Astheno-Khler and strong KT General results metrics Bismut - - PowerPoint PPT Presentation

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 1

Astheno-Kähler and strong KT metrics

“XVII International Fall Workshop on Geometry and Physics” – 3-6 September 2008 Anna Fino Dipartimento di Matematica Università di Torino

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 2

1

General results 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

2

Construction of examples 6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 3

Bismut connection On any Hermitian manifold (M2n, J, g) ∃! connection ∇B such that ∇Bg = 0 (metric) ∇BJ = 0 (Hermitian) c(X, Y, Z) = g(X, T B(Y, Z)) 3-form where T B is the torsion of ∇B. ∇B = ∇LC + 1

2c is the Bismut connection and c = −JdJF,

where F = g(J·, ·) is the associated Kähler form. ∇B is also called a KT connection and in general Hol(∇B) ⊆ U(n).

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 3

Bismut connection On any Hermitian manifold (M2n, J, g) ∃! connection ∇B such that ∇Bg = 0 (metric) ∇BJ = 0 (Hermitian) c(X, Y, Z) = g(X, T B(Y, Z)) 3-form where T B is the torsion of ∇B. ∇B = ∇LC + 1

2c is the Bismut connection and c = −JdJF,

where F = g(J·, ·) is the associated Kähler form. ∇B is also called a KT connection and in general Hol(∇B) ⊆ U(n).

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 3

Bismut connection On any Hermitian manifold (M2n, J, g) ∃! connection ∇B such that ∇Bg = 0 (metric) ∇BJ = 0 (Hermitian) c(X, Y, Z) = g(X, T B(Y, Z)) 3-form where T B is the torsion of ∇B. ∇B = ∇LC + 1

2c is the Bismut connection and c = −JdJF,

where F = g(J·, ·) is the associated Kähler form. ∇B is also called a KT connection and in general Hol(∇B) ⊆ U(n).

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 4

Strong KT and astheno-Kähler metrics c = 0 ⇐ ⇒ ∇B = ∇LC ⇐ ⇒ (M, J, g) is Kähler

Definition

A Hermitian structure (J, g) on M2n is said to be strong Kähler with torsion (strong KT) if dc = 0, i.e. if ∂∂F = 0.

Definition (Jost, Yau)

(J, g) on M2n is called astheno-Kähler if ∂∂ F n−2 = 0. 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.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 4

Strong KT and astheno-Kähler metrics c = 0 ⇐ ⇒ ∇B = ∇LC ⇐ ⇒ (M, J, g) is Kähler

Definition

A Hermitian structure (J, g) on M2n is said to be strong Kähler with torsion (strong KT) if dc = 0, i.e. if ∂∂F = 0.

Definition (Jost, Yau)

(J, g) on M2n is called astheno-Kähler if ∂∂ F n−2 = 0. 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.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 4

Strong KT and astheno-Kähler metrics c = 0 ⇐ ⇒ ∇B = ∇LC ⇐ ⇒ (M, J, g) is Kähler

Definition

A Hermitian structure (J, g) on M2n is said to be strong Kähler with torsion (strong KT) if dc = 0, i.e. if ∂∂F = 0.

Definition (Jost, Yau)

(J, g) on M2n is called astheno-Kähler if ∂∂ F n−2 = 0. 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.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 4

Strong KT and astheno-Kähler metrics c = 0 ⇐ ⇒ ∇B = ∇LC ⇐ ⇒ (M, J, g) is Kähler

Definition

A Hermitian structure (J, g) on M2n is said to be strong Kähler with torsion (strong KT) if dc = 0, i.e. if ∂∂F = 0.

Definition (Jost, Yau)

(J, g) on M2n is called astheno-Kähler if ∂∂ F n−2 = 0. 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.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 4

Strong KT and astheno-Kähler metrics c = 0 ⇐ ⇒ ∇B = ∇LC ⇐ ⇒ (M, J, g) is Kähler

Definition

A Hermitian structure (J, g) on M2n is said to be strong Kähler with torsion (strong KT) if dc = 0, i.e. if ∂∂F = 0.

Definition (Jost, Yau)

(J, g) on M2n is called astheno-Kähler if ∂∂ F n−2 = 0. 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.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 5

Link with balanced metrics θ = Jd∗F Lee form If θ = 0, the metric is said to be balanced (semi-kähler or cosymplectic) ⇐ ⇒ dF n−1 = 0.

