^{2n}\left(BG{L}_{N}\left(ℂ\right),{\Gamma }_{𝒟}\left(n\right)\right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}n\le 4.$

In particular, this gives explicit formulas for Cech cocycles for the topological Chern classes.

Nous introduisons des complexes exponentiels de faisceaux sur une variété. Il s’agit de résolutions des faisceaux (Tate-twistés) constants de nombres rationnels généralisant la suite exacte courte exponentielle. Il existe des applications canoniques de ces complexes vers le complexe de de Rham. A l’aide de celles-ci, et en calculant la cohomologie de Deligne rationnelle, nous introduisons de nouveaux complexes que nous appelons complexes de Deligne exponentiels. L’avantage de ces derniers est qu’au moins au point générique d’une variété complexe on peut définir l’application de régulateur de Beilinson vers la cohomologie de Deligne rationnelle au niveau des complexes. En particulier, nous définissons des morphismes de périodes à l’aide desquels nous construisons des homomorphismes entre les complexes motiviques et les complexes de Deligne exponentiels en un point générique. En combinant cette construction avec celle des classes de Chern à coefficients dans des bicomplexes, nous obtenons une formule explicite, à l’aide de polylogarithmes, pour les classes de Chern à valeurs dans la cohomologie de Deligne rationnelle, en degré $\le 4$.

Published online:
DOI: 10.5802/afst.1507
Goncharov, A. B. 1

1 Department of Mathematics, Yale University, New Haven, CT 06511, USA
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Goncharov, A. B. Exponential complexes, period morphisms, and characteristic classes. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 2-3, pp. 619-681. doi : 10.5802/afst.1507. http://www.numdam.org/articles/10.5802/afst.1507/

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