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Connexions et classes caractéristiques de Beilinson. (Connections and Beilinson characteristic classes). (French) Zbl 0695.14003
Algebraic $$K$$-theory and algebraic number theory, Proc. Semin., Honolulu/Hawaii 1987, Contemp. Math. 83, 349-376 (1989).
[For the entire collection see Zbl 0655.00010.]
Denote by $$\Lambda$$ a subring of $${\mathbb{R}}$$ such that $$\Lambda \otimes_{{\mathbb{Z}}}{\mathbb{Q}}$$ is a field and write $$\Lambda$$ (i) for the group $$(2\pi i)^ i\Lambda \subset {\mathbb{C}}$$. Let $$X=(X_ n)$$ be a smooth pointed (strict) simplicial scheme of finite type (over $${\mathbb{C}})$$ with fat geometric realization $$| X|$$. Also, let $$\bar X$$ be a compactification of X such that $$Y=\bar X\setminus X$$ is a simplicial normal crossings divisor. Then one can associate to X (or to the pair $$(X,\bar X))$$ a complex $$A^{\bullet}(X)$$ of complex differential forms with filtration $$F^ iA^{\bullet}(X)$$, a subcomplex $$C_ m^{\bullet}(X)$$, $$m\in {\mathbb{N}}$$, and the complexes $$S^{\bullet}(X,\Lambda (i))$$, respectively $$S^{\bullet}(X)$$, of differentiable singular cochains with coefficients in $$\Lambda$$ (i), respectively $${\mathbb{C}}$$. Using on the one hand the $$F^ iA^{\bullet}(X)$$, and on the other hand the $$C_ m^{\bullet}(X)$$, one obtains two explicit morphisms $$\Phi$$, resp. $$\Phi_ m$$, with common source $$A^{\bullet}(X)\oplus S^{\bullet}(X,\Lambda (i))$$ and targets $$(A^{\bullet}(X)/F^ iA^{\bullet}(X))\oplus S^{\bullet}(X)$$, resp. $$(A^{\bullet}(X)/C_ m^{\bullet}(X))\oplus S^{\bullet}(X)$$. The cohomology spaces $$H^ k$$ of the corresponding cone complexes (twisted by [-1]) are mutually isomorphic, and besides, they are isomorphic with the usual Deligne-Beilinson cohomology $$H^ k_{{\mathcal D}}(X,\Lambda (i))$$ if $$m+k=2i$$. The data of $$A^{\bullet}(X)$$ is equivalent to the one of the simplicial de Rham complex $$A^{\bullet}(| X|)$$ on $$| X|$$, so a (pointed) differentiable map $$f:\quad | X| \to BU=\lim_{a} \lim_{b}({\mathfrak G}_{a,b}),$$ where $${\mathfrak G}_{a,b}$$ is the complex Graßmannian of a-dimensional subspaces of $${\mathbb{C}}^{a+b}$$, gives rise to elements $$ch_ i(f)\in A^{2i}(X)$$ induced by the Chern classes of the universal bundle with unitary connection on the $${\mathfrak G}_{a,b}$$. The group $${\mathcal K}_ 0(X)$$ is defined as the abelian group generated by pairs (f,u) with f: $$| X| \to BU$$ a (pointed) differentiable map and $$u=(u_ i)$$ is a family of elements $$u_ i\in A^{2i-1}(X)/(F^ iA^{2i-1}(X)\oplus dA^{2i-2}(A))$$ such that $$du_ i\equiv ch_ i(f)\quad mod\quad F^ iA^{2i}(X),$$ subject to two rather evident relations assuring homotopy invariance and giving the group structure. For a smooth projective variety X, $${\mathcal K}_ 0(X)$$ is just Karoubi’s multiplicative K-group consisting of triples (E,D,$$\omega)$$, where E is a complex differentiable vector bundle on X with differentiable connection D and $$\omega =(\omega_ i)$$, $$\omega_ i\in A^{2i-1}/C_ 0^{2i-1}(X)\oplus dA^{2i-2}$$, such that $$d\omega_ i\equiv ch_ i(D)\quad mod\quad C_ 0^{2i}(X),$$ subject to two relations of the kind mentioned above. Also, one may define higher $${\mathcal K}_ m(X)$$ by $${\mathcal K}_ m(X)={\mathcal K}_ 0(X\times S^ m)/{\mathcal K}_ 0(X)\oplus {\mathcal K}_ 0(S^ m)$$, $$m\geq 1$$, where $$S^ m$$ is the finite simplicial set obtained by triangulating the m-sphere $$\Sigma^ m$$. One has bilinear, associative and anticommutative products $${\mathcal K}_ m(X)\times {\mathcal K}_ n(X)\to {\mathcal K}_{m+n}(X)$$. The $${\mathcal K}_ m(X)$$, $$m\geq 1$$, can be interpreted as the homotopy groups $$\pi_ m$$ of a space that can be explicitly described. There is an Atiyah-Hirzebruch spectral sequence with $$E_ 2$$-terms equal to Deligne-Beilinson cohomology and abutting to the higher $${\mathcal K}'s$$ (for suitable indices). Using results of V. Schechtman on the representability of algebraic K-groups of schemes, one obtains canonical natural morphisms $$\rho_ m: K_ m(X)\to {\mathcal K}_ m(X)$$ from algebraic to multiplicative K-theory.
The main results can now be formulated as follows: One can construct explicitly natural maps $$c_ i:\quad {\mathcal K}_ 0(X)\to H_{{\mathcal D}}^{2i}(X,\Lambda (i)),$$ such that their composition with $$H_{{\mathcal D}}^{2i}(X,\Lambda (i))\to H^{2i}(X,\Lambda (i))$$ is equal to the composition of $${\mathcal K}_ 0(X)\to K_ c^{top}(| X|)$$ with the i-th topological Chern class of X. For the higher $${\mathcal K}'s$$ one can construct $$c_{i,k}:\quad {\mathcal K}_ m(X)\to H^ k_{{\mathcal D}}(X,\Lambda (i)),$$ $$k+m=2i$$, such that for a smooth scheme of finite type the composed morphism $$c_{i,k}\circ \rho_ m:\quad K_ m(X)\to H^ k_{{\mathcal D}}(X,\Lambda (i))$$ coincides with Beilinson’s Chern class map $$c^ B_{i,k}:\quad K_ m(X)\to H^ k_{{\mathcal D}}(X,\Lambda (i)).$$ For a smooth projective variety X with flat algebraic vector bundle E the secondary Chern classes of E (Chern, Cheeger, Simons) can be compared with $$c^ B_ i: K_ 0(X)\to H_{{\mathcal D}}^{2i}(X,\Lambda (i))$$.
Reviewer: W.Hulsbergen

##### MSC:
 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 55N15 Topological $$K$$-theory 14F40 de Rham cohomology and algebraic geometry 53C05 Connections, general theory