Theorem (Alexandrov, Ivanov)

(M2n, J, g), n > 2 can be strong KT (non Kähler) only if θ = 0.

Theorem (Matsuo, Takahashi)

(M2n, J, g), n > 2 can be astheno-Kähler (non Kähler) only if θ = 0.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 5

Link with balanced metrics θ = Jd∗F Lee form If θ = 0, the metric is said to be balanced (semi-kähler or cosymplectic) ⇐ ⇒ dF n−1 = 0.

Theorem (Alexandrov, Ivanov)

(M2n, J, g), n > 2 can be strong KT (non Kähler) only if θ = 0.

Theorem (Matsuo, Takahashi)

(M2n, J, g), n > 2 can be astheno-Kähler (non Kähler) only if θ = 0.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 5

Link with balanced metrics θ = Jd∗F Lee form If θ = 0, the metric is said to be balanced (semi-kähler or cosymplectic) ⇐ ⇒ dF n−1 = 0.

Theorem (Alexandrov, Ivanov)

(M2n, J, g), n > 2 can be strong KT (non Kähler) only if θ = 0.

Theorem (Matsuo, Takahashi)

(M2n, J, g), n > 2 can be astheno-Kähler (non Kähler) only if θ = 0.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 5

Link with balanced metrics θ = Jd∗F Lee form If θ = 0, the metric is said to be balanced (semi-kähler or cosymplectic) ⇐ ⇒ dF n−1 = 0.

Theorem (Alexandrov, Ivanov)

(M2n, J, g), n > 2 can be strong KT (non Kähler) only if θ = 0.

Theorem (Matsuo, Takahashi)

(M2n, J, g), n > 2 can be astheno-Kähler (non Kähler) only if θ = 0.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 6

Link with standard metrics

Definition

A Hermitian structure (J, g) on a M2n is called standard if ∂∂F n−1 = 0 or equivalently if the Lee form θ is co-closed.

Theorem (Gauduchon)

For a compact complex manifold a standard metric can be found in the conformal class of any given Hermitian metric. If n = 2 ⇒ standard = strong KT If n > 2 a strong KT g is standard ⇔ |dF|2 = (n − 1)|θ ∧ F|2.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 6

Link with standard metrics

Definition

A Hermitian structure (J, g) on a M2n is called standard if ∂∂F n−1 = 0 or equivalently if the Lee form θ is co-closed.

Theorem (Gauduchon)

For a compact complex manifold a standard metric can be found in the conformal class of any given Hermitian metric. If n = 2 ⇒ standard = strong KT If n > 2 a strong KT g is standard ⇔ |dF|2 = (n − 1)|θ ∧ F|2.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 6

Link with standard metrics

Definition

A Hermitian structure (J, g) on a M2n is called standard if ∂∂F n−1 = 0 or equivalently if the Lee form θ is co-closed.

Theorem (Gauduchon)

For a compact complex manifold a standard metric can be found in the conformal class of any given Hermitian metric. If n = 2 ⇒ standard = strong KT If n > 2 a strong KT g is standard ⇔ |dF|2 = (n − 1)|θ ∧ F|2.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 6

Link with standard metrics

Definition

A Hermitian structure (J, g) on a M2n is called standard if ∂∂F n−1 = 0 or equivalently if the Lee form θ is co-closed.

Theorem (Gauduchon)

For a compact complex manifold a standard metric can be found in the conformal class of any given Hermitian metric. If n = 2 ⇒ standard = strong KT If n > 2 a strong KT g is standard ⇔ |dF|2 = (n − 1)|θ ∧ F|2.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 7

Holonomy of ∇B in SU(n) If θ is exact (closed), the metric is said to be (locally) conformally balanced.

Theorem (Ivanov, Papadopolous)

A conformally balanced strong KT structure (J, g) on a compact manifod M2n with Hol(∇B) ⊆ SU(n) is necessarily Kähler and therefore it is a Calabi-Yau structure.

Theorem (–, Tomassini)

A conformally balanced astheno-Kähler structure (J, g) on a compact manifod M2n, n ≥ 3, with Hol(∇B) ⊆ SU(n) is necessarily Kähler and therefore it is a Calabi-Yau structure. ∃ examples of locally conformally balanced astheno-Kähler (non-Kähler) manifolds (M6, J, g) with Hol(∇B) ⊆ SU(3).

