Type one generalized Calabi–Yaus

Bailey, Michael; Cavalcanti, Gil R.; Gualtieri, Marco

doi:10.1016/j.geomphys.2017.03.012

Cited by 6 publications

(6 citation statements)

References 11 publications

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“…Any compact type‐1 generalized Calabi–Yau manifold, such as

D_{J}

, fibers over the torus

T^{2}

. Moreover, the semilocal form of a stable generalized complex structure around its type change locus is given by its linearization along

D_{J}

, which is the stable generalized complex structure naturally present on the normal bundle to this type‐1 generalized Calabi–Yau manifold.…”

Section: Stable Generalized Complex and Log‐symplectic Structuresmentioning

confidence: 99%

“…Moreover, the semilocal form of a stable generalized complex structure around its type change locus is given by its linearization along

D_{J}

, which is the stable generalized complex structure naturally present on the normal bundle to this type‐1 generalized Calabi–Yau manifold. We will not elaborate on this further here and instead refer to .…”

Section: Stable Generalized Complex and Log‐symplectic Structuresmentioning

confidence: 99%

“…After perturbing each structure, this can be achieved using one‐forms on

D

and

Z

induced by the geometric structures (that is, they can be made proper). For a stable generalized complex structure

scriptJ

on X, these one‐forms are

false({Res}_{r}, {Res}_{θ} false) false(ω_{X} false) \in {normalΩ}^{1} false(D false) \times {normalΩ}^{1} false(D false)

, where

ω_{X}

is the elliptic symplectic structure obtained from

scriptJ

using Theorem , see . In the log‐Poisson case, the one‐form used instead is

{Res}_{Z} false(ω_{Y} false)

(see ), with

ω_{Y}

the log‐symplectic structure induced by the log‐Poisson structure using Proposition .…”

Section: Constructing Stable Generalized Complex Structuresmentioning

confidence: 99%

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Fibrations and stable generalized complex structures

Cavalcanti

Klaasse

2018

Proc. London Math. Soc.

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A generalized complex structure is called stable if its defining anticanonical section vanishes transversally, on a codimension-two submanifold. Alternatively, it is a zero elliptic residue symplectic structure in the elliptic tangent bundle associated to this submanifold. We develop Gompf-Thurston symplectic techniques adapted to Lie algebroids, and use these to construct stable generalized complex structures out of log-symplectic structures. In particular we introduce the notion of a boundary Lefschetz fibration for this purpose and describe how they can be obtained from genus one Lefschetz fibrations over the disc. ContentsLet X be a 2n-dimensional manifold equipped with a closed three-form H ∈ Ω 3 cl (X). Recall that the double tangent bundle TX := T X ⊕ T * X is a Courant algebroid whose anchor is the projection p : TX → T X. It carries a natural pairing V + ξ,This takes (TX, H) to (TX, H + dB), leading to theŠevera class [H] ∈ H 3 (X; R) determining TX up to Courant isomorphism. The Courant automorphisms of TX are generated by the diffeomorphisms and closed B-field transformations.Definition 2.1. A generalized complex structure on (X, H) is a complex structure J on TX that is orthogonal with respect to ·, · , and whose +i-eigenbundle is involutive underThere is an alternative definition of a generalized complex structure using spinors. To state it, recall that sections v = V + ξ ∈ Γ(TX) of the double tangent bundle act on differential forms via Clifford multiplication, given byDefinition 2.2. A generalized complex structure on (X, H) is given by a complex line bundle K J ⊂ ∧ • T * C X pointwise generated by a differential form ρ = e B+iω ∧ Ω with Ω a decomposable complex k-form, satisfying Ω ∧ Ω ∧ ω n−k = 0, and such that dρ + H ∧ ρ = v · ρ for any local section ρ ∈ Γ(K J ) and some v ∈ Γ(TX).Both definitions are related using that K J = Ann(E J ) is the annihilator under the Clifford action of E J , the +i-eigenbundle of J . The bundle K J is called the canonical bundle of J . For later use, we further introduce the analogue of a Calabi-Yau manifold in generalized geometry. Denote by d H = d + H∧ the H-twisted de Rham differential.Example 2.4. The following provide examples of generalized complex structures on (X, 0).• Let ω be a symplectic structure on X. Then K Jω := e iω defines a generalized complex structure J ω . • Let J be a complex structure on X with canonical bundle K J = ∧ n,0 T * X. Then K JJ := K J defines a generalized complex structure J J . • Let P ∈ Γ(∧ 2,0 T X) a holomorphic Poisson structure with respect to a complex structure J. Then K JP,J := e P K J defines a generalized complex structure J P,J .The automorphisms J ω , J J and J P,J are given by, with π = Im(P ):We next introduce the type of a generalized complex structure J , which colloquially provides a measure for how many complex directions there are. The type is an integer-valued upper semicontinuous function on X whose parity is locally constant.Definition 2.5. Let J be a generalized complex structure on X. The type of J is a m...

