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 \(c^ B_ i: K_ 0(X)\to H_{{\mathcal D}}^{2i}(X,\Lambda (i))\).

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 |