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 7

Holonomy of ∇B in SU(n) If θ is exact (closed), the metric is said to be (locally) conformally balanced.

Theorem (Ivanov, Papadopolous)

A conformally balanced strong KT structure (J, g) on a compact manifod M2n with Hol(∇B) ⊆ SU(n) is necessarily Kähler and therefore it is a Calabi-Yau structure.

Theorem (–, Tomassini)

A conformally balanced astheno-Kähler structure (J, g) on a compact manifod M2n, n ≥ 3, with Hol(∇B) ⊆ SU(n) is necessarily Kähler and therefore it is a Calabi-Yau structure. ∃ examples of locally conformally balanced astheno-Kähler (non-Kähler) manifolds (M6, J, g) with Hol(∇B) ⊆ SU(3).

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 7

Holonomy of ∇B in SU(n) If θ is exact (closed), the metric is said to be (locally) conformally balanced.

Theorem (Ivanov, Papadopolous)

A conformally balanced strong KT structure (J, g) on a compact manifod M2n with Hol(∇B) ⊆ SU(n) is necessarily Kähler and therefore it is a Calabi-Yau structure.

Theorem (–, Tomassini)

A conformally balanced astheno-Kähler structure (J, g) on a compact manifod M2n, n ≥ 3, with Hol(∇B) ⊆ SU(n) is necessarily Kähler and therefore it is a Calabi-Yau structure. ∃ examples of locally conformally balanced astheno-Kähler (non-Kähler) manifolds (M6, J, g) with Hol(∇B) ⊆ SU(3).

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 7

Holonomy of ∇B in SU(n) If θ is exact (closed), the metric is said to be (locally) conformally balanced.

Theorem (Ivanov, Papadopolous)

A conformally balanced strong KT structure (J, g) on a compact manifod M2n with Hol(∇B) ⊆ SU(n) is necessarily Kähler and therefore it is a Calabi-Yau structure.

Theorem (–, Tomassini)

A conformally balanced astheno-Kähler structure (J, g) on a compact manifod M2n, n ≥ 3, with Hol(∇B) ⊆ SU(n) is necessarily Kähler and therefore it is a Calabi-Yau structure. ∃ examples of locally conformally balanced astheno-Kähler (non-Kähler) manifolds (M6, J, g) with Hol(∇B) ⊆ SU(3).

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 7

Holonomy of ∇B in SU(n) If θ is exact (closed), the metric is said to be (locally) conformally balanced.

Theorem (Ivanov, Papadopolous)

A conformally balanced strong KT structure (J, g) on a compact manifod M2n with Hol(∇B) ⊆ SU(n) is necessarily Kähler and therefore it is a Calabi-Yau structure.

Theorem (–, Tomassini)

A conformally balanced astheno-Kähler structure (J, g) on a compact manifod M2n, n ≥ 3, with Hol(∇B) ⊆ SU(n) is necessarily Kähler and therefore it is a Calabi-Yau structure. ∃ examples of locally conformally balanced astheno-Kähler (non-Kähler) manifolds (M6, J, g) with Hol(∇B) ⊆ SU(3).

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 8

If Hol(∇B) ⊆ SU(n) and M is compact ⇒ c1(M) = 0.

Conjecture (Gutowski, Ivanov, Papadopoulos)

Any compact complex manifold (M, J) with c1(M) = 0 admits a Hermitian metric such that Hol(∇B) ⊆ SU(n). A counter-example of the above conjecture has been found, showing that the condition is also not stable under small deformations of the complex structure [–, Grantcharov].

Theorem (–, Grantcharov)

Let (M2n, J) be a compact complex manifold with a holomorphic non-vanishing (n, 0)-form. If the Ricci tensor of the Bismut connection of some Hermitian metric g vanishes, (M, J, g) is conformally balanced and in particular admits a balanced metric. The previous theorem can be applied to any nilmanifold Γ\G with a left-invariant complex structure.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 8

If Hol(∇B) ⊆ SU(n) and M is compact ⇒ c1(M) = 0.

