Balanced Hermitian geometry on nilmanifolds

<strong>Balanced</strong> <strong>Hermitian</strong> <strong>geometry</strong>on nilmanifoldsLuis UgarteUniv. Zaragoza – I.U.M.A.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 1 / 28

OutlineGoal: invariant balanced <strong>Hermitian</strong> <strong>geometry</strong> of nilmanifolds ofdimension 6, deformation of balanced metrics and holonomy of theassociated Bismut connection. As an application, solutions of theStrominger system with respect to the Bismut and Chern connection.Invariant complex structures on nilmanifoldsL. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 2 / 28

OutlineGoal: invariant balanced <strong>Hermitian</strong> <strong>geometry</strong> of nilmanifolds ofdimension 6, deformation of balanced metrics and holonomy of theassociated Bismut connection. As an application, solutions of theStrominger system with respect to the Bismut and Chern connection.Invariant complex structures on nilmanifolds<strong>Balanced</strong> <strong>Hermitian</strong> <strong>geometry</strong> in six dimensionsL. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 2 / 28

OutlineGoal: invariant balanced <strong>Hermitian</strong> <strong>geometry</strong> of nilmanifolds ofdimension 6, deformation of balanced metrics and holonomy of theassociated Bismut connection. As an application, solutions of theStrominger system with respect to the Bismut and Chern connection.Invariant complex structures on nilmanifolds<strong>Balanced</strong> <strong>Hermitian</strong> <strong>geometry</strong> in six dimensionsDeformation of balanced metricsL. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 2 / 28

OutlineGoal: invariant balanced <strong>Hermitian</strong> <strong>geometry</strong> of nilmanifolds ofdimension 6, deformation of balanced metrics and holonomy of theassociated Bismut connection. As an application, solutions of theStrominger system with respect to the Bismut and Chern connection.Invariant complex structures on nilmanifolds<strong>Balanced</strong> <strong>Hermitian</strong> <strong>geometry</strong> in six dimensionsDeformation of balanced metricsHolonomy of the Bismut connectionL. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 2 / 28

OutlineGoal: invariant balanced <strong>Hermitian</strong> <strong>geometry</strong> of nilmanifolds ofdimension 6, deformation of balanced metrics and holonomy of theassociated Bismut connection. As an application, solutions of theStrominger system with respect to the Bismut and Chern connection.Invariant complex structures on nilmanifolds<strong>Balanced</strong> <strong>Hermitian</strong> <strong>geometry</strong> in six dimensionsDeformation of balanced metricsHolonomy of the Bismut connectionApplications to heterotic string theoryL. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 2 / 28

OutlineGoal: invariant balanced <strong>Hermitian</strong> <strong>geometry</strong> of nilmanifolds ofdimension 6, deformation of balanced metrics and holonomy of theassociated Bismut connection. As an application, solutions of theStrominger system with respect to the Bismut and Chern connection.Invariant complex structures on nilmanifolds<strong>Balanced</strong> <strong>Hermitian</strong> <strong>geometry</strong> in six dimensionsDeformation of balanced metricsHolonomy of the Bismut connectionApplications to heterotic string theory(Joint work with M. Fernández, S. Ivanov and R. Villacampa.)L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 2 / 28

NilmanifoldsDef.: A nilmanifold is a compact quotient Γ\G of a simply connectednilpotent Lie group G by a lattice Γ of maximal rank.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 3 / 28

NilmanifoldsDef.: A nilmanifold is a compact quotient Γ\G of a simply connectednilpotent Lie group G by a lattice Γ of maximal rank.The Lie algebra g of G is s-step nilpotent, i.e. the ascending centralseriesg 0 = {0}, g l = {X ∈ g | [X, g] ⊆ g l−1 } , l ≥ 1,satisfies {0} ⊂ g 1 ⊂ · · · ⊂ g s−1 ≠ g s = g.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 3 / 28