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“…Any compact type‐1 generalized Calabi–Yau manifold, such as

D_{J}

, fibers over the torus

T^{2}

. Moreover, the semilocal form of a stable generalized complex structure around its type change locus is given by its linearization along

D_{J}

, which is the stable generalized complex structure naturally present on the normal bundle to this type‐1 generalized Calabi–Yau manifold.…”

Section: Stable Generalized Complex and Log‐symplectic Structuresmentioning

confidence: 99%

“…Moreover, the semilocal form of a stable generalized complex structure around its type change locus is given by its linearization along

D_{J}

Section: Stable Generalized Complex and Log‐symplectic Structuresmentioning

confidence: 99%

“…After perturbing each structure, this can be achieved using one‐forms on

D

and

Z

induced by the geometric structures (that is, they can be made proper). For a stable generalized complex structure

scriptJ

on X, these one‐forms are

false({Res}_{r}, {Res}_{θ} false) false(ω_{X} false) \in {normalΩ}^{1} false(D false) \times {normalΩ}^{1} false(D false)

, where

ω_{X}

is the elliptic symplectic structure obtained from

scriptJ

using Theorem , see . In the log‐Poisson case, the one‐form used instead is

{Res}_{Z} false(ω_{Y} false)

(see ), with

ω_{Y}

the log‐symplectic structure induced by the log‐Poisson structure using Proposition .…”

Section: Constructing Stable Generalized Complex Structuresmentioning

confidence: 99%

See 1 more Smart Citation

Fibrations and stable generalized complex structures

Cavalcanti

Klaasse

2018

Proc. London Math. Soc.

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“…There are extensions of this statement when D is not cooriented (see [32,33]). Moreover, this result is known in the context of stable generalized complex geometry due to [7,12]. There are three residue maps, namely Res r , Res θ and Res q , so that the pure spinor above differs slightly from Proposition 5.51.…”

Section: 3mentioning

confidence: 83%

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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“…Namely, Z becomes cosymplectic ( [19]), while D inherits a 2-cosymplectic structure if it is coorientable (see [23]). 1 Both structures can be after slight perturbation assumed to be proper ( [10,28], and [2,8]), resulting in induced fibration maps Z → S 1 and D → T 2 (which themselves also form obstructions).…”

Section: Theorem 32 ([11]mentioning

confidence: 99%

Obstructions for Symplectic Lie Algebroids

Klaasse

2018

Preprint

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Several generically-nondegenerate Poisson structures can be effectively studied as symplectic structures on naturally associated Lie algebroids. Relevant examples of this phenomenon include log-, elliptic, b k -, scattering and elliptic-log Poisson structures.In this paper we discuss topological obstructions to the existence of such Poisson structures (obtained through their symplectic Lie algebroids) of several different flavors, namely coming from cohomology, characteristic classes, and Seiberg-Witten theory.

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Type one generalized Calabi–Yaus

Cited by 6 publications

References 11 publications

Fibrations and stable generalized complex structures

Fibrations and stable generalized complex structures

Poisson structures of divisor-type

Obstructions for Symplectic Lie Algebroids

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