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24
Algebraic geometry over model categories  A general approach to derived algebraic geometry
, 2001
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Higher and derived stacks: a global overview
, 2005
"... These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, JulyAugust 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the sub ..."
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Cited by 21 (6 self)
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These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, JulyAugust 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the subject, starting from the motivations and the foundational material, passing through some examples and basic notions, and ending with some more recent developments and open questions.
The construction problem in Kähler geometry
, 2004
"... One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a Kähler metric leads to all sorts of Hodge theoretic ..."
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Cited by 11 (1 self)
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One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a Kähler metric leads to all sorts of Hodge theoretical restrictions on the homotopy types of algebraic varieties. On the other hand, a sparse collection of examples shows that the remaining liberty is nontrivially large. Paradoxically, with all of this information, the research field remains as wide open as it was many decades ago, because the gap between the known restrictions, and the known examples of what can occur, only seems to grow wider and wider the more closely we look at it. In spite of the differentialgeometric nature of the questions and methods, the origins of the situation are very algebraic. We look at subvarieties of projective space over the complex numbers. The main overarching problem in algebraic geometry is to understand the classification of algebrogeometric objects. The topology of the usual complexvalued points of a variety plays
Compact Kähler manifolds with elliptic homotopy type
 Adv. Math. 224 (2010) 1167–1182. MR2628808, Zbl 1198.32008
"... Abstract. Simply connected compact Kähler manifolds of dimension up to three with elliptic homotopy type are characterized in terms of their Hodge diamonds. This is applied to classify the simply connected Kähler surfaces and Fano threefolds with elliptic homotopy type. 1. ..."
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Cited by 1 (0 self)
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Abstract. Simply connected compact Kähler manifolds of dimension up to three with elliptic homotopy type are characterized in terms of their Hodge diamonds. This is applied to classify the simply connected Kähler surfaces and Fano threefolds with elliptic homotopy type. 1.
TOWARDS NON–ABELIAN P –ADIC HODGE THEORY IN THE GOOD REDUCTION CASE
"... Abstract. We develop a non–abelian version of P –adic Hodge Theory for varieties (possible open with “nice compactification”) with good reduction. This theory yields in particular a comparison between smooth p–adic sheaves and F –isocrystals on the level of certain Tannakian categories, p–adic Hodge ..."
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Abstract. We develop a non–abelian version of P –adic Hodge Theory for varieties (possible open with “nice compactification”) with good reduction. This theory yields in particular a comparison between smooth p–adic sheaves and F –isocrystals on the level of certain Tannakian categories, p–adic Hodge theory for relative Malcev completions of fundamental groups and their Lie algebras, and gives information about the action of Galois on fundamental groups. Contents
Proalgebraic homotopy types
, 2008
"... The purpose of this paper is to generalise Sullivan’s rational homotopy theory to nonnilpotent spaces, in such a way as to be amenable to Hodge theory. The proalgebraic homotopy type (over a field k of characteristic zero) of a pointed space (X, x) is constructed as a certain prokalgebraic compl ..."
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The purpose of this paper is to generalise Sullivan’s rational homotopy theory to nonnilpotent spaces, in such a way as to be amenable to Hodge theory. The proalgebraic homotopy type (over a field k of characteristic zero) of a pointed space (X, x) is constructed as a certain prokalgebraic completion of Kan’s loop group of X. This has the property that the proalgebraic fundamental group is the proalgebraic completion of π1(X, x), while the nth proalgebraic 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 proalgebraic homotopy type, replacing groups by groupoids. If X is simply connected, its proalgebraic homotopy type is equivalent to Sullivan’s rational homotopy type. Toën’s schematic homotopy type can be recovered from the proalgebraic homotopy type, as the proalgebraic 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.
Proalgebraic homotopy types
, 2008
"... The purpose of this paper is to generalise Sullivan’s rational homotopy theory to nonnilpotent 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 nonnilpotent 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 MaurerCartan 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 nonnilpotent 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 nonnilpotent 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 MaurerCartan equations, convergent spectral sequences comparing schematic homotopy groups with cohomology of the universal semisimple local system, and a generalisation of the BauesLemaire conjecture. 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,