NilmanifoldsDef.: A nilmanifold is a compact quotient Γ\G of a simply connectednilpotent Lie group G by a lattice Γ of maximal rank.The Lie algebra g of G is s-step nilpotent, i.e. the ascending centralseriesg 0 = {0}, g l = {X ∈ g | [X, g] ⊆ g l−1 } , l ≥ 1,satisfies {0} ⊂ g 1 ⊂ · · · ⊂ g s−1 ≠ g s = g.Nilmanifolds have proved to be useful in constructing a rich and widevariety of examples of compact complex manifolds possessingadditional geometric structures with interesting properties.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 3 / 28

NilmanifoldsDef.: A nilmanifold is a compact quotient Γ\G of a simply connectednilpotent Lie group G by a lattice Γ of maximal rank.The Lie algebra g of G is s-step nilpotent, i.e. the ascending centralseriesg 0 = {0}, g l = {X ∈ g | [X, g] ⊆ g l−1 } , l ≥ 1,satisfies {0} ⊂ g 1 ⊂ · · · ⊂ g s−1 ≠ g s = g.Nilmanifolds have proved to be useful in constructing a rich and widevariety of examples of compact complex manifolds possessingadditional geometric structures with interesting properties.Γ\G carries a Kähler metric if and only if it is a torus [BG][BG] C. Benson, C. Gordon: Kähler and symplectic structures on nilmanifolds, Topology 27 (1988),513-518.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 3 / 28

Invariant complex structures on nilmanifoldsWe consider complex nilmanifolds (M = Γ\G, J) endowed with aninvariant complex structure J, i.e. J comes from a left-invariantintegrable almost complex structure on G.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 4 / 28

Invariant complex structures on nilmanifoldsWe consider complex nilmanifolds (M = Γ\G, J) endowed with aninvariant complex structure J, i.e. J comes from a left-invariantintegrable almost complex structure on G.The Lie algebra g of G has an endomorphism J : g −→ g such thatJ 2 = −Id, and[JX, JY ] = J[JX, Y ] + J[X, JY ] + [X, Y ], X, Y ∈ g.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 4 / 28

Invariant complex structures on nilmanifoldsWe consider complex nilmanifolds (M = Γ\G, J) endowed with aninvariant complex structure J, i.e. J comes from a left-invariantintegrable almost complex structure on G.The Lie algebra g of G has an endomorphism J : g −→ g such thatJ 2 = −Id, and[JX, JY ] = J[JX, Y ] + J[X, JY ] + [X, Y ], X, Y ∈ g.Equivalently,d(g 1,0 ) ⊂ ∧ 2,0 (g ∗ ) ⊕ ∧ 1,1 (g ∗ )L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 4 / 28

Invariant complex structures on nilmanifoldsWe consider complex nilmanifolds (M = Γ\G, J) endowed with aninvariant complex structure J, i.e. J comes from a left-invariantintegrable almost complex structure on G.The Lie algebra g of G has an endomorphism J : g −→ g such thatJ 2 = −Id, and[JX, JY ] = J[JX, Y ] + J[X, JY ] + [X, Y ], X, Y ∈ g.Equivalently,d(g 1,0 ) ⊂ ∧ 2,0 (g ∗ ) ⊕ ∧ 1,1 (g ∗ )In general the terms g l are not invariant under J.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 4 / 28

Invariant complex structures on nilmanifoldsWe consider complex nilmanifolds (M = Γ\G, J) endowed with aninvariant complex structure J, i.e. J comes from a left-invariantintegrable almost complex structure on G.The Lie algebra g of G has an endomorphism J : g −→ g such thatJ 2 = −Id, and[JX, JY ] = J[JX, Y ] + J[X, JY ] + [X, Y ], X, Y ∈ g.Equivalently,d(g 1,0 ) ⊂ ∧ 2,0 (g ∗ ) ⊕ ∧ 1,1 (g ∗ )In general the terms g l are not invariant under J. We define theascending series a l (J) compatible with J: a 0 (J) = {0},a l (J)={X ∈ g | [X, g] ⊆ a l−1 (J) and [JX, g] ⊆ a l−1 (J)}, l ≥1.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 4 / 28

