Astheno-Kähler and strong KT metrics Anna Fino Astheno-Kähler and strong KT General results metrics Bismut connection Definition of strong KT and astheno-Kähler metrics Link with balanced metrics Link with standard metrics Holonomy of ∇ B in SU ( n ) Blow-ups Resolution of orbifolds “XVII International Fall Workshop on Geometry and Physics” – Construction of 3-6 September 2008 examples 6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example Anna Fino 1 Dipartimento di Matematica Università di Torino
Astheno-Kähler and strong KT metrics Anna Fino General results 1 General results Bismut connection Bismut connection Definition of strong KT and astheno-Kähler metrics Definition of strong KT and astheno-Kähler metrics Link with balanced metrics Link with standard metrics Link with balanced metrics Holonomy of ∇ B in SU ( n ) Link with standard metrics Blow-ups Resolution of orbifolds Holonomy of ∇ B in SU ( n ) Construction of examples Blow-ups 6-dimensional strong KT nilmanifolds Resolution of orbifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example Construction of examples 2 6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 2
Astheno-Kähler and Bismut connection strong KT metrics Anna Fino General results Bismut connection Definition of strong KT and On any Hermitian manifold ( M 2 n , J , g ) ∃ ! connection ∇ B such astheno-Kähler metrics Link with balanced metrics that Link with standard metrics Holonomy of ∇ B in SU ( n ) ∇ B g = 0 ( metric ) Blow-ups ∇ B J = 0 ( Hermitian ) Resolution of orbifolds c ( X , Y , Z ) = g ( X , T B ( Y , Z )) 3-form Construction of examples 6-dimensional strong KT where T B is the torsion of ∇ B . nilmanifolds 8-dimensional astheno-Kähler nilmanifolds ∇ B = ∇ LC + 1 A simply-connected 2 c is the Bismut connection and c = − JdJF , example where F = g ( J · , · ) is the associated Kähler form. ∇ B is also called a KT connection and in general Hol ( ∇ B ) ⊆ U ( n ) . 3
Astheno-Kähler and Bismut connection strong KT metrics Anna Fino General results Bismut connection Definition of strong KT and On any Hermitian manifold ( M 2 n , J , g ) ∃ ! connection ∇ B such astheno-Kähler metrics Link with balanced metrics that Link with standard metrics Holonomy of ∇ B in SU ( n ) ∇ B g = 0 ( metric ) Blow-ups ∇ B J = 0 ( Hermitian ) Resolution of orbifolds c ( X , Y , Z ) = g ( X , T B ( Y , Z )) 3-form Construction of examples 6-dimensional strong KT where T B is the torsion of ∇ B . nilmanifolds 8-dimensional astheno-Kähler nilmanifolds ∇ B = ∇ LC + 1 A simply-connected 2 c is the Bismut connection and c = − JdJF , example where F = g ( J · , · ) is the associated Kähler form. ∇ B is also called a KT connection and in general Hol ( ∇ B ) ⊆ U ( n ) . 3
Astheno-Kähler and Bismut connection strong KT metrics Anna Fino General results Bismut connection Definition of strong KT and On any Hermitian manifold ( M 2 n , J , g ) ∃ ! connection ∇ B such astheno-Kähler metrics Link with balanced metrics that Link with standard metrics Holonomy of ∇ B in SU ( n ) ∇ B g = 0 ( metric ) Blow-ups ∇ B J = 0 ( Hermitian ) Resolution of orbifolds c ( X , Y , Z ) = g ( X , T B ( Y , Z )) 3-form Construction of examples 6-dimensional strong KT where T B is the torsion of ∇ B . nilmanifolds 8-dimensional astheno-Kähler nilmanifolds ∇ B = ∇ LC + 1 A simply-connected 2 c is the Bismut connection and c = − JdJF , example where F = g ( J · , · ) is the associated Kähler form. ∇ B is also called a KT connection and in general Hol ( ∇ B ) ⊆ U ( n ) . 3
Astheno-Kähler and Strong KT and astheno-Kähler metrics strong KT metrics Anna Fino ⇒ ∇ B = ∇ LC ⇐ c = 0 ⇐ ⇒ ( M , J , g ) is Kähler General results Bismut connection Definition of strong KT and Definition astheno-Kähler metrics Link with balanced metrics A Hermitian structure ( J , g ) on M 2 n is said to be strong Kähler Link with standard metrics Holonomy of ∇ B in SU ( n ) with torsion (strong KT) if dc = 0, i.e. if ∂∂ F = 0. Blow-ups Resolution of orbifolds Construction of examples Definition (Jost, Yau) 6-dimensional strong KT nilmanifolds ( J , g ) on M 2 n is called astheno-Kähler if ∂∂ F n − 2 = 0. 8-dimensional astheno-Kähler nilmanifolds A simply-connected example If n = 2 ⇒ any Hermitian metric is astheno-Kähler. If n = 3 ⇒ strong KT= astheno-Kähler. • If ∃ a astheno-Kähler metric on a compact complex manifold, then any holomorphic 1-form must be closed [Jost-Yau]. ⇒ a complex parallelizable manifold ( M , J ) cannot admit any astheno-Kähler metric compatible with J . 4
