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.

TheBottconnection The Bott connection 1/28

The Bott connection 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 2/28

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 Let ( A , J ) be a Lie pair . The Atiyah class of the Lie pair is a cohomology class α J ∈ H 1 ( 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 ∇ a b ∇ J -flat . ¯ ⇒ The Bott connection 5/28

Atiyah class of a Lie pair The Bott connection 6/28

Atiyah class of a Lie pair 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 Bott connection 7/28

TheAtiyahclassofa holomorphicvectorbundle The Atiyah class of a holomorphic vector bundle 8/28

Holomorphic vector bundles 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 = D 1 , 0 + D 0 , 1 : Γ( TM C ) × Γ( E ) → Γ( E ) such that D 0 , 1 is flat. The Atiyah class of a holomorphic vector bundle 9/28

Holomorphic connections Let E → M be a holomorphic vector bundle. A C -linear connection ∇ : Γ( TM ) × Γ( E ) → Γ( E ) is holomorphic if ∇ X e is holomorphic for X ∈ X ( M ) a local holomorphic vector field and e ∈ Γ U ( E ) a local holomorphic section. The Atiyah class of a holomorphic vector bundle 10/28

Holomorphic Lie algebroids 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 . The Atiyah class of a holomorphic vector bundle 11/28

Infinitesimalideals andtheAtiyahclass Infinitesimal ideals and the Atiyah class 12/28

Infinitesimal ideals Definition (JL-Ortiz 14, Hawkins 07) Let ( q : A → M , ρ, [ · , · ]) be a Lie algebroid, F M ⊆ TM an involutive subbundle, J ⊆ A a subalgebroid over M such that ρ ( J ) ⊆ F M and ∇ a flat F M -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 ∇ F M -flat. The triple ( F M , J , ∇ ) is an infinitesimal ideal system in A . Infinitesimal ideals and the Atiyah class 13/28

Infinitesimal ideals Infinitesimal ideals and the Atiyah class 14/28

Atiyah class of an infinitesimal ideal Let ( F M , J , ∇ i ) be an infinitesimal ideal in A . The Atiyah class α ∈ H 1 ( F M , Hom( TM / F M , End( A / J ))) of the infinitesimal ideal is a cohomology class that vanishes if and only if there exists an extension ∇ : X ( M ) × Γ( A ) → Γ( A ) of ∇ i such that � ¯ a ∈ Γ( A / J ) ∇ i -flat and ∇ X a ∇ i -flat . ⇒ ¯ X ∈ Γ( TM / F M ) ∇ F M -flat Infinitesimal ideals and the Atiyah class 15/28

Reducible algebroid Theorem Let ( F M , J , ∇ ) be an infinitesimal ideal in a Lie algebroid A → M. If the quotient vector bundle A ′ := ( A / J ) / ∇ → M / F M =: M ′ exists, then the Atiyah class of the infinitesimal ideal vanishes. In other words... Theorem Let ( F M , J , ∇ ) be an infinitesimal ideal in a Lie algebroid A → M. If ( F M , J , ∇ ) integrates to an ideal, then the Atiyah class of ( F M , J , ∇ ) vanishes. Infinitesimal ideals and the Atiyah class 16/28

Foliatedprincipalbundles Foliated principal bundles 17/28

Foliated principal bundles A principal G -bundle π : P → M is described infinitesimally by its Atiyah sequence 0 → g P → TP G → TM → 0 . A principal foliation on π : P → M is an involutive subbundle F ⊆ TP that is G -invariant; T p Φ g F ( p ) = F ( pg ) for all p ∈ P , g ∈ G , and with F π := F ∩ T π P of constant rank. Foliated principal bundles 18/28

The infinitesimal ideals Let g be the Lie algebra of the Lie group G . Then there is an ideal i ⊆ g such that F π ( p ) = { x P ( p ) | x ∈ i } for all p ∈ P . The associated bundle i P is a naive ideal in TP / G . Foliated principal bundles 19/28

The Atiyah class(es) Foliated principal bundles 20/28

LiepairsandtheAtiyahclass Lie pairs and the Atiyah class 22/28

( A , J ) a Lie pair; F M ⊆ TM involutive subbundle with ρ ( J ) ⊆ F M and a flat F M -connection ∇ on A / J . Then ρ ⋆ : Ω • ( F M , Hom( TM / F M , End( A / J ))) → Ω • ( J , Hom( A / J , End( A / J ))) is defined by ( ρ ⋆ ω )( j 1 , . . . , j p )( a 1 , a 2 ) = ω ( ρ ( j 1 ) , . . . , ρ ( j p ))( ρ ( a 1 ))( a 2 ) . a = ∇ J If ∇ ρ ( j ) ¯ j ¯ a for all j ∈ Γ( J ) , a ∈ Γ( A ) , then d ∇ J ◦ ρ ⋆ = ρ ⋆ ◦ d ∇ Hom . Lie pairs and the Atiyah class 23/28

Theorem (JL 19) If ( F M , J , ∇ ) is an infinitesimal ideal in A, then the image under ρ ⋆ of its Atiyah class α ∈ H 1 d ∇ Hom ( F M , Hom( TM / F M , End( A / J ))) is the Atiyah class α J ∈ H 1 d ∇ J ( J , Hom( A / J , End( A / J ))) of the Lie pair. Lie pairs and the Atiyah class 24/28

Obstruction result. Theorem (JL 19) Let ( A , J ) be a Lie pair. If ( A , J ) is an ideal pair, then α J = 0 . Lie pairs and the Atiyah class 25/28

References References 26/28

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 27/28

Thank you for your attention! References 28/28