TheAtiyahclassandideal systems by M. Jotz Lean Friday Fish - - PowerPoint PPT Presentation

TheAtiyahclassandideal systems

by

• M. Jotz Lean

Friday Fish Seminar — Utrecht, 14.8.2020

Contents

• 1. The Bott connection.

The “universal” example.

• 2. The Atiyah class of a holomorphic vector bundle.

Holomorphic vector bundles and holomorphic Lie algebroids as infinitesimal ideal systems.

• 3. Infinitesimal ideals and the Atiyah class.

Example: The Atiyah class(es) of a foliated principal bundle.

• 4. Lie pairs and the Atiyah class.

Obstruction to a Lie pair carrying an ideal pair.

The Bott connection 1/28

TheBottconnection

The Bott connection

The Bott connection 2/28

Let A → M be a Lie algebroid and J ⊆ A a subalgebroid. The Bott connection is the flat J-connection on A/J defined by ∇J : Γ(J) × Γ(A/J) → Γ(A/J), ∇J

j ¯

a = [j, a].

The Bott connection and foliations

The Bott connection 3/28

The Bott connection – properties

The Bott connection 4/28

Atiyah class of a Lie pair

The Bott connection 5/28

Let (A, J) be a Lie pair. The Atiyah class of the Lie pair is a cohomology class αJ ∈ H1(J, Hom(A/J, End(A/J))). If it vanishes, there exists an extension ∇: Γ(A) × Γ(A) → Γ(A) of ∇J such that ¯ a, ¯ b ∈ Γ(A/J) ∇J-flat ⇒ ∇ab ∇J-flat.

Atiyah class of a Lie pair

The Bott connection 6/28

Atiyah class of a Lie pair

The Bott connection 7/28

Chen, Stiénon, Xu 2016: From Atiyah Classes to Homotopy Leibniz Algebras. Laurent-Gengoux, Stiénon, Xu 2014: Poincaré–Birkhoff–Witt isomorphisms and Kapranov dg-manifolds.

The Atiyah class of a holomorphic vector bundle 8/28

TheAtiyahclassofa holomorphicvectorbundle

Holomorphic vector bundles

The Atiyah class of a holomorphic vector bundle 9/28

Theorem (Kobayashi)

Let E → M be a complex vector bundle over a complex manifold M. Then E is a holomorphic vector bundle if and only if there exists a C-linear connection D = D1,0 + D0,1 : Γ(TMC) × Γ(E) → Γ(E) such that D0,1 is flat.

Holomorphic connections

The Atiyah class of a holomorphic vector bundle 10/28

Let E → M be a holomorphic vector bundle. A C-linear connection ∇: Γ(TM) × Γ(E) → Γ(E) is holomorphic if ∇Xe is holomorphic for X ∈ X(M) a local holomorphic vector field and e ∈ ΓU(E) a local holomorphic section.

Holomorphic Lie algebroids

The Atiyah class of a holomorphic vector bundle 11/28

A holomorphic Lie algebroid is a holomorphic vector bundle A → M with a Lie algebroid structure (ρ, [· , ·]) such that [A, A] ⊆ A, and ρ(A) ⊆ X.

Infinitesimal ideals and the Atiyah class 12/28

Infinitesimalideals andtheAtiyahclass

Infinitesimal ideals

Infinitesimal ideals and the Atiyah class 13/28

Definition (JL-Ortiz 14, Hawkins 07)

Let (q: A → M, ρ, [· , ·]) be a Lie algebroid, FM ⊆ TM an involutive subbundle, J ⊆ A a subalgebroid over M such that ρ(J) ⊆ FM and ∇ a flat FM-connection on A/J with the following properties:

• 1. If a ∈ Γ(A) is ∇-flat, then [a, j] ∈ Γ(J) for all j ∈ Γ(J).
• 2. If a, b ∈ Γ(A) are ∇-flat, then [a, b] is also ∇-flat.
• 3. If a ∈ Γ(A) is ∇-flat, then ρ(a) is ∇FM-flat.

The triple (FM, J, ∇) is an infinitesimal ideal system in A.

