### Pro-algebraic homotopy types

, 2008

"... The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, in such a way as to be amenable to Hodge theory. The pro-algebraic homotopy type (over a field k of characteristic zero) of a pointed space (X, x) is constructed as a certain pro-k-algebraic compl ..."

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The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, in such a way as to be amenable to Hodge theory. The pro-algebraic homotopy type (over a field k of characteristic zero) of a pointed space (X, x) is constructed as a certain pro-k-algebraic completion of Kan’s loop group of X. This has the property that the pro-algebraic fundamental group is the pro-algebraic completion of π1(X, x), while the nth pro-algebraic homotopy group is the completion of πn(X, x) ⊗Z k with respect to its π1(X, x)-subrepresentations of finite codimension. There is also a notion of unpointed pro-algebraic homotopy type, replacing groups by groupoids. If X is simply connected, its pro-algebraic homotopy type is equivalent to Sullivan’s rational homotopy type. Toën’s schematic homotopy type can be recovered from the pro-algebraic homotopy type, as the pro-algebraic homotopy type of X is equivalent to the homotopy type of the cochain algebra with coefficients in the universal semisimple local system on X. As an application, we show that the proalgebraic homotopy groups of a compact Kähler manifold have a canonical weight decomposition, which can be recovered explicitly from the cohomology ring of the universal semisimple local system.

### Pro-algebraic homotopy types

, 2008

"... The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, providing an alternative approach to defining Toën’s schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Mau ..."

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The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, providing an alternative approach to defining Toën’s schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Maurer-Cartan equations and convergent spectral sequences comparing schematic homotopy groups with cohomology of the universal semisimple local system. For compact Kähler manifolds, the schematic homotopy groups can be described explicitly in terms of this cohomology ring, giving them canonical weight decompositions. There are also notions of minimal models, unpointed homotopy types and algebraic automorphism groups. For a space with algebraically good fundamental group and higher homotopy groups of finite rank, the schematic homotopy groups are shown to be πn(X) ⊗Z k.

### Contents

, 2008

"... The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, in such a way as to be amenable to Hodge theory. The pro-algebraic homotopy type (over a field k of characteristic zero) of a pointed space (X, x) is constructed as a certain pro-k-algebraic compl ..."

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The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, in such a way as to be amenable to Hodge theory. The pro-algebraic homotopy type (over a field k of characteristic zero) of a pointed space (X, x) is constructed as a certain pro-k-algebraic completion of Kan’s loop group of X. There is also a notion of unpointed pro-algebraic homotopy type, replacing groups by groupoids. If X is simply connected, its pro-algebraic homotopy type is equivalent to Sullivan’s rational homotopy type. Toën’s schematic homotopy type is equivalent to the pro-algebraic homotopy type, with the pro-algebraic homotopy groups isomorphic to the schematic homotopy groups. There are spectral sequences relating pro-algebraic homotopy groups to the cohomology ring of the universal semisimple local system. For compact Kähler manifolds, the pro-algebraic homotopy groups can be described explicitly in terms of this cohomology ring. There is a notion of minimal models for pro-algebraic homotopy types, which allows us to study their automorphisms.

### Pro-algebraic homotopy types

, 2008

"... The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, providing an alternative approach to defining Toën’s schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Mau ..."

Abstract
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The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, providing an alternative approach to defining Toën’s schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Maurer-Cartan equations and convergent spectral sequences comparing schematic homotopy groups with cohomology of the universal semisimple local system. For compact Kähler manifolds, the schematic homotopy groups can be described explicitly in terms of this cohomology ring, giving them canonical weight decompositions. There are also notions of minimal models, unpointed homotopy types and algebraic automorphism groups. For a space with algebraically good fundamental group and higher homotopy groups of finite rank, the schematic homotopy groups are shown to be πn(X) ⊗Z k.

### Pro-algebraic homotopy types

, 2008

"... The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, in such a way as to be amenable to Hodge theory. The pro-algebraic homotopy type (over a field k of characteristic zero) of a pointed space (X, x) is constructed as a certain pro-k-algebraic compl ..."

Abstract
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The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, in such a way as to be amenable to Hodge theory. The pro-algebraic homotopy type (over a field k of characteristic zero) of a pointed space (X, x) is constructed as a certain pro-k-algebraic completion of Kan’s loop group of X. There is also a notion of unpointed pro-algebraic homotopy type, replacing groups by groupoids. If X is simply connected, its pro-algebraic homotopy type is equivalent to Sullivan’s rational homotopy type. Toën’s schematic homotopy type is equivalent to the pro-algebraic homotopy type, with the pro-algebraic homotopy groups isomorphic to the schematic homotopy groups. There are spectral sequences relating pro-algebraic homotopy groups to the cohomology ring of the universal semisimple local system. For compact Kähler manifolds, the pro-algebraic homotopy groups can be described explicitly in terms of this cohomology ring. There is a notion of minimal models for pro-algebraic homotopy types, which allows us to study their automorphisms.

### Contents

, 2008

"... The deformation theory of representations of the fundamental group of a smooth variety ..."

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The deformation theory of representations of the fundamental group of a smooth variety

### A non-abelian Weil conjecture

, 2008

"... Let X be a smooth or proper variety defined over a finite field. The étale fundamental group π1(X, ¯x) of X is a normal subgroup of the Weil group, so conjugation gives it a Weil action. We consider the pro-Ql-algebraic completion of π1(X, ¯x) as a non-abelian Weil representation. Deligne’s conjectu ..."

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Let X be a smooth or proper variety defined over a finite field. The étale fundamental group π1(X, ¯x) of X is a normal subgroup of the Weil group, so conjugation gives it a Weil action. We consider the pro-Ql-algebraic completion of π1(X, ¯x) as a non-abelian Weil representation. Deligne’s conjecture and the Weil conjectures imply that this affine group scheme is mixed, in the sense that its structure sheaf is a mixed Weil representation. When X is smooth, weight restrictions apply, affecting the possibilities for the structure of this group. This gives new examples of