Compositio Mathematica

Research Article

Algebraic and topological aspects of the schematization functor

L. Katzarkova1, T. Panteva2 and B. Toëna3

a1 Department of Mathematics, University of Miami, 1365 Memorial Drive, Ungar 515, Coral Gables, FL 33146, USA (email:

a2 Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA (email:

a3 Laboratoire Emile Picard, Université Paul Sabatier Bat 1R2, 31062 Toulouse Cedex 9, France (email:


We study some basic properties of schematic homotopy types and the schematization functor. We describe two different algebraic models for schematic homotopy types, namely cosimplicial Hopf alegbras and equivariant cosimplicial algebras, and provide explicit constructions of the schematization functor for each of these models. We also investigate some standard properties of the schematization functor that are helpful for describing the schematization of smooth projective complex varieties. In a companion paper, these results are used in the construction of a non-abelian Hodge structure on the schematic homotopy type of a smooth projective variety.

(Received April 14 2008)

(Revised December 16 2008)

(Accepted September 22 2008)

(Online publication May 13 2009)

2000 Mathematics Subject Classification

  • 14C30;
  • 32J27;
  • 55P62


  • schematic homotopy types;
  • homotopy theory;
  • non-abelian Hodge theory


L. Katzarkov was partially supported by an NSF FRG grant DMS-0652633, an NSF research grant DMS-0600800, and an FWF grant P20778. T. Pantev was partially supported by NSF research grants DMS-0403884 and DMS-0700446, and by the NSF RTG grant DMS-0636606.