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- Série 6 : Tome 25 (2016)

- no. 2-3

- p. 619-681

- Suivant

- Abstract (VO)

- Résumé

We introduce a weight n exponential complex of sheaves ℚ E • (n) on a manifold X:

<table class="formula"><tr><td class="formula-inner"><span class="mathjax-formula" data-tex="\[ {\mathcal{O}} (n-1) \longrightarrow {\mathcal{O}}^* \otimes {\mathcal{O}} (n-2) \longrightarrow ... \longrightarrow \otimes ^{n-1}{\mathcal{O}} ^*\otimes {\mathcal{O}} \longrightarrow \otimes ^{n }{\mathcal{O}}^*.\quad (1) \]"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>𝒪</mi><mrow><mo>(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mo>⟶</mo><msup><mrow><mi>𝒪</mi></mrow><mo>*</mo></msup><mo>⊗</mo><mi>𝒪</mi><mrow><mo>(</mo><mi>n</mi><mo>-</mo><mn>2</mn><mo>)</mo></mrow><mo>⟶</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>⟶</mo><msup><mo>⊗</mo><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><msup><mrow><mi>𝒪</mi></mrow><mo>*</mo></msup><mo>⊗</mo><mi>𝒪</mi><mo>⟶</mo><msup><mo>⊗</mo><mi>n</mi></msup><msup><mrow><mi>𝒪</mi></mrow><mo>*</mo></msup><mo>.</mo><mspace width="1em" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span></td><td class="formula-label"></td></tr></table>

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

<table class="formula"><tr><td class="formula-inner"><span class="mathjax-formula" data-tex="\[ \mathbb{Z}(1) \longrightarrow {\mathcal{O}} \stackrel{\rm exp}{\longrightarrow } {\mathcal{O}}^*,\quad \mathbb{Z}(1):= 2\pi i\mathbb{Z}. \]"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ℤ</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>⟶</mo><mi>𝒪</mi><mover><mo>⟶</mo><mi> exp </mi></mover><msup><mrow><mi>𝒪</mi></mrow><mo>*</mo></msup><mo>,</mo><mspace width="1em" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mi>ℤ</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mn>2</mn><mi>π</mi><mi>i</mi><mi>ℤ</mi><mo>.</mo></mrow></math></span></td><td class="formula-label"></td></tr></table>

There is a canonical map from the complex ℚ E • (n) to the de Rham complex Ω • of X. Using it, we define a weight n exponential Deligne complex, calculating rational Deligne cohomology:

<table class="formula"><tr><td class="formula-inner"><span class="mathjax-formula" data-tex="\[ \Gamma _{\mathcal{D}}(X; n):= {\rm Cone}\Bigl (\mathbb{Q}^\bullet _{\bf E}(n) \oplus F^{\ge n}\Omega ^\bullet \longrightarrow \Omega ^\bullet \Bigr )[-1]. \]"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>Γ</mi><mi>𝒟</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>;</mo><mi>n</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mi> Cone </mi><mfenced close=")" open="(" separators=""><msubsup><mi>ℚ</mi><mi mathvariant="bold">E</mi><mo>•</mo></msubsup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>⊕</mo><msup><mi>F</mi><mrow><mo>≥</mo><mi>n</mi></mrow></msup><msup><mi>Ω</mi><mo>•</mo></msup><mo>⟶</mo><msup><mi>Ω</mi><mo>•</mo></msup></mfenced><mrow><mo>[</mo><mo>-</mo><mn>1</mn><mo>]</mo></mrow><mo>.</mo></mrow></math></span></td><td class="formula-label"></td></tr></table>

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

<table class="formula"><tr><td class="formula-inner"><span class="mathjax-formula" data-tex="\begin{multline} \text{\emph {a weight $n$ motivic complex of $\mathcal{X}$}}\ \longrightarrow \\ \text{\emph {the weight $n$ exponential complex of $\mathcal{X}$}}. (2) \end{multline}"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable displaystyle="true"><mtr><mtd columnalign="left"><mrow><mtext>a</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>weight</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mi>n</mi><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>motivic</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>complex</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>of</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mi>𝒳</mi><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mo>⟶</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mtext>the</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>weight</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mi>n</mi><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>exponential</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>complex</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>of</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mi>𝒳</mi><mo>.</mo><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></span></td><td class="formula-label"></td></tr></table>

