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43
Survey of Oka theory
, 2011
"... Oka theory has its roots in the classical Oka principle in complex analysis. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. Following a brief review of Stein manifolds, we discuss the recently introduced category of Ok ..."
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Cited by 23 (9 self)
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Oka theory has its roots in the classical Oka principle in complex analysis. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. Following a brief review of Stein manifolds, we discuss the recently introduced category of Oka manifolds and Oka maps. We consider geometric sufficient conditions for being Oka, the most important of which is ellipticity, introduced by Gromov. We explain how Oka manifolds and maps naturally fit into an abstract homotopytheoretic framework. We describe recent applications and some key open problems. This article is a much expanded version of the lecture given by the firstnamed author
PRESENTING HIGHER STACKS AS SIMPLICIAL SCHEMES
, 2009
"... We show that an ngeometric stack may be regarded as a special kind of simplicial scheme, namely a Duskin nhypergroupoid in affine schemes, where surjectivity is defined in terms of covering maps, yielding Artin nstacks, DeligneMumford nstacks and nschemes as the notion of covering varies. T ..."
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Cited by 9 (4 self)
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We show that an ngeometric stack may be regarded as a special kind of simplicial scheme, namely a Duskin nhypergroupoid in affine schemes, where surjectivity is defined in terms of covering maps, yielding Artin nstacks, DeligneMumford nstacks and nschemes as the notion of covering varies. This formulation adapts to most HAG contexts, so in particular works for derived nstacks (replacing rings with simplicial rings). We exploit this to describe quasicoherent sheaves and complexes on these stacks, and to draw comparisons with Kontsevich’s dgschemes. As an application, we show how the
Principal ∞bundles – General theory
, 2012
"... The theory of principal bundles makes sense in any ∞topos, such as the ∞topos of topological, of smooth, or of otherwise geometric ∞groupoids/∞stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fib ..."
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Cited by 4 (3 self)
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The theory of principal bundles makes sense in any ∞topos, such as the ∞topos of topological, of smooth, or of otherwise geometric ∞groupoids/∞stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure ∞group G these Gprincipal ∞bundles reproduce the theories of ordinary principal bundles, of bundle gerbes/principal 2bundles and of bundle 2gerbes and generalize these to their further higher and equivariant analogs. The induced associated ∞bundles subsume the notion of Giraud’s gerbes, Breen’s 2gerbes, Lurie’s ngerbes, and generalize these to the notion of nonabelian ∞gerbes; which are the universal local coefficient bundles for nonabelian twisted cohomology. We discuss here this general abstract theory of principal ∞bundles, observing that it is intimately related to the axioms of Giraud, ToënVezzosi, Rezk and Lurie that characterize ∞toposes. A central result is a natural equivalence between principal ∞bundles and intrinsic nonabelian cocycles, implying the classification of principal
Grothendieck topologies from unique factorisation systems
, 2009
"... This article presents a way to associate a Grothendieck site structure to a category endowed with a unique factorisation system of its arrows. In particular this recovers the Zariski and Etale topologies and others related to Voevodsky’s cdstructures. As unique factorisation systems are also freque ..."
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Cited by 3 (0 self)
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This article presents a way to associate a Grothendieck site structure to a category endowed with a unique factorisation system of its arrows. In particular this recovers the Zariski and Etale topologies and others related to Voevodsky’s cdstructures. As unique factorisation systems are also frequent outside algebraic geometry, the same construction applies to some new contexts, where it is related with known structures defined otherwise. The paper details algebraic geometrical situations and sketches only the other contexts.
Mixed motives and quotient stacks: Abelian varieties, preprint, available at my webpage
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