Astheno-Kähler and Strong KT and astheno-Kähler metrics strong KT metrics Anna Fino ⇒ ∇ B = ∇ LC ⇐ c = 0 ⇐ ⇒ ( M , J , g ) is Kähler General results Bismut connection Definition of strong KT and Definition astheno-Kähler metrics Link with balanced metrics A Hermitian structure ( J , g ) on M 2 n is said to be strong Kähler Link with standard metrics Holonomy of ∇ B in SU ( n ) with torsion (strong KT) if dc = 0, i.e. if ∂∂ F = 0. Blow-ups Resolution of orbifolds Construction of examples Definition (Jost, Yau) 6-dimensional strong KT nilmanifolds ( J , g ) on M 2 n is called astheno-Kähler if ∂∂ F n − 2 = 0. 8-dimensional astheno-Kähler nilmanifolds A simply-connected example If n = 2 ⇒ any Hermitian metric is astheno-Kähler. If n = 3 ⇒ strong KT= astheno-Kähler. • If ∃ a astheno-Kähler metric on a compact complex manifold, then any holomorphic 1-form must be closed [Jost-Yau]. ⇒ a complex parallelizable manifold ( M , J ) cannot admit any astheno-Kähler metric compatible with J . 4
Astheno-Kähler and Strong KT and astheno-Kähler metrics strong KT metrics Anna Fino ⇒ ∇ B = ∇ LC ⇐ c = 0 ⇐ ⇒ ( M , J , g ) is Kähler General results Bismut connection Definition of strong KT and Definition astheno-Kähler metrics Link with balanced metrics A Hermitian structure ( J , g ) on M 2 n is said to be strong Kähler Link with standard metrics Holonomy of ∇ B in SU ( n ) with torsion (strong KT) if dc = 0, i.e. if ∂∂ F = 0. Blow-ups Resolution of orbifolds Construction of examples Definition (Jost, Yau) 6-dimensional strong KT nilmanifolds ( J , g ) on M 2 n is called astheno-Kähler if ∂∂ F n − 2 = 0. 8-dimensional astheno-Kähler nilmanifolds A simply-connected example If n = 2 ⇒ any Hermitian metric is astheno-Kähler. If n = 3 ⇒ strong KT= astheno-Kähler. • If ∃ a astheno-Kähler metric on a compact complex manifold, then any holomorphic 1-form must be closed [Jost-Yau]. ⇒ a complex parallelizable manifold ( M , J ) cannot admit any astheno-Kähler metric compatible with J . 4
Astheno-Kähler and Strong KT and astheno-Kähler metrics strong KT metrics Anna Fino ⇒ ∇ B = ∇ LC ⇐ c = 0 ⇐ ⇒ ( M , J , g ) is Kähler General results Bismut connection Definition of strong KT and Definition astheno-Kähler metrics Link with balanced metrics A Hermitian structure ( J , g ) on M 2 n is said to be strong Kähler Link with standard metrics Holonomy of ∇ B in SU ( n ) with torsion (strong KT) if dc = 0, i.e. if ∂∂ F = 0. Blow-ups Resolution of orbifolds Construction of examples Definition (Jost, Yau) 6-dimensional strong KT nilmanifolds ( J , g ) on M 2 n is called astheno-Kähler if ∂∂ F n − 2 = 0. 8-dimensional astheno-Kähler nilmanifolds A simply-connected example If n = 2 ⇒ any Hermitian metric is astheno-Kähler. If n = 3 ⇒ strong KT= astheno-Kähler. • If ∃ a astheno-Kähler metric on a compact complex manifold, then any holomorphic 1-form must be closed [Jost-Yau]. ⇒ a complex parallelizable manifold ( M , J ) cannot admit any astheno-Kähler metric compatible with J . 4
Astheno-Kähler and Strong KT and astheno-Kähler metrics strong KT metrics Anna Fino ⇒ ∇ B = ∇ LC ⇐ c = 0 ⇐ ⇒ ( M , J , g ) is Kähler General results Bismut connection Definition of strong KT and Definition astheno-Kähler metrics Link with balanced metrics A Hermitian structure ( J , g ) on M 2 n is said to be strong Kähler Link with standard metrics Holonomy of ∇ B in SU ( n ) with torsion (strong KT) if dc = 0, i.e. if ∂∂ F = 0. Blow-ups Resolution of orbifolds Construction of examples Definition (Jost, Yau) 6-dimensional strong KT nilmanifolds ( J , g ) on M 2 n is called astheno-Kähler if ∂∂ F n − 2 = 0. 8-dimensional astheno-Kähler nilmanifolds A simply-connected example If n = 2 ⇒ any Hermitian metric is astheno-Kähler. If n = 3 ⇒ strong KT= astheno-Kähler. • If ∃ a astheno-Kähler metric on a compact complex manifold, then any holomorphic 1-form must be closed [Jost-Yau]. ⇒ a complex parallelizable manifold ( M , J ) cannot admit any astheno-Kähler metric compatible with J . 4
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