×

Cosimplicial objects in algebraic geometry. (English) Zbl 0916.14006

Goerss, P. G. (ed.) et al., Algebraic \(K\)-theory and algebraic topology. Proceedings of the NATO Advanced Study Institute, Lake Louise, Alberta, Canada, December 12–16, 1991. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 407, 287-327 (1993).
Let \(X\) be an arc-connected and locally arc-connected topological space and let \(I\) be the unit interval. Applying the connected component functor to each fibre of the fibration of the total space map \((I,X)\) over \(X\times X\), \(P(w)=(w(0),w(1))\), we get a local system of sets (Poincaré groupoid) over \(X\times X\). This construction does not have a straightforward generalization to algebraic varieties over any field. Using cosimplicial schemes we construct the analog of the bundle of fundamental groups in algebraic geometry and we equip it with an integrable connection. We recover in this way constructions given by R. M. Hain and S. Zucker [Invent. Math. 88, 83-124 (1987; Zbl 0622.14007)] over complex numbers. However our construction applies to smooth schemes defined over any field of characteristic zero. The cosimplicial schemes can be used to introduce mixed Hodge structures on homotopy groups as it was done by other methods by J. W. Morgan [Publ. Math., Inst. Hautes Étud. Sci. 48, 137-204 (1978; Zbl 0401.14003)], R. M. Hain [“Iterated integrals and homotopy periods”, Mem. Am. Math. Soc. 291 (1984; Zbl 0539.55002)] and V. Navarro Aznar [Invent. Math. 90, 11-76 (1987; Zbl 0639.14002)]. They seem to be specially suited to treat motives associated to fundamental groups, higher homotopy groups and other topological invariants. For example we can define motivic \(\pi_1\) and also motivic fundamental groupoids for any smooth algebraic veriety \(X\), while P. Deligne [in: Galois groups over \(\mathbb{Q}\), Proc. Workshop, Berkeley 1987, Publ., Math. Sci. Res. Inst. 16, 79-297 (1989; Zbl 0742.14022)] needs the restriction that \(H^1(\overline X,{\mathcal O})=0\) where \(\overline X\) is a smooth compactification of \(X\).
We partly calculate the Betti lattice in the algebraic fundamental group for the projective line minus three points.
For the entire collection see [Zbl 0880.00040].

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
14F40 de Rham cohomology and algebraic geometry
14A20 Generalizations (algebraic spaces, stacks)
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
PDFBibTeX XMLCite