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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 17 (2 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
On the homotopy theory of ntypes
 Homology, Homotopy Appl
"... Abstract. An ntruncated model structure on simplicial (pre)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine’s intermediate model structures we construct such an ntype model structure via Bo ..."
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Abstract. An ntruncated model structure on simplicial (pre)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine’s intermediate model structures we construct such an ntype model structure via BousfieldFriedlander localization and exhibit useful generating sets of trivial cofibrations. Injectively fibrant objects in these categories are called nhyperstacks. The whole setup can consequently be viewed as a description of the homotopy theory of higher hyperstacks. More importantly, we construct analogous ntruncations on simplicial groupoids and prove a Quillen equivalence between these settings. We achieve a classification of ntypes of simplicial presheaves in terms of (n −1)types of presheaves of simplicial groupoids. Our classification holds for general n. Therefore this can also be viewed as the homotopy theory of (pre)sheaves of (weak) higher groupoids. Contents
HOMOTOPY TOPOI AND EQUIVARIANT ELLIPTIC COHOMOLOGY
, 2005
"... We use the language of homotopy topoi, as developed by Lurie [17], Rezk [21], Simpson [23], and TöenVezossi [24], in order to provide a common foundation for equivariant homotopy theory and derived algebraic geometry. In particular, we obtain the categories of Gspaces, for a topological group G, a ..."
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Cited by 2 (0 self)
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We use the language of homotopy topoi, as developed by Lurie [17], Rezk [21], Simpson [23], and TöenVezossi [24], in order to provide a common foundation for equivariant homotopy theory and derived algebraic geometry. In particular, we obtain the categories of Gspaces, for a topological group G, and Eschemes, for an E∞ring spectrum E, as full topological subcategories of the homotopy topoi associated to sheaves of spaces on certain small topological sites. This allows for a particularly elegant construction of the equivariant elliptic cohomology associated to an oriented elliptic curve A and a compact abelian Lie group G as an essential geometric morphism of homotopy topoi. It follows that our definition satisfies a conceptually simpler homotopytheoretic analogue of the GinzburgKapranovVasserot axioms [8], which allows us to calculate the cohomology of the equivariant Gspectra S V associated to representations V of G. iii To my parents.
GENERALIZED BROWN REPRESENTABILITY IN HOMOTOPY CATEGORIES
, 2005
"... Brown representability approximates the homotopy ..."
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GENERALIZED BROWN REPRESENTABILITY IN HOMOTOPY CATEGORIES
, 2008
"... Brown representability approximates the homotopy ..."
ON HOMOTOPY VARIETIES
, 2005
"... Given an algebraic theory T, a homotopy Talgebra is a simplicial set where all equations from T hold up to homotopy. All homotopy Talgebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties. We will also study homotopy mo ..."
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Given an algebraic theory T, a homotopy Talgebra is a simplicial set where all equations from T hold up to homotopy. All homotopy Talgebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties. We will also study homotopy models of limit theories which leads to homotopy locally presentable categories. These were recently considered by Simpson, Lurie, Toën and Vezzosi.