ArXiv:1203.0493 Gil Cavalcanti Utrecht University Poisson Geometry in Mathematics and Physics Utrecht 2012 ��� ��������
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Introduction — SKT structures K¨ ahler structure with torsion : Hermitian structure ( g , I ) with a connection ∇ with Tor ( ∇ ) = H ∈ Ω 3 ( M ) such that ∇ g = ∇ I = 0 . K¨ ahler structure with strong torsion (SKT): dH = 0. SKT structure: Hermitian structure ( g , I ) such that dd c ω = 0 . H = d c ω . SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Examples K¨ ahler manifolds ; Compact even dimensional Lie groups ; Compact complex surfaces (Gauduchon); Instanton moduli space over compact complex surfaces; Instanton moduli space over Hermitian manifolds with Gauduchon metrics; Classification on 6-nilmanifolds ; SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations K¨ ahler vs. SKT K¨ ahler SKT Decomposition of cohomology � Hodge theory � Fr¨ olicher spectral seq. degenerates � Formality � Unobstructed deformations � Fino, Parton and Salamon. Families of strong KT structures in six dimensions. Comment. Math. Helv. 2004. (arXiv:math/0209259) SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations K¨ ahler vs. SKT K¨ ahler SKT ✪ Decomposition of cohomology � ✪ Hodge theory � ✪ Fr¨ olicher spectral seq. degenerates � ✪ Formality � ✪ Unobstructed deformations � Fino, Parton and Salamon. Families of strong KT structures in six dimensions. Comment. Math. Helv. 2004. (arXiv:math/0209259) SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations K¨ ahler vs. SKT K¨ ahler SKT ✪ Decomposition of cohomology � ✪ Hodge theory � ✪ Fr¨ olicher spectral seq. degenerates � ✪ Formality � ✪ ✪ Unobstructed deformations Fino, Parton and Salamon. Families of strong KT structures in six dimensions. Comment. Math. Helv. 2004. (arXiv:math/0209259) SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations K¨ ahler vs. SKT K¨ ahler SKT Decomposition of cohomology � � Hodge theory � � Fr¨ olicher spectral seq. degenerates � � ✪ Formality � ✪ ✪ Unobstructed deformations SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations K¨ ahler vs. GK vs. SKT K¨ ahler GK SKT Decomposition of cohomology � � � Hodge theory � � � Fr¨ olicher spectral seq. degenerates � � � ✪ ✪ Formality � ✪ ✪ ✪ Unobstructed deformations SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Main insights SKT structures are also generalized structures ` a la Hitchin. Description of intrinsic torsion of generalized almost Hermitian structures. SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Outline of Topics Geometry of T ⊕ T ∗ 1 Nijenhuis tensor and intrinsic torsion 2 Hodge theory 3 Deformations 4 SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Geometry of T ⊕ T ∗ — Natural pairing Natural pairing � X + ξ, Y + η � = 1 2 ( η ( X ) + ξ ( Y )) . Action of T ⊕ T ∗ on ∧ • T ∗ : ( X + ξ ) · ϕ = ι X ϕ + ξ ∧ ϕ. Extends to an action of Clif ( T ⊕ T ∗ ) on ∧ • T ∗ : v · ( v · ϕ ) = � v , v � ϕ. ∧ • T ∗ ❀ spinors. Spin invariant pairing: ( · , · ) Ch : ∧ • T ∗ ⊗ ∧ • T ∗ − → ∧ top T ∗ . SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Geometry of T ⊕ T ∗ — Generalized metric A generalized metric is an orthogonal, self-adjoint bundle isomorphism: G : T ⊕ T ∗ − → T ⊕ T ∗ ; such that �G v , v � > 0 . G − 1 = G t = G G 2 = Id . ⇒ G is determined by its + 1-eigenspace V + . SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Geometry of T ⊕ T ∗ — Generalized Hodge star Generalized metric + orientation ⇒ generalized Hodge star ⋆ ( ϕ, ⋆ϕ ) Ch > 0 . ⋆ 2 = ( − 1 ) m ( m − 1 ) . 