Nilpotent complex structuresDef. [CFGU 1 ]: J is said to be nilpotent if a t (J) = g for some t,that is, {0} ⊂ a 1 (J) ⊂ · · · ⊂ a t−1 (J) ≠ a t (J) = g.[CFGU1] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte: Compact nilmanifolds with nilpotent complexstructure: Dolbeault cohomology, Trans. Amer. Math. Soc. 352 (2000), 5405-5433.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 5 / 28

Nilpotent complex structuresDef. [CFGU 1 ]: J is said to be nilpotent if a t (J) = g for some t,that is, {0} ⊂ a 1 (J) ⊂ · · · ⊂ a t−1 (J) ≠ a t (J) = g.Particular examples of nilpotent complex structures:[CFGU1] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte: Compact nilmanifolds with nilpotent complexstructure: Dolbeault cohomology, Trans. Amer. Math. Soc. 352 (2000), 5405-5433.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 5 / 28

Nilpotent complex structuresDef. [CFGU 1 ]: J is said to be nilpotent if a t (J) = g for some t,that is, {0} ⊂ a 1 (J) ⊂ · · · ⊂ a t−1 (J) ≠ a t (J) = g.Particular examples of nilpotent complex structures:Abelian complex structures: [JX, JY ] = [X, Y ].a l (J) = g l , l ≥1 ⇒ J nilpotent (g 1,0 abelian Lie algebra).[CFGU1] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte: Compact nilmanifolds with nilpotent complexstructure: Dolbeault cohomology, Trans. Amer. Math. Soc. 352 (2000), 5405-5433.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 5 / 28

Nilpotent complex structuresDef. [CFGU 1 ]: J is said to be nilpotent if a t (J) = g for some t,that is, {0} ⊂ a 1 (J) ⊂ · · · ⊂ a t−1 (J) ≠ a t (J) = g.Particular examples of nilpotent complex structures:Abelian complex structures: [JX, JY ] = [X, Y ].a l (J) = g l , l ≥1 ⇒ J nilpotent (g 1,0 abelian Lie algebra).Complex-parallelizable structures: [JX, Y ] = J[X, Y ].a l (J) = g l , l ≥1 ⇒ J nilpotent (g complex Lie algebra).[CFGU1] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte: Compact nilmanifolds with nilpotent complexstructure: Dolbeault cohomology, Trans. Amer. Math. Soc. 352 (2000), 5405-5433.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 5 / 28

Complex structure equationsNilpotent structuresdω k ∈ ∧2 〈ω 1 , . . . , ω k−1 , ω 1 , . . . , ω k−1 〉, k = 1, . . . , n.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 6 / 28

Complex structure equationsNilpotent structuresdω k ∈ ∧2 〈ω 1 , . . . , ω k−1 , ω 1 , . . . , ω k−1 〉, k = 1, . . . , n.Abeliand(g 1,0 ) ⊂ ∧ 1,1 (g ∗ )L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 6 / 28

Complex structure equationsNilpotent structuresdω k ∈ ∧2 〈ω 1 , . . . , ω k−1 , ω 1 , . . . , ω k−1 〉, k = 1, . . . , n.Abeliand(g 1,0 ) ⊂ ∧ 1,1 (g ∗ )Complex-parallelizabled(g 1,0 ) ⊂ ∧ 2,0 (g ∗ )L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 6 / 28

Complex structure equationsNilpotent structuresdω k ∈ ∧2 〈ω 1 , . . . , ω k−1 , ω 1 , . . . , ω k−1 〉, k = 1, . . . , n.Abeliand(g 1,0 ) ⊂ ∧ 1,1 (g ∗ )Complex-parallelizabled(g 1,0 ) ⊂ ∧ 2,0 (g ∗ )Theorem [S]J is a complex structure on a NLA g if and only if g 1,0 has a basis{ω k } n k=1 such that dω k ∈ I(ω 1 , . . . , ω k−1 ), k = 1, . . . , n.[S] S. Salamon: Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311-333.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 6 / 28