Conjecture (Gutowski, Ivanov, Papadopoulos)

Any compact complex manifold (M, J) with c1(M) = 0 admits a Hermitian metric such that Hol(∇B) ⊆ SU(n). A counter-example of the above conjecture has been found, showing that the condition is also not stable under small deformations of the complex structure [–, Grantcharov].

Theorem (–, Grantcharov)

Let (M2n, J) be a compact complex manifold with a holomorphic non-vanishing (n, 0)-form. If the Ricci tensor of the Bismut connection of some Hermitian metric g vanishes, (M, J, g) is conformally balanced and in particular admits a balanced metric. The previous theorem can be applied to any nilmanifold Γ\G with a left-invariant complex structure.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 8

If Hol(∇B) ⊆ SU(n) and M is compact ⇒ c1(M) = 0.

Conjecture (Gutowski, Ivanov, Papadopoulos)

Any compact complex manifold (M, J) with c1(M) = 0 admits a Hermitian metric such that Hol(∇B) ⊆ SU(n). A counter-example of the above conjecture has been found, showing that the condition is also not stable under small deformations of the complex structure [–, Grantcharov].

Theorem (–, Grantcharov)

Let (M2n, J) be a compact complex manifold with a holomorphic non-vanishing (n, 0)-form. If the Ricci tensor of the Bismut connection of some Hermitian metric g vanishes, (M, J, g) is conformally balanced and in particular admits a balanced metric. The previous theorem can be applied to any nilmanifold Γ\G with a left-invariant complex structure.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 8

If Hol(∇B) ⊆ SU(n) and M is compact ⇒ c1(M) = 0.

Conjecture (Gutowski, Ivanov, Papadopoulos)

Any compact complex manifold (M, J) with c1(M) = 0 admits a Hermitian metric such that Hol(∇B) ⊆ SU(n). A counter-example of the above conjecture has been found, showing that the condition is also not stable under small deformations of the complex structure [–, Grantcharov].

Theorem (–, Grantcharov)

Let (M2n, J) be a compact complex manifold with a holomorphic non-vanishing (n, 0)-form. If the Ricci tensor of the Bismut connection of some Hermitian metric g vanishes, (M, J, g) is conformally balanced and in particular admits a balanced metric. The previous theorem can be applied to any nilmanifold Γ\G with a left-invariant complex structure.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 9

Theorem (–, Grantcharov)

If M = Γ\G admits a left-invariant complex structure J and F is a Kähler form of a non-invariant Hermitian metric g ⇒

α(A1, ..., A2n−2) = Z

M

F n−1|m(A1|m, ...A2n−2|m)dµ,

is equal to ˜ F n−1 for some Kähler form ˜ F of a left-invariant Hermitian metric ˜

• g. If dF n−1 = 0 ⇒ d ˜

F n−1 = 0. If g is strong KT, then ˜ g is strong KT [Ugarte]

Example

The Iwasawa manifold Γ\H3

C, where

HC

3 =

8 < : @ 1 z1 z3 1 z2 1 1 A : zj ∈ C 9 = ; ,

has a left- invariant complex structure J0 which does not admit any compatible balanced metric ⇒ (Γ\H3

C, J0) does not admit any compatible metric with

Hol(∇B) ⊆ SU(3).

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 9

Theorem (–, Grantcharov)

If M = Γ\G admits a left-invariant complex structure J and F is a Kähler form of a non-invariant Hermitian metric g ⇒

α(A1, ..., A2n−2) = Z

M

F n−1|m(A1|m, ...A2n−2|m)dµ,

is equal to ˜ F n−1 for some Kähler form ˜ F of a left-invariant Hermitian metric ˜

• g. If dF n−1 = 0 ⇒ d ˜

F n−1 = 0. If g is strong KT, then ˜ g is strong KT [Ugarte]

Example

The Iwasawa manifold Γ\H3

C, where

HC

3 =

8 < : @ 1 z1 z3 1 z2 1 1 A : zj ∈ C 9 = ; ,

has a left- invariant complex structure J0 which does not admit any compatible balanced metric ⇒ (Γ\H3

C, J0) does not admit any compatible metric with

Hol(∇B) ⊆ SU(3).

