TY - JOUR

T1 - Schematic homotopy types and non-abelian Hodge theory

AU - Katzarkov, L.

AU - Pantev, T.

AU - Toën, B.

N1 - Funding Information:
The first author was partially supported by NSF Career Award DMS-9875383 and an A.P. Sloan research fellowship. The second author was Partially supported by NSF Grants DMS-0099715 and DMS-0403884 and an A.P. Sloan research fellowship. This journal is ©c Foundation Compositio Mathematica 2008.

PY - 2008/5

Y1 - 2008/5

N2 - We use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the schematization functor X → (X ⊗ ℂ)sch, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a Hodge decomposition on (X ⊗ ℂ) sch. This Hodge decomposition is encoded in an action of the discrete group ℂ×δ on the object (X ⊗ ℂ) sch and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group, and, in the simply connected case, the Hodge decomposition on the complexified homotopy groups. We show that our Hodge decomposition satisfies a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As applications we construct new examples of homotopy types which are not realizable as complex projective manifolds and we prove a formality theorem for the schematization of a complex projective manifold.

AB - We use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the schematization functor X → (X ⊗ ℂ)sch, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a Hodge decomposition on (X ⊗ ℂ) sch. This Hodge decomposition is encoded in an action of the discrete group ℂ×δ on the object (X ⊗ ℂ) sch and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group, and, in the simply connected case, the Hodge decomposition on the complexified homotopy groups. We show that our Hodge decomposition satisfies a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As applications we construct new examples of homotopy types which are not realizable as complex projective manifolds and we prove a formality theorem for the schematization of a complex projective manifold.

KW - Homotopy type

KW - Non-abelian Hodge theory

UR - http://www.scopus.com/inward/record.url?scp=52949126744&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=52949126744&partnerID=8YFLogxK

U2 - 10.1112/S0010437X07003351

DO - 10.1112/S0010437X07003351

M3 - Article

AN - SCOPUS:52949126744

VL - 144

SP - 582

EP - 632

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 3

ER -