Dimension 4Complex equations{dω 1 = 0dω 2 = A 12 ω 12 + A 1¯1ω 1¯1 + A 1¯2ω 1¯2L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 7 / 28

Dimension 4Complex equations{dω 1 = 0dω 2 = A 12 ω 12 + A 1¯1ω 1¯1 + A 1¯2ω 1¯2reduce to dω 1 = 0, dω 2 = ɛ ω 1¯1 (ɛ = 0, 1).L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 7 / 28

Dimension 4Complex equations{dω 1 = 0dω 2 = A 12 ω 12 + A 1¯1ω 1¯1 + A 1¯2ω 1¯2reduce to dω 1 = 0, dω 2 = ɛ ω 1¯1 (ɛ = 0, 1).PropositionA 4-dimensional NLA g has a complex structure if and only if it isisomorphic to (0, 0, 0, 0) or (0, 0, 0, 12).Moreover, any complex structure J is abelian.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 7 / 28

Dimension 4Complex equations{dω 1 = 0dω 2 = A 12 ω 12 + A 1¯1ω 1¯1 + A 1¯2ω 1¯2reduce to dω 1 = 0, dω 2 = ɛ ω 1¯1 (ɛ = 0, 1).PropositionA 4-dimensional NLA g has a complex structure if and only if it isisomorphic to (0, 0, 0, 0) or (0, 0, 0, 12).Moreover, any complex structure J is abelian.Notation: g ∼ = (0, 0, 0, 12) means that ∃ basis {α 1 , α 2 , α 3 , α 4 } for g ∗such that dα 1 = dα 2 = dα 3 = 0, dα 4 = α 1 ∧ α 2 ;L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 7 / 28

Dimension 4Complex equations{dω 1 = 0dω 2 = A 12 ω 12 + A 1¯1ω 1¯1 + A 1¯2ω 1¯2reduce to dω 1 = 0, dω 2 = ɛ ω 1¯1 (ɛ = 0, 1).PropositionA 4-dimensional NLA g has a complex structure if and only if it isisomorphic to (0, 0, 0, 0) or (0, 0, 0, 12) .Moreover, any complex structure J is abelian.Notation: g ∼ = (0, 0, 0, 12) means that ∃ basis {α 1 , α 2 , α 3 , α 4 } for g ∗such that dα 1 = dα 2 = dα 3 = 0, dα 4 = α 1 ∧ α 2 ;Proof: if ɛ = 1 then ω 1 = α 1 + iα 2 , ω 2 = α 3 + iα 4 .L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 7 / 28

Dimension 6Generic complex equations:⎧dω 1 = 0,⎪⎨ dω 2 = A 12 ω 12 + A 13 ω 13 + A 1¯1ω 1¯1 + A 1¯2ω 1¯2 + A 1¯3ω 1¯3 ,⎪⎩dω 3 = B 12 ω 12 + B 13 ω 13 + B 1¯1ω 1¯1 + B 1¯2ω 1¯2 + B 1¯3ω 1¯3+ B 23 ω 23 + B 2¯1ω 2¯1 + B 2¯2ω 2¯2 + B 2¯3ω 2¯3 ,where A 12 , A 13 , A 1¯1,A 1¯2,A 1¯3,B 12 , . . . , B 2¯3∈ C.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 8 / 28

Reduced equations in dimension 6 [UV 1 ]Let J be a complex structure on a NLA g of dimension 6.[UV1] L. Ugarte, R. Villacampa: Non-nilpotent complex <strong>geometry</strong> of nilmanifolds and heterotic supersymmetry,arXiv:0912.5110v1 [math.DG]L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 9 / 28

Reduced equations in dimension 6 [UV 1 ]Let J be a complex structure on a NLA g of dimension 6.⎧dω 1 = 0,⎪⎨ dω 2 = ɛ ω 1¯1 ,J nilpotent:dω 3 = ρ ω 12 + (1−ɛ)A ω 1¯1 + B ω 1¯2⎪⎩+ C ω 2¯1 + (1−ɛ)D ω 2¯2,where A, B, C, D ∈ C and ɛ, ρ ∈ {0, 1};[UV1] L. Ugarte, R. Villacampa: Non-nilpotent complex <strong>geometry</strong> of nilmanifolds and heterotic supersymmetry,arXiv:0912.5110v1 [math.DG]L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 9 / 28