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 9

Theorem (–, Grantcharov)

If M = Γ\G admits a left-invariant complex structure J and F is a Kähler form of a non-invariant Hermitian metric g ⇒

α(A1, ..., A2n−2) = Z

M

F n−1|m(A1|m, ...A2n−2|m)dµ,

is equal to ˜ F n−1 for some Kähler form ˜ F of a left-invariant Hermitian metric ˜

• g. If dF n−1 = 0 ⇒ d ˜

F n−1 = 0. If g is strong KT, then ˜ g is strong KT [Ugarte]

Example

The Iwasawa manifold Γ\H3

C, where

HC

3 =

8 < : @ 1 z1 z3 1 z2 1 1 A : zj ∈ C 9 = ; ,

has a left- invariant complex structure J0 which does not admit any compatible balanced metric ⇒ (Γ\H3

C, J0) does not admit any compatible metric with

Hol(∇B) ⊆ SU(3).

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 9

Theorem (–, Grantcharov)

If M = Γ\G admits a left-invariant complex structure J and F is a Kähler form of a non-invariant Hermitian metric g ⇒

α(A1, ..., A2n−2) = Z

M

F n−1|m(A1|m, ...A2n−2|m)dµ,

is equal to ˜ F n−1 for some Kähler form ˜ F of a left-invariant Hermitian metric ˜

• g. If dF n−1 = 0 ⇒ d ˜

F n−1 = 0. If g is strong KT, then ˜ g is strong KT [Ugarte]

Example

The Iwasawa manifold Γ\H3

C, where

HC

3 =

8 < : @ 1 z1 z3 1 z2 1 1 A : zj ∈ C 9 = ; ,

has a left- invariant complex structure J0 which does not admit any compatible balanced metric ⇒ (Γ\H3

C, J0) does not admit any compatible metric with

Hol(∇B) ⊆ SU(3).

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 10

Blow-ups The blow-up of a Kähler manifold at a point or along a compact complex submanifold is still Kähler [Blanchard].

Theorem (–, Tomassini)

The blow-up of a strong KT manifold at a point or along a compact complex submanifold is still strong KT. The same theorem holds for Hermitian manifolds satisfying ∂∂ F = 0 , ∂∂ F 2 = 0 (⇒ astheno-Kähler.) By using a deep extension and regularity result for positive or negative plurisubharmonic currents by Alessandrini and Bassanelli

Theorem (–, Tomassini)

If M2n \ {p}, n ≥ 2, admits a strong KT metric, then there exists a strong KT metric on M2n. This theorem is a generalization of Miyaoka’s extension result.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 10

Blow-ups The blow-up of a Kähler manifold at a point or along a compact complex submanifold is still Kähler [Blanchard].

Theorem (–, Tomassini)

The blow-up of a strong KT manifold at a point or along a compact complex submanifold is still strong KT. The same theorem holds for Hermitian manifolds satisfying ∂∂ F = 0 , ∂∂ F 2 = 0 (⇒ astheno-Kähler.) By using a deep extension and regularity result for positive or negative plurisubharmonic currents by Alessandrini and Bassanelli

Theorem (–, Tomassini)

If M2n \ {p}, n ≥ 2, admits a strong KT metric, then there exists a strong KT metric on M2n. This theorem is a generalization of Miyaoka’s extension result.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 10

Blow-ups The blow-up of a Kähler manifold at a point or along a compact complex submanifold is still Kähler [Blanchard].

Theorem (–, Tomassini)

The blow-up of a strong KT manifold at a point or along a compact complex submanifold is still strong KT. The same theorem holds for Hermitian manifolds satisfying ∂∂ F = 0 , ∂∂ F 2 = 0 (⇒ astheno-Kähler.) By using a deep extension and regularity result for positive or negative plurisubharmonic currents by Alessandrini and Bassanelli

Theorem (–, Tomassini)

If M2n \ {p}, n ≥ 2, admits a strong KT metric, then there exists a strong KT metric on M2n. This theorem is a generalization of Miyaoka’s extension result.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 10

Blow-ups The blow-up of a Kähler manifold at a point or along a compact complex submanifold is still Kähler [Blanchard].

