Holonomic Poisson Manifolds and Deformations of Elliptic Algebras

Pym, Brent; Schedler, Travis

doi:10.1093/oso/9780198802020.003.0028

Cited by 11 publications

(9 citation statements)

References 49 publications

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“…Earlier, Hitchin [8] and Fiorenza and Manetti [4] had proven unobstructedness for certain deformation directions, namely, those corresponding to the Kähler class itself. Also, Pym [16] has introduced the notion of elliptic Poisson structures and more generally, Pym and Schedler [18] then introduced the notion of holonomic Poisson manifolds, where the degeneracy divisor is reduced but not necessarily normal crossings, and have studied their deformations focusing on questions of local finite dimensionality.…”

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confidence: 99%

A Bogomolov Unobstructedness Theorem for Log-Symplectic Manifolds in General Position

Ran¹

2018

J. Inst. Math. Jussieu

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We consider compact Kählerian manifolds X of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure Π which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor D(Π). We prove that (X, Π) has unobsrtuced deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on H 2 of the open symplectic manifold X \ D(Π), and in fact coincides with this H 2 provided the Hodge number h 2,0 X = 0, and finally that the degeneracy locus D(Π) deforms locally trivially under deformations of (X, Π).

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confidence: 99%

A Bogomolov Unobstructedness Theorem for Log-Symplectic Manifolds in General Position

Ran¹

2018

J. Inst. Math. Jussieu

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“…When X is smooth and the Poisson structure is a classical Poisson structure of degree 0, the D X -module u ! (O X ) has been considered in [PS18] where it was used to define the notion of holonomic Poisson varieties and to get finiteness results for Poisson cohomology. The GRR formula 6.2.2, and more generally the general formalism of crystals along derived foliations, provides a way to extend these notions and results to shifted Poisson structures.…”

Section: An Interesting Feature Of This Situation Is That Any Morphismmentioning

confidence: 99%

Algebraic foliations and derived geometry II: the Grothendieck-Riemann-Roch theorem

Toën,

Vezzosi

2020

Preprint

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This is the second of series of papers on the study of foliations in the setting of derived algebraic geometry based on the central notion of derived foliations. We introduce sheaf-like coefficients for derived foliations, called quasi-coherent crystals, and construct a certain sheaf of dg-algebras of differential operators along a given derived foliation, with the property that quasi-coherent crystals can be interpreted as modules over this sheaf of differential operators. We use this interpretation in order to introduce the notion of good filtrations on quasi-coherent crystals, and define the notion of characteristic cycle. Finally, we prove a Grothendieck-Riemann-Roch (GRR) formula expressing that formation of characteristic cycles is compatible with push-forwards along proper and quasi-smooth morphisms. Several examples and applications are deduced from this, e.g. a GRR formula for D-modules on possibly singular schemes. BERTRAND TOËN AND GABRIELE VEZZOSI 6. GRR for derived foliations 31 6.1. The GRR formula 32 6.2. The non-proper case: Fredholm crystals 33 7. Examples and applications 35 7.1. Grothendieck-Riemann-Roch for derived D-modules. 35 7.2. Smooth Lie algebroids 36 7.3. Shifted Poisson structures 37 7.4. A foliated index formula 37 References 39

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“…Let us describe the modular foliation locally, as in [67]. Let π ∈ Poiss(X) and let µ be a local volume form.…”

Section: The Modular Foliationmentioning

confidence: 99%

“…Given local trivializing sections ν ∈ Γ(det(A * )) and µ ∈ Γ(det(T * X)), note that ν ⊗ µ ∈ Γ(det(A * ) ⊗ det(T * X)) can serve as a nonvanishing section Example 5.43. As in [67,Example 3.1], given π = f ∂ x ∧∂ y on (R 2 , (x, y)), its modular vector field associated to µ = ∂ x ∧ ∂ y is given by V π,µ = (∂ x f )∂ y − (∂ y f )∂ x . The symplectic foliation is given by the subset where f is nonvanishing, and the points where f vanishes.…”

Section: The Modular Foliationmentioning

confidence: 99%

Poisson structures of divisor-type

Klaasse

2018

Preprint

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Poisson structures of divisor-type are those whose degeneracy can be captured by a divisor ideal, which is a locally principal ideal sheaf with nowhere-dense quotient support. This is a large class of Poisson structures which includes all generically-nondegenerate Poisson structures, such as log-, b k -, elliptic, elliptic-log, and scattering Poisson structures.Divisor ideals are used to define almost-injective Lie algebroids of derivations preserving them, to which these Poisson structures can be lifted, often nondegenerately so. The resulting symplectic Lie algebroids can be studied using tools from symplectic geometry. In this paper we develop an effective framework for the study of Poisson structures of divisor-type (also called almost-regular Poisson structures) and their Lie algebroids. We provide lifting criteria for such Poisson structures, develop the language of divisors on smooth manifolds, and discuss residue maps to extract information from Lie algebroid forms along their degeneraci loci. Contents 1. Introduction 1 2. Divisors on smooth manifolds 5 3. Lie algebroids of divisor-type 8 4. Poisson structures of divisor-type 25 5. Residue maps and the modular foliation 39 References 55 2010 Mathematics Subject Classification. 53D17. 1 2 RALPH L. KLAASSEwe develop a framework for the study of Poisson structures of divisor-type. Below we outline the ideas underlying this framework, and mention our main results.The techniques and language developed in this paper are used to various establish results for almost-regular Poisson structures, including the computation of Poisson cohomology [32], determining the adjoint symplectic groupoids integrating them [33], and developing topological obstructions to their existence [31]. Moreover, this framework was used in [10] in the development of Gompf-Thurston methods for symplectic Lie algebroids.Poisson structures of divisor-type. In this paper we adapt the language of divisors to the context of smooth manifolds, following their use in [10,12]. A divisor consist of a line bundle together with a section whose zero set is nowhere dense. Each such pair (L, s) specifies a divisor ideal I s ∶= s(Γ(L * )) ⊆ C ∞ (X). Abstractly, divisor ideals are locally principal ideals I for which the support Z(I) ⊆ X of the quotient sheaf C ∞ (X) I is nowhere dense, whereThese ideals keep track of degenerative behavior along this support, and are an extension of the concept of a Cartier divisor in algebraic and complex geometry to the smooth setting.Poisson structures naturally lead to divisors: Given π ∈ Poiss(X) such that dim(X) = 2n, consider its Pfaffian ∧ n π ∈ Γ(det(T X)). Note that π is nondegenerate exactly at those points x ∈ X where ∧ n π x ≠ 0. Thus π is generically nondegenerate if and only if the pair (det(T X), ∧ n π) is a divisor. In this case we denote this pair by div(π), with associated divisor ideal I π ⊆ C ∞ (X). This ideal captures the behavior of π along Z(π) = Z(I π ), called the degeneracy locus of π, consisting of points where the rank of π is not maximal.In gen...

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Holonomic Poisson Manifolds and Deformations of Elliptic Algebras

Cited by 11 publications

References 49 publications

A Bogomolov Unobstructedness Theorem for Log-Symplectic Manifolds in General Position

A Bogomolov Unobstructedness Theorem for Log-Symplectic Manifolds in General Position

Algebraic foliations and derived geometry II: the Grothendieck-Riemann-Roch theorem

Poisson structures of divisor-type

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