Reduced equations in dimension 6 [UV 1 ]Let J be a complex structure on a NLA g of dimension 6.⎧dω 1 = 0,⎪⎨ dω 2 = ɛ ω 1¯1 ,J nilpotent:dω 3 = ρ ω 12 + (1−ɛ)A ω 1¯1 + B ω 1¯2⎪⎩+ C ω 2¯1 + (1−ɛ)D ω 2¯2,where A, B, C, D ∈ C and ɛ, ρ ∈ {0, 1};⎧⎪⎨ dω 1 = 0,J non-nilpotent: dω⎪⎩2 = ω 13 + ω 1¯3 ,dω 3 = i ɛ ω 1¯1 ± i(ω 1¯2 − ω 2¯1),where ɛ = 0, 1.[UV1] L. Ugarte, R. Villacampa: Non-nilpotent complex <strong>geometry</strong> of nilmanifolds and heterotic supersymmetry,arXiv:0912.5110v1 [math.DG]L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 9 / 28

Dimension 6Theorem [S, CFGU 2 ]A 6-dimensional NLA g has a complex structure if and only if it isisomorphic to:h 1 = (0,0,0,0,0,0),h 2 = (0,0,0,0,12,34),h 3 = (0,0,0,0,0,12+34),h 4 = (0,0,0,0,12,14+23),h 5 = (0,0,0,0,13+42,14+23),h 6 = (0,0,0,0,12,13),h 7 = (0,0,0,12,13,23),h 8 = (0,0,0,0,0,12),h 9 = (0,0,0,0,12,14+25),h 10 = (0,0,0,12,13,14),h 11 = (0,0,0,12,13,14+23),h 12 = (0,0,0,12,13,24),h 13 = (0,0,0,12,13+14,24),h 14 = (0,0,0,12,14,13+42),h 15 = (0,0,0,12,13+42,14+23),h 16 = (0,0,0,12,14,24),h − 19 = (0,0,0,12,23,14−35),h + 26 = (0,0,12,13,23,14+25).[CFGU2] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte: Nilpotent complex structures on compact nilmanifolds,Rend. Circ. Mat. Palermo 49 (1997), 83-100.[S] S. Salamon: Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311-333.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 10 / 28

Dimension 6Theorem [S, CFGU 2 ]A 6-dimensional NLA g has a complex structure if and only if it isisomorphic to:h 1 = (0,0,0,0,0,0),h 2 = (0,0,0,0,12,34),h 3 = (0,0,0,0,0,12+34),h 4 = (0,0,0,0,12,14+23),h 5 = (0,0,0,0,13+42,14+23),h 6 = (0,0,0,0,12,13),h 7 = (0,0,0,12,13,23),h 8 = (0,0,0,0,0,12),h 9 = (0,0,0,0,12,14+25),h 10 = (0,0,0,12,13,14),h 11 = (0,0,0,12,13,14+23),h 12 = (0,0,0,12,13,24),h 13 = (0,0,0,12,13+14,24),h 14 = (0,0,0,12,14,13+42),h 15 = (0,0,0,12,13+42,14+23),h 16 = (0,0,0,12,14,24),h − 19 = (0,0,0,12,23,14−35),h + 26 = (0,0,12,13,23,14+25).[CFGU2] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte: Nilpotent complex structures on compact nilmanifolds,Rend. Circ. Mat. Palermo 49 (1997), 83-100.[S] S. Salamon: Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311-333.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 10 / 28