Theorem (–, Tomassini)

The blow-up of a strong KT manifold at a point or along a compact complex submanifold is still strong KT. The same theorem holds for Hermitian manifolds satisfying ∂∂ F = 0 , ∂∂ F 2 = 0 (⇒ astheno-Kähler.) By using a deep extension and regularity result for positive or negative plurisubharmonic currents by Alessandrini and Bassanelli

Theorem (–, Tomassini)

If M2n \ {p}, n ≥ 2, admits a strong KT metric, then there exists a strong KT metric on M2n. This theorem is a generalization of Miyaoka’s extension result.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 11

Resolution of orbifolds

Definition

A complex orbifold is a singular complex manifold M of dimension n such that each singularity p is locally isomorphic to U/G, where U is an open set of Cn, G is a finite subgroup of GL(n, C) acting linearly on U with the only one fixed point p. Moreover, real codim of {singular points of M} ≥ 2. By using the Hironaka result that the singularities of a complex algebraic variety can be resolved by a finite number of blow-ups.

Theorem (–, Tomassini)

Let (M, J) be a complex orbifold of complex dimension n endowed with a J-Hermitian strong KT metric g. Then there exists a strong KT resolution. The same result holds for Hermitian orbifolds satisfying ∂∂ F = 0 , ∂∂ F 2 = 0.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 11

Resolution of orbifolds

Definition

A complex orbifold is a singular complex manifold M of dimension n such that each singularity p is locally isomorphic to U/G, where U is an open set of Cn, G is a finite subgroup of GL(n, C) acting linearly on U with the only one fixed point p. Moreover, real codim of {singular points of M} ≥ 2. By using the Hironaka result that the singularities of a complex algebraic variety can be resolved by a finite number of blow-ups.

Theorem (–, Tomassini)

Let (M, J) be a complex orbifold of complex dimension n endowed with a J-Hermitian strong KT metric g. Then there exists a strong KT resolution. The same result holds for Hermitian orbifolds satisfying ∂∂ F = 0 , ∂∂ F 2 = 0.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 12

6-dimensional strong KT nilmanifolds

Theorem (-, Parton, Salamon)

M6 = G/Γ nilmanifold, J a left-invariant complex structure, g any compatible metric. Then (J, g) strong KT ⇔ ∃ a basis (αi) of (1, 0)-forms such that    dα1 = dα2 = 0, dα3 = Aα1 ∧ α2 + Bα2 ∧ α2 + Cα1 ∧ α1+ Dα1 ∧ α2 + Eα1 ∧ α2 with |A|2 + |D|2 + |E|2 + 2Re (BC) = 0. The nilpotent Lie group G has to be 2-step and the existence of a strong KT metric depends only on the complex structure.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 12

6-dimensional strong KT nilmanifolds

Theorem (-, Parton, Salamon)

M6 = G/Γ nilmanifold, J a left-invariant complex structure, g any compatible metric. Then (J, g) strong KT ⇔ ∃ a basis (αi) of (1, 0)-forms such that    dα1 = dα2 = 0, dα3 = Aα1 ∧ α2 + Bα2 ∧ α2 + Cα1 ∧ α1+ Dα1 ∧ α2 + Eα1 ∧ α2 with |A|2 + |D|2 + |E|2 + 2Re (BC) = 0. The nilpotent Lie group G has to be 2-step and the existence of a strong KT metric depends only on the complex structure.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 12

6-dimensional strong KT nilmanifolds

Theorem (-, Parton, Salamon)

M6 = G/Γ nilmanifold, J a left-invariant complex structure, g any compatible metric. Then (J, g) strong KT ⇔ ∃ a basis (αi) of (1, 0)-forms such that    dα1 = dα2 = 0, dα3 = Aα1 ∧ α2 + Bα2 ∧ α2 + Cα1 ∧ α1+ Dα1 ∧ α2 + Eα1 ∧ α2 with |A|2 + |D|2 + |E|2 + 2Re (BC) = 0. The nilpotent Lie group G has to be 2-step and the existence of a strong KT metric depends only on the complex structure.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 13

Example

Consider the 6-dimensional nilpotent Lie group with structure equations

• dei = 0 , i = 1, . . . , 5

de6 = e12 + e34 . and the complex structure J defined by η1 = e1 + ie2, η2 = e3 + ie4, η3 = e5 + ie6. Take Γ ⊂ G such that J is rational on M = Γ/G ⇒

• any holomorphic 1-form on M is d-closed since

H1,0

∂ (M, J) = span < η1, η2 >

and

• (M, J) does not admit any strong KT metric compatible with J.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 13