2 m ( m − 1 ) SD forms = − i -eigenspace; 2 m ( m − 1 ) ASD forms = i -eigenspace; 2 SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Geometry of T ⊕ T ∗ — Gen. almost complex structure Generalized almost complex structure: J : T ⊕ T ∗ − J 2 = − Id ; → T ⊕ T ∗ ; J is orthogonal. J ⇔ L ⊂ ( T ⊕ T ∗ ) ⊗ C , maximal isotropic L ∩ L = { 0 } . J t = J − 1 = −J ⇒ J ∈ ∧ 2 ( T ⊕ T ∗ ) = spin ( T ⊕ T ∗ ) . J splits ∧ • T ∗ into its ik -eigenspaces: ∧ • T ∗ C M = ⊕ − n ≤ k ≤ n U k . SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Geometry of T ⊕ T ∗ — Gen. almost Hermitian str. Generalized almost Hermitian structure: ( G , J 1 ) GJ 1 = J 1 G . J 2 = GJ 1 is a gcs and J 2 J 1 = J 1 J 2 . ( T ⊕ T ∗ ) ⊗ C = V 1 , 0 + ⊕ V 0 , 1 + ⊕ V 1 , 0 − ⊕ V 0 , 1 − . C M = ⊕ p , q U p J 1 ∩ U q ∧ • T ∗ J 2 . SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Geometry of T ⊕ T ∗ — Gen. almost Hermitian str. U 0 , 3 U − 1 , 2 U 1 , 2 U − 2 , 1 U 0 , 1 U 2 , 1 U − 3 , 0 U − 1 , 0 U 1 , 0 U 3 , 0 U − 2 , − 1 U 0 , − 1 U 2 , − 1 U − 1 , − 2 U 1 , − 2 U 0 , − 3 Spaces U p , q on a 6-dimensional generalized almost Hermitian structure. SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Geometry of T ⊕ T ∗ — Gen. almost Hermitian str. Generalized almost Hermitian structure: ( G , J 1 ) GJ 1 = J 1 G . J 2 = GJ 1 is a gcs and J 2 J 1 = J 1 J 2 . ( T ⊕ T ∗ ) ⊗ C = V 1 , 0 + ⊕ V 0 , 1 + ⊕ V 1 , 0 − ⊕ V 0 , 1 − . C M = ⊕ p , q U p J 1 ∩ U q ∧ • T ∗ J 2 . π J 1 π J 2 ⋆ = − e 2 e 2 ⋆ | U p , q = − i p + q . SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Geometry of T ⊕ T ∗ — Gen. almost Hermitian str. U 0 , 3 U − 1 , 2 U 1 , 2 U − 2 , 1 U 0 , 1 U 2 , 1 U − 3 , 0 U − 1 , 0 U 1 , 0 U 3 , 0 U − 2 , − 1 U 0 , − 1 U 2 , − 1 U − 1 , − 2 U 1 , − 2 U 0 , − 3 SD and ASD forms on a 6-dimensional generalized almost Hermitian structure. SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Geometry of T ⊕ T ∗ — Gen. almost Hermitian str. U 0 , 3 U − 1 , 2 U 1 , 2 U − 2 , 1 U 0 , 1 U 2 , 1 U − 3 , 0 U − 1 , 0 U 1 , 0 U 3 , 0 U − 2 , − 1 U 0 , − 1 U 2 , − 1 U − 1 , − 2 U 1 , − 2 U 0 , − 3 SD and ASD forms on a 6-dimensional generalized almost Hermitian structure. SKT geometry Cavalcanti
Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Geometry of T ⊕ T ∗ — Courant bracket Courant bracket [ [ X + ξ, Y + η ] ] H = [ X , Y ] + L X η − ι Y d ξ − ι Y ι X H . ] H · ϕ = {{ v 1 , d H } , v 2 } · ϕ. [ [ v 1 , v 2 ] SKT geometry Cavalcanti
� � � Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Nijenhuis tensor and intrinsic torsion Given a gacs J , define → Ω 0 ( M ; C ) N : Γ( L ) × Γ( L ) × Γ( L ) − N ( v 1 , v 2 , v 3 ) = − 2 � [ [ v 1 , v 2 ] ] , v 3 � . J is integrable iff N ≡ 0 . N ∈ Γ( ∧ 3 L ) . N ∂ J 1 � U k − 3 U k − 2 U k − 1 U k U k + 1 U k + 2 U k + 3 N ∂ J 1 SKT geometry Cavalcanti
� � � � � � � � � � � � � � Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Nijenhuis tensor and intrinsic torsion U p − 3 , q + 3 U p − 1 , q + 3 U p + 1 , q + 3 U p + 3 , q + 3 N − N + N 1 N 2 U p − 3 , q + 1 U p − 1 , q + 1 U p + 1 , q + 1 U p + 3 , q + 1 δ − δ + N 3 N 4 U p , q N 3 N 4 � δ − � δ + U p − 3 , q − 1 U p − 1 , q − 1 U p + 1 , q − 1 U p + 3 , q − 1 N 2 N 1 N − N + U p − 3 , q − 3 U p − 1 , q − 3 U p + 1 , q − 3 U p + 3 , q − 3 Components of d H for a generalized almost Hermitian structure. SKT geometry Cavalcanti
� � � � � � � Introduction Outline Geometry of T ⊕ T ∗ Intrinsic torsion Hodge theory Deformations Nijenhuis tensor and intrinsic torsion U p − 1 , q + 3 U p + 1 , q + 3 N 1 N 2 U p − 1 , q + 1 U p + 1 , q + 1 δ − δ + U p , q δ − � δ + U p − 3 , q − 1 U p − 1 , q − 1 N 2 N 1 U p − 1 , q − 3 U p + 1 , q − 3 Components of d H for a generalized Hermitian structure. SKT geometry Cavalcanti
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