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
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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 -