Dimension 6Theorem [S, CFGU 2 ]A 6-dimensional NLA g has a complex structure if and only if it isisomorphic to:h 1 = (0,0,0,0,0,0),h 2 = (0,0,0,0,12,34) ,h 3 = (0,0,0,0,0,12+34),h 4 = (0,0,0,0,12,14+23),h 5 = (0,0,0,0,13+42,14+23),h 6 = (0,0,0,0,12,13),h 7 = (0,0,0,12,13,23),h 8 = (0,0,0,0,0,12),h 9 = (0,0,0,0,12,14+25),h 10 = (0,0,0,12,13,14),h 11 = (0,0,0,12,13,14+23),h 12 = (0,0,0,12,13,24),h 13 = (0,0,0,12,13+14,24),h 14 = (0,0,0,12,14,13+42),h 15 = (0,0,0,12,13+42,14+23),h 16 = (0,0,0,12,14,24),h − 19 = (0,0,0,12,23,14−35),h + 26 = (0,0,12,13,23,14+25).[CFGU2] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte: Nilpotent complex structures on compact nilmanifolds,Rend. Circ. Mat. Palermo 49 (1997), 83-100.[S] S. Salamon: Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311-333.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 10 / 28

Dimension 6Theorem [S, CFGU 2 ]A 6-dimensional NLA g has a complex structure if and only if it isisomorphic to:h 1 = (0,0,0,0,0,0),h 2 = (0,0,0,0,12,34),h 3 = (0,0,0,0,0,12+34) ,h 4 = (0,0,0,0,12,14+23),h 5 = (0,0,0,0,13+42,14+23),h 6 = (0,0,0,0,12,13),h 7 = (0,0,0,12,13,23),h 8 = (0,0,0,0,0,12),h 9 = (0,0,0,0,12,14+25),h 10 = (0,0,0,12,13,14),h 11 = (0,0,0,12,13,14+23),h 12 = (0,0,0,12,13,24),h 13 = (0,0,0,12,13+14,24),h 14 = (0,0,0,12,14,13+42),h 15 = (0,0,0,12,13+42,14+23),h 16 = (0,0,0,12,14,24),h − 19 = (0,0,0,12,23,14−35),h + 26 = (0,0,12,13,23,14+25).[CFGU2] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte: Nilpotent complex structures on compact nilmanifolds,Rend. Circ. Mat. Palermo 49 (1997), 83-100.[S] S. Salamon: Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311-333.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 10 / 28

Dimension 6Theorem [S, CFGU 2 ]A 6-dimensional NLA g has a complex structure if and only if it isisomorphic to:h 1 = (0,0,0,0,0,0),h 2 = (0,0,0,0,12,34),h 3 = (0,0,0,0,0,12+34),h 4 = (0,0,0,0,12,14+23),h 5 = (0,0,0,0,13+42,14+23) ,h 6 = (0,0,0,0,12,13),h 7 = (0,0,0,12,13,23),h 8 = (0,0,0,0,0,12),h 9 = (0,0,0,0,12,14+25),h 10 = (0,0,0,12,13,14),h 11 = (0,0,0,12,13,14+23),h 12 = (0,0,0,12,13,24),h 13 = (0,0,0,12,13+14,24),h 14 = (0,0,0,12,14,13+42),h 15 = (0,0,0,12,13+42,14+23),h 16 = (0,0,0,12,14,24),h − 19 = (0,0,0,12,23,14−35),h + 26 = (0,0,12,13,23,14+25).[CFGU2] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte: Nilpotent complex structures on compact nilmanifolds,Rend. Circ. Mat. Palermo 49 (1997), 83-100.[S] S. Salamon: Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311-333.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 10 / 28