We show that it gives rise to a map of complexes

<table class="formula"><tr><td class="formula-inner"><span class="mathjax-formula" data-tex="\begin{multline} \text{\emph {a weight $n$ motivic complex of $\mathcal{X}$}}\ \longrightarrow \\ \text{\emph {the weight $n$ exponential Deligne complex of $\mathcal{X}$}}.(3) \end{multline}"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable displaystyle="true"><mtr><mtd columnalign="left"><mrow><mtext>a</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>weight</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mi>n</mi><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>motivic</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>complex</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>of</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mi>𝒳</mi><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mo>⟶</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mtext>the</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>weight</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mi>n</mi><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>exponential</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>Deligne</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>complex</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mtext>of</mtext><mspace width="4pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mi>𝒳</mi><mo>.</mo><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></span></td><td class="formula-label"></td></tr></table>

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≤4. Equivalently, we get an explicit construction for the universal Chern class in the rational Deligne cohomology

<table class="formula"><tr><td class="formula-inner"><span class="mathjax-formula" data-tex="\[ c_n^{\mathcal{D}}\in H^{2n}(BGL_N(\mathbb{C}), \Gamma _{\mathcal{D}}(n)), ~~~~n\le 4. \]"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>c</mi><mi>n</mi><mi>𝒟</mi></msubsup><mo>∈</mo><msup><mi>H</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>B</mi><mi>G</mi><msub><mi>L</mi><mi>N</mi></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow><mo>,</mo><msub><mi>Γ</mi><mi>𝒟</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo><mspace width="3.33333pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mspace width="3.33333pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mspace width="3.33333pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mspace width="3.33333pt" xmlns:mml="http://www.w3.org/1998/Math/MathML"></mspace><mi>n</mi><mo>≤</mo><mn>4</mn><mo>.</mo></mrow></math></span></td><td class="formula-label"></td></tr></table>

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 ≤4.

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A. B. Goncharov 1

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[1] Atiyah (M.).— Convexity and commuting hamiltonians. Michale Atiyah Collected papers, Vol 5 (1982). | DOI

[2] Beilinson (A.A.).— Higher regulators and values of L-functions. VINITI 24, (1984). 181-238. J. Soviet Math. 30 p. 2036-2070 (1985). | DOI | Zbl

[3] Beilinson (A.A.).— Height pairings between algebraic cycles Lect. Notes in Math 1289, p. 1-26 (1987). | DOI

[4] Beilinson (A.A.), Deligne (P.).— Interpretation motivique de la conjecture de Zagier reliant polylogarithms et regulateurs. Proc. Proc. of Symposia in Pure Math., Volume 55, Part 2. p. 97-122. | DOI | Zbl

[5] Beilinson (A.A.), MacPherson (R.), Schechtman (V.V.).— Notes on motivic cohomology, Duke Math. J. 54, p. 679-710 (1987). | MR | DOI | Zbl

[6] Beilinson (A.A.), Goncharov (A.B.), Schechtman (V.V.), Varchenko (A.N.).— Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles on the plane. The Grothendieck Festschrift, Vol. I, p. 135-172, Progr. Math., 86, Birkhäuser Boston, Boston, MA, (1990). | DOI

[7] Bloch (S.).— K 2 and algebraic cycles. Annals of Math. 99, p. 349-379 (1974). | DOI | Zbl

[8] Bloch (S.).— Applications of the dilogarithm function in algebraic K-theory and algebraic geometry. Proc. Int. Symposium on Algebraic Geometry, Kyoto, 1977. Kinokuniyo Book Store Ltd., Tokyo, Japan.

[9] Bloch (S.).— Higher regulators, Algebraic K-theory, and Zeta functions of elliptic curves. Irvine lecture notes. CRM Monograph series, AMS 2000. Original preprint of 1978. | DOI

[10] Bloch (S.), Kriz (I.).— Mixed Tate motives. Ann of Math. 140, p. 557-605 (1994). | MR | DOI | Zbl

[11] Dupont (J.), Sah (S.-H.).— Scissor congruences II. J. Pure Appl. Algebra 25, p. 159-195 (1982). | DOI

[12] Fock (V.V.), Goncharov (A.B.).— Moduli spaces of local systems and Higher Teichmuller theory. Publ. Math. IHES, n. 103 (2006) 1-212. ArXiv math.AG/0311149. | DOI

[13] Fock (V.V.), Goncharov (A.B.).— The quantum dilogarithm and representations of quantum cluster varieties. 175, Inventiones Math., p. 223-286 (2009). arXiv:math/0702397.

