A differential graded model for derived analytic geometry

Pridham, J. P.

doi:10.1016/j.aim.2019.106922

Cited by 8 publications

(6 citation statements)

References 35 publications

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“…There has appeared quite many works in this direction in the literature mainly motivated by derived algebraic geometry of Lurie and Toën-Vezzosi [39,53]. See, for instance, [51,13,14,31,10,48] for the C ∞setting and [49,47] for the analytic-setting. Our approach is based on the geometry of bundles of positively graded curved L ∞ [1]-algebras, or equivalently, dg manifolds of positive amplitudes.…”

Section: Derived Manifoldsmentioning

confidence: 99%

Derived Differentiable Manifolds

Behrend¹,

Liao²,

Xu³

2020

Preprint

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We develop the theory of derived differential geometry in terms of bundles of curved L∞[1]-algebras, i.e. dg manifolds of positive amplitudes. We prove the category of derived manifolds is a category of fibrant objects. We construct a factorization of the diagonal using path spaces. First we construct an infinite-dimensional factorization using actual path spaces motivated by the AKSZ construction, then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient is the homotopy transfer theorem for curved L∞[1]-algebras.We also prove the inverse function theorem for derived manifolds, and investigate the relationship between weak equivalence and quasiisomorphism for derived manifolds.

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Section: Derived Manifoldsmentioning

confidence: 99%

Derived Differentiable Manifolds

Behrend¹,

Liao²,

Xu³

2020

Preprint

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“…The next definition is essentially similar to the notions used in [Pir15,Pri20]. Note however that in the non-connective case, our notions of holomorphic cdga differs slightly from the one in loc.…”

Section: Quick Reminder On Derived Analytic Geometrymentioning

confidence: 97%

“…We recall that we can define a set valued symmetric operad hol, of holomorphic rings as follows (see [Pir15,Pri20,Por19] and [CR13,Nui12] for the C ∞ -version). The set hol(n) is defined to be the set of holomorphic functions of C n , and the operadic structure is defined by substitution in a natural manner (n = 0 is included here, the operad hol is unital).…”

Section: Quick Reminder On Derived Analytic Geometrymentioning

confidence: 99%

Algebraic foliations and derived geometry: the Riemann–Hilbert correspondence

Toën

Vezzosi

2022

Sel. Math. New Ser.

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This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations defined in terms of differential ideals in the algebra of forms. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety X is formally integrable at any point, and, if we suppose that its singular locus has codimension $$\ge 2$$ ≥ 2 , its analytification is a locally integrable singular foliation on the associated complex manifold $$X^h$$ X h . We then introduce the derived category of perfect crystals on a quasi-smooth rigid derived foliation on X, and prove a Riemann-Hilbert correspondence for them when X is proper. We discuss several examples and applications.

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“…They arise naturally in a variety of situations in differential geometry, Lie theory, representation theory and homotopy algebras [27,54,52,53,20]. They are closely related to the emerging fields of derived differential geometry [5,11,12,23,39,40,48] and higher Lie algebroids [4,7,8,9,19,21,22,43,54,52,46] (see also [45,Letters 7 and 8]).…”

Section: Introductionmentioning

confidence: 99%

Dg manifolds, formal exponential maps and homotopy Lie algebras

Seol,

Stiénon,

2021

Preprint

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This paper is devoted to the study of the relation between 'formal exponential maps,' the Atiyah class, and Kapranov L∞[1] algebras associated with dg manifolds in the C ∞ context. Given a dg manifold, we prove that a 'formal exponential map' exists if and only if the Atiyah class vanishes. Inspired by Kapranov's construction of a homotopy Lie algebra associated with the holomorphic tangent bundle of a complex manifold, we prove that the space of vector fields on a dg manifold admits an L∞[1] algebra structure, unique up to isomorphism, whose unary bracket is the Lie derivative w.r.t. the homological vector field, whose binary bracket is a 1-cocycle representative of the Atiyah class, and whose higher multibrackets can be computed by a recursive formula. For the dg manifold (T 0,1 X [1], ∂) arising from a complex manifold X, we prove that this L∞[1] algebra structure is quasi-isomorphic to the standard L∞[1] algebra structure on the Dolbeault complex Ω 0,• (T 1,0 X ).

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A differential graded model for derived analytic geometry

Cited by 8 publications

References 35 publications

Derived Differentiable Manifolds

Derived Differentiable Manifolds

Algebraic foliations and derived geometry: the Riemann–Hilbert correspondence

Dg manifolds, formal exponential maps and homotopy Lie algebras

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