Dimension 6Theorem [S, CFGU 2 ]A 6-dimensional NLA g has a complex structure if and only if it isisomorphic to:h 1 = (0,0,0,0,0,0),h 2 = (0,0,0,0,12,34),h 3 = (0,0,0,0,0,12+34),h 4 = (0,0,0,0,12,14+23),h 5 = (0,0,0,0,13+42,14+23),h 6 = (0,0,0,0,12,13),h 7 = (0,0,0,12,13,23),h 8 = (0,0,0,0,0,12),h 9 = (0,0,0,0,12,14+25),h 10 = (0,0,0,12,13,14),h 11 = (0,0,0,12,13,14+23),h 12 = (0,0,0,12,13,24),h 13 = (0,0,0,12,13+14,24),h 14 = (0,0,0,12,14,13+42),h 15 = (0,0,0,12,13+42,14+23),h 16 = (0,0,0,12,14,24),h − 19 = (0,0,0,12,23,14−35),h + 26 = (0,0,12,13,23,14+25).[CFGU2] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte: Nilpotent complex structures on compact nilmanifolds,Rend. Circ. Mat. Palermo 49 (1997), 83-100.[S] S. Salamon: Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), 311-333.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 10 / 28

<strong>Balanced</strong> <strong>Hermitian</strong> structuresDef.: <strong>Hermitian</strong> structure on a manifold M 2n is a pair (J, g), J complexstructure and g Riemannian metric such that g(JX, JY ) = g(X, Y ).L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 11 / 28

<strong>Balanced</strong> <strong>Hermitian</strong> structuresDef.: <strong>Hermitian</strong> structure on a manifold M 2n is a pair (J, g), J complexstructure and g Riemannian metric such that g(JX, JY ) = g(X, Y ).Fundamental 2-form: F(X, Y ) = g(X, JY ).L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 11 / 28

<strong>Balanced</strong> <strong>Hermitian</strong> structuresDef.: <strong>Hermitian</strong> structure on a manifold M 2n is a pair (J, g), J complexstructure and g Riemannian metric such that g(JX, JY ) = g(X, Y ).Fundamental 2-form: F(X, Y ) = g(X, JY ).Lee 1-form: θ = 11−nJδF, where δ is the g-adjoint of d.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 11 / 28

<strong>Balanced</strong> <strong>Hermitian</strong> structuresDef.: <strong>Hermitian</strong> structure on a manifold M 2n is a pair (J, g), J complexstructure and g Riemannian metric such that g(JX, JY ) = g(X, Y ).Fundamental 2-form: F(X, Y ) = g(X, JY ).Lee 1-form: θ = 11−nJδF, where δ is the g-adjoint of d.Bismut connection [B]: ∇ B unique connection for which g and J areparallel with torsion T : g(X, T (Y , Z )) = JdF(X, Y , Z ).[B] J.-M. Bismut, A local index theorem for non-Kähler manifolds, Math. Ann. 284 (1989), 681-699.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 11 / 28

<strong>Balanced</strong> <strong>Hermitian</strong> structuresDef.: <strong>Hermitian</strong> structure on a manifold M 2n is a pair (J, g), J complexstructure and g Riemannian metric such that g(JX, JY ) = g(X, Y ).Fundamental 2-form: F(X, Y ) = g(X, JY ).Lee 1-form: θ = 11−nJδF, where δ is the g-adjoint of d.Bismut connection [B]: ∇ B unique connection for which g and J areparallel with torsion T : g(X, T (Y , Z )) = JdF(X, Y , Z ).Special class [M]: (J, g) on M 2n is called <strong>Balanced</strong> <strong>Hermitian</strong> if theLee form θ ≡ 0. Equivalently, F n−1 is closed.[B] J.-M. Bismut, A local index theorem for non-Kähler manifolds, Math. Ann. 284 (1989), 681-699.[M] M.L. Michelsohn, On the existence of special metrics in complex <strong>geometry</strong>, Acta Math. 149 (1982), 261-295.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 11 / 28