[14] Gabrielov (A.), Gelfand (I.M.), Losik (M.).— Combinatorial computation of characteristic classes I, II, Funct. Anal. i ego pril. 9, p. 12-28 (1975), ibid 9, p. 5-26 (1975). MR0410758 (53 14504a). | DOI

[15] Gelfand (I.M.), MacPherson (R.).— Geometry in Grassmannians and a generalisation of the dilogarithm, Adv. in Math., 44, p. 279-312 (1982). MR0658730 (84b:57014). | MR | DOI | Zbl

[16] Goncharov (A.B.).— Geometry of configurations, polylogarithms, and motivic cohomology. Adv. Math. 114, no. 2, p. 197-318 (1995). | MR | DOI | Zbl

[17] Goncharov (A.B.).— Explicit construction of characteristic classes. Advances in Soviet Mathematics, 16, v. 1, Special volume dedicated to I.M.Gelfand’s 80th birthday, p. 169-210 (1993). | DOI | Zbl

[18] Goncharov (A.B.).— Chow polylogarithms and regulators. Math. Res. Letters, 2, p. 99-114 (1995). | MR | DOI | Zbl

[19] Goncharov (A.B.).— Polylogarithms and motivic Galois groups. Proc. of Symposia in Pure Math., Volume 55, Part 2. p. 43-96. | DOI | Zbl

[20] Goncharov (A.B.).— Deninger’s conjecture on L-function of elliptic curve at s=3. ArXiv:alg-geom/9512016. | DOI

[21] Goncharov (A.B.).— Volumes of hyperbolic manifolds and mixed Tate motives JAMS vol. 12, N2, p. 569-618 (1999). | MR | DOI | Zbl

[22] Goncharov (A.B.).— Geometry of the trilogarithm and the motivic Lie algebra of a field. Regulators in analysis, geometry and number theory, p. 127-165, Progr. Math., 171, Birkhäuser Boston, Boston, MA (2000).

[23] Goncharov (A.B.).— Polylogarithms, regulators and Arakelov motivic complexes. JAMS J. Amer. Math. Soc. 18, no. 1, p. 1-60 (2005). | MR | DOI | Zbl

[24] Goncharov (A.B.).— Hodge correlators ArXive:math/0803.0297. | MR | DOI

[25] Goncharov (A.B.).— A simple construction of Grassmannian polylogarithms. Adv. in Math 2014. arXive | MR | DOI

[26] Hain (R.), MacPherson (R.).— Higher Logarithms, Ill. J. of Math,, vol. 34, N2, p. 392-475 (1990). | MR | DOI | Zbl

[27] Hanamura (M.), MacPherson (R.).— Geometric construction of polylogarithms, Duke Math. J. 70, p. 481-516 (1993). | MR | DOI | Zbl

[28] Hanamura (M.), MacPherson (R.).— Geometric construction of polylogarithms, II, Progress in Math. vol. 132, p. 215-282 (1996). | DOI | Zbl

[29] MacPherson (R.).— Gabrielov, Gelfand, Losic combinatorial formula for the first Pontryagin class Seminar Bourbaki 1976. MR0521763 (81a:57022). | MR | Numdam | DOI

[30] Milnor (J.).— Introduction to algebraic K-theory. Annals of mathematical studies 72, Princeton University Press. (1971-. | DOI

[31] Suslin (A.A.).— K-theory of a field and the Bloch group, Proc. Steklov Inst. Math. 4, p. 217 239 (1991).

[32] Suslin (A.A.).— Homology of GL n , characteristic classes and Milnor’s K-theory. Trudy Mat. Inst. Steklov, 165, p. 188-204 (1984).

[33] Yusin (B. V.).— Sur les formes S p,q apparaissant dans le calcul combinatoire de la deuxiéme classe de Pontriaguine par la méthode de Gabrielov, Gelfand et Losik. (French) C. R. Acad. Sci. Paris Sér. I Math. 292, no. 13, p. 641-644 (1981). | MR | Zbl

[34] Zagier (D.).— Polylogarithms, zeta-functions, and algebraic K-theory of fields. Progress Math Vol 89. Birkhauser, Boston, MA, p. 392-430 (1991). | DOI

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