Example

Consider the 6-dimensional nilpotent Lie group with structure equations

• dei = 0 , i = 1, . . . , 5

de6 = e12 + e34 . and the complex structure J defined by η1 = e1 + ie2, η2 = e3 + ie4, η3 = e5 + ie6. Take Γ ⊂ G such that J is rational on M = Γ/G ⇒

• any holomorphic 1-form on M is d-closed since

H1,0

∂ (M, J) = span < η1, η2 >

and

• (M, J) does not admit any strong KT metric compatible with J.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 14

8-dimensional astheno-Kähler nilmanifolds Consider the complex nilmanifold (M8 = Γ\G, J) defined by

8 > > < > > : dηj = 0, j = 1, 2, 3, dη4 = a1 η1 ∧ η2 + a2η1 ∧ η3 + a3η1 ∧ η1 + a4 η1 ∧ η2 + a5 η1 ∧ η3 +a6 η2 ∧ η3 + a7 η2 ∧ η1 + a8 η2 ∧ η2 + a9 η2 ∧ η3 + a10 η3 ∧ η1 +a11 η3 ∧ η1 + a12 η3 ∧ η3, aj ∈ Q[i]. Theorem (–, Tomassini)

The Hermitian metric g = i

2

4

j=1 ηj ⊗ ηj + ηj ⊗ ηj on (M8, J) is

astheno-Kähler if and only if

|a1|2 + |a2|2 + |a4|2 + |a5|2 + |a6|2 + |a7|2 + |a8|2+ |a9|2 + |a10|2 + |a11|2 = 2ℜe (a3a8 + a3a12 + a8a12).

If a8 = 0 and |a4|2 + |a11|2 = 0, then g is not strong KT. The astheno-Kähler condition depends on the metric and there is no relation between astheno-Kähler and strong KT condition for n > 3.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 14

8-dimensional astheno-Kähler nilmanifolds Consider the complex nilmanifold (M8 = Γ\G, J) defined by

8 > > < > > : dηj = 0, j = 1, 2, 3, dη4 = a1 η1 ∧ η2 + a2η1 ∧ η3 + a3η1 ∧ η1 + a4 η1 ∧ η2 + a5 η1 ∧ η3 +a6 η2 ∧ η3 + a7 η2 ∧ η1 + a8 η2 ∧ η2 + a9 η2 ∧ η3 + a10 η3 ∧ η1 +a11 η3 ∧ η1 + a12 η3 ∧ η3, aj ∈ Q[i]. Theorem (–, Tomassini)

The Hermitian metric g = i

2

4

j=1 ηj ⊗ ηj + ηj ⊗ ηj on (M8, J) is

astheno-Kähler if and only if

|a1|2 + |a2|2 + |a4|2 + |a5|2 + |a6|2 + |a7|2 + |a8|2+ |a9|2 + |a10|2 + |a11|2 = 2ℜe (a3a8 + a3a12 + a8a12).

If a8 = 0 and |a4|2 + |a11|2 = 0, then g is not strong KT. The astheno-Kähler condition depends on the metric and there is no relation between astheno-Kähler and strong KT condition for n > 3.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 14

8-dimensional astheno-Kähler nilmanifolds Consider the complex nilmanifold (M8 = Γ\G, J) defined by

8 > > < > > : dηj = 0, j = 1, 2, 3, dη4 = a1 η1 ∧ η2 + a2η1 ∧ η3 + a3η1 ∧ η1 + a4 η1 ∧ η2 + a5 η1 ∧ η3 +a6 η2 ∧ η3 + a7 η2 ∧ η1 + a8 η2 ∧ η2 + a9 η2 ∧ η3 + a10 η3 ∧ η1 +a11 η3 ∧ η1 + a12 η3 ∧ η3, aj ∈ Q[i]. Theorem (–, Tomassini)

The Hermitian metric g = i

2

4

j=1 ηj ⊗ ηj + ηj ⊗ ηj on (M8, J) is

astheno-Kähler if and only if

|a1|2 + |a2|2 + |a4|2 + |a5|2 + |a6|2 + |a7|2 + |a8|2+ |a9|2 + |a10|2 + |a11|2 = 2ℜe (a3a8 + a3a12 + a8a12).