<strong>Balanced</strong> <strong>Hermitian</strong> structuresDef.: <strong>Hermitian</strong> structure on a manifold M 2n is a pair (J, g), J complexstructure and g Riemannian metric such that g(JX, JY ) = g(X, Y ).Fundamental 2-form: F(X, Y ) = g(X, JY ).Lee 1-form: θ = 11−nJδF, where δ is the g-adjoint of d.Bismut connection [B]: ∇ B unique connection for which g and J areparallel with torsion T : g(X, T (Y , Z )) = JdF(X, Y , Z ).Special class [M]: (J, g) on M 2n is called <strong>Balanced</strong> <strong>Hermitian</strong> if theLee form θ ≡ 0. Equivalently, F n−1 is closed.Intersections in dimension 2n ≥ 6 [AI]:[B] J.-M. Bismut, A local index theorem for non-Kähler manifolds, Math. Ann. 284 (1989), 681-699.[M] M.L. Michelsohn, On the existence of special metrics in complex <strong>geometry</strong>, Acta Math. 149 (1982), 261-295.[AI] B. Alexandrov, S. Ivanov: Vanishing theorems on <strong>Hermitian</strong> manifolds, Differential Geom. Appl. 14 (2001), 251-265.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 11 / 28

<strong>Balanced</strong> <strong>Hermitian</strong> structuresDef.: <strong>Hermitian</strong> structure on a manifold M 2n is a pair (J, g), J complexstructure and g Riemannian metric such that g(JX, JY ) = g(X, Y ).Fundamental 2-form: F(X, Y ) = g(X, JY ).Lee 1-form: θ = 11−nJδF, where δ is the g-adjoint of d.Bismut connection [B]: ∇ B unique connection for which g and J areparallel with torsion T : g(X, T (Y , Z )) = JdF(X, Y , Z ).Special class [M]: (J, g) on M 2n is called <strong>Balanced</strong> <strong>Hermitian</strong> if theLee form θ ≡ 0. Equivalently, F n−1 is closed.Intersections in dimension 2n ≥ 6 [AI]:BH ∩ LCK = K BH ∩ KT = K LCK ∩ KT = K[B] J.-M. Bismut, A local index theorem for non-Kähler manifolds, Math. Ann. 284 (1989), 681-699.[M] M.L. Michelsohn, On the existence of special metrics in complex <strong>geometry</strong>, Acta Math. 149 (1982), 261-295.[AI] B. Alexandrov, S. Ivanov: Vanishing theorems on <strong>Hermitian</strong> manifolds, Differential Geom. Appl. 14 (2001), 251-265.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 11 / 28

6-nilmanifolds with BH structureProposition [U]A compact nilmanifold M 6 = Γ\G admits a balanced metric compatiblewith an invariant complex structure J if and only if the Lie algebra of Gis isomorphic to h 1 ,. . .,h 6 or h − 19 .[U] L. Ugarte: <strong>Hermitian</strong> structures on six-dimensional nilmanifolds, Transform. Groups 12 (2007), 175-202.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 12 / 28

6-nilmanifolds with BH structureProposition [U]A compact nilmanifold M 6 = Γ\G admits a balanced metric compatiblewith an invariant complex structure J if and only if the Lie algebra of Gis isomorphic to h 1 ,. . .,h 6 or h − 19 .Remark: [B]: Symmetrization process on M = Γ\G: given anyQ : X(M) × · · · × X(M) −→ C ∞ (M), define Q ν : g × · · · × g −→ R byQ ν (X 1 , . . . , X k ) = ∫ p∈M Q p(X 1 | p , . . . , X k | p ) ν[U] L. Ugarte: <strong>Hermitian</strong> structures on six-dimensional nilmanifolds, Transform. Groups 12 (2007), 175-202.[B] F.A. Belgun: On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), 1-40.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 12 / 28

6-nilmanifolds with BH structureProposition [U]A compact nilmanifold M 6 = Γ\G admits a balanced metric compatiblewith an invariant complex structure J if and only if the Lie algebra of Gis isomorphic to h 1 ,. . .,h 6 or h − 19 .Remark: [B]: Symmetrization process on M = Γ\G: given anyQ : X(M) × · · · × X(M) −→ C ∞ (M), define Q ν : g × · · · × g −→ R byQ ν (X 1 , . . . , X k ) = ∫ p∈M Q p(X 1 | p , . . . , X k | p ) ν[FG]: dF 2 = 0 =⇒ d(F 2 ) ν = 0;J invariant =⇒ (F 2 ) ν is strictly positive (2, 2)-form