Infinitesimal ideals

Infinitesimal ideals and the Atiyah class 14/28

Atiyah class of an infinitesimal ideal

Infinitesimal ideals and the Atiyah class 15/28

Let (FM, J, ∇i) be an infinitesimal ideal in A. The Atiyah class α ∈ H1(FM, Hom(TM/FM, End(A/J)))

• f the infinitesimal ideal is a cohomology class that vanishes if and
• nly if there exists an extension ∇: X(M) × Γ(A) → Γ(A) of ∇i such

that ¯ a ∈ Γ(A/J) ∇i-flat and ¯ X ∈ Γ(TM/FM) ∇FM-flat ⇒ ∇Xa ∇i-flat.

Reducible algebroid

Infinitesimal ideals and the Atiyah class 16/28

Theorem

Let (FM, J, ∇) be an infinitesimal ideal in a Lie algebroid A → M. If the quotient vector bundle A′ := (A/J)/∇ → M/FM =: M′ exists, then the Atiyah class of the infinitesimal ideal vanishes. In other words...

Theorem

Let (FM, J, ∇) be an infinitesimal ideal in a Lie algebroid A → M. If (FM, J, ∇) integrates to an ideal, then the Atiyah class of (FM, J, ∇) vanishes.

Foliated principal bundles 17/28

Foliatedprincipalbundles

Foliated principal bundles

Foliated principal bundles 18/28

A principal G-bundle π: P → M is described infinitesimally by its Atiyah sequence 0 → gP → TP G → TM → 0. A principal foliation on π: P → M is an involutive subbundle F ⊆ TP

that is G-invariant; TpΦgF(p) = F(pg) for all p ∈ P, g ∈ G, and with Fπ := F ∩ TπP of constant rank.

The infinitesimal ideals

Foliated principal bundles 19/28

Let g be the Lie algebra of the Lie group G. Then there is an ideal i ⊆ g such that Fπ(p) = {xP(p) | x ∈ i} for all p ∈ P. The associated bundle iP is a naive ideal in TP/G.

The Atiyah class(es)

Foliated principal bundles 20/28

Lie pairs and the Atiyah class 22/28

LiepairsandtheAtiyahclass

Lie pairs and the Atiyah class 23/28

(A, J) a Lie pair; FM ⊆ TM involutive subbundle with ρ(J) ⊆ FM and a flat FM-connection ∇ on A/J. Then

ρ⋆ : Ω•(FM, Hom(TM/FM, End(A/J))) → Ω•(J, Hom(A/J, End(A/J)))

is defined by (ρ⋆ω)(j1, . . . , jp)(a1, a2) = ω(ρ(j1), . . . , ρ(jp))(ρ(a1))(a2). If ∇ρ(j)¯ a = ∇J

j ¯

a for all j ∈ Γ(J), a ∈ Γ(A), then d∇J ◦ ρ⋆ = ρ⋆ ◦ d∇Hom.

Lie pairs and the Atiyah class 24/28

Theorem (JL 19)

If (FM, J, ∇) is an infinitesimal ideal in A, then the image under ρ⋆ of its Atiyah class α ∈ H1

d∇Hom(FM, Hom(TM/FM, End(A/J)))

is the Atiyah class αJ ∈ H1

d∇J(J, Hom(A/J, End(A/J)))

• f the Lie pair.

Obstruction result.

Lie pairs and the Atiyah class 25/28

Theorem (JL 19)

Let (A, J) be a Lie pair. If (A, J) is an ideal pair, then αJ = 0.

References 26/28

References

References 27/28

Foliated groupoids and infinitesimal ideal systems, with Cristian Ortiz, "Indagationes Mathematicae " (2014), Volume 25, Number 5, 1019-1053. VB-algebroids morphisms and representations up to homotopy, with Thiago Drummond and Cristian Ortiz, "Differential Geometry and its Applications" (2015), Volume 40, 332-357. Obstructions to representations up to homotopy and ideals, arXiv:1905.10237, 2019. Infinitesimal ideal systems and the Atiyah class, arXiv:1910.04492, 2019, new version very soon! Linear generalised complex structures, with Malte Heuer, in preparation (2020).

References 28/28

Thank you for your attention!