If a8 = 0 and |a4|2 + |a11|2 = 0, then g is not strong KT. The astheno-Kähler condition depends on the metric and there is no relation between astheno-Kähler and strong KT condition for n > 3.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 15

A simply-connected example Let T2n = R2n/Z2n and σ the involution on T2n induced by σ ((x1, . . . , x2n)) = (−x1, . . . , −x2n). Consider on T2n the complex structure J defined by

• η1 = dx1 + i (f dxn + dxn+1) ,

ηj = dxj + i dxn+j , j = 2, . . . , n, where f = f(xn, x2n) is a C∞, Z2n-periodic and even function. Then

• (T2n/σ, J) is a complex orbifold with singular point set

S =

• x + Z2n | x ∈ 1

2Z2n

.

• The J-Hermitian metric g = 1

2

n

j=1

• ηj ⊗ ηj + ηj ⊗ ηj

is strong KT and ∂∂F 2 = 0.

• The strong KT resolution is simply-connected.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 15

A simply-connected example Let T2n = R2n/Z2n and σ the involution on T2n induced by σ ((x1, . . . , x2n)) = (−x1, . . . , −x2n). Consider on T2n the complex structure J defined by

• η1 = dx1 + i (f dxn + dxn+1) ,

ηj = dxj + i dxn+j , j = 2, . . . , n, where f = f(xn, x2n) is a C∞, Z2n-periodic and even function. Then

• (T2n/σ, J) is a complex orbifold with singular point set

S =

• x + Z2n | x ∈ 1

2Z2n

.

• The J-Hermitian metric g = 1

2

n

j=1

• ηj ⊗ ηj + ηj ⊗ ηj

is strong KT and ∂∂F 2 = 0.

• The strong KT resolution is simply-connected.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 15

A simply-connected example Let T2n = R2n/Z2n and σ the involution on T2n induced by σ ((x1, . . . , x2n)) = (−x1, . . . , −x2n). Consider on T2n the complex structure J defined by

• η1 = dx1 + i (f dxn + dxn+1) ,

ηj = dxj + i dxn+j , j = 2, . . . , n, where f = f(xn, x2n) is a C∞, Z2n-periodic and even function. Then

• (T2n/σ, J) is a complex orbifold with singular point set

S =

• x + Z2n | x ∈ 1

2Z2n

.

• The J-Hermitian metric g = 1

2

n

j=1

• ηj ⊗ ηj + ηj ⊗ ηj

is strong KT and ∂∂F 2 = 0.

• The strong KT resolution is simply-connected.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 15

A simply-connected example Let T2n = R2n/Z2n and σ the involution on T2n induced by σ ((x1, . . . , x2n)) = (−x1, . . . , −x2n). Consider on T2n the complex structure J defined by

• η1 = dx1 + i (f dxn + dxn+1) ,

ηj = dxj + i dxn+j , j = 2, . . . , n, where f = f(xn, x2n) is a C∞, Z2n-periodic and even function. Then

• (T2n/σ, J) is a complex orbifold with singular point set

S =

• x + Z2n | x ∈ 1

2Z2n

.

• The J-Hermitian metric g = 1

2

n

j=1

• ηj ⊗ ηj + ηj ⊗ ηj

is strong KT and ∂∂F 2 = 0.

• The strong KT resolution is simply-connected.

Astheno-Kähler and strong KT metrics Anna Fino General results

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

Construction of examples

6-dimensional strong KT nilmanifolds 8-dimensional astheno-Kähler nilmanifolds A simply-connected example 15

A simply-connected example Let T2n = R2n/Z2n and σ the involution on T2n induced by σ ((x1, . . . , x2n)) = (−x1, . . . , −x2n). Consider on T2n the complex structure J defined by

• η1 = dx1 + i (f dxn + dxn+1) ,

ηj = dxj + i dxn+j , j = 2, . . . , n, where f = f(xn, x2n) is a C∞, Z2n-periodic and even function. Then

• (T2n/σ, J) is a complex orbifold with singular point set

S =

• x + Z2n | x ∈ 1

2Z2n

.

• The J-Hermitian metric g = 1

2

n

j=1

• ηj ⊗ ηj + ηj ⊗ ηj

is strong KT and ∂∂F 2 = 0.

• The strong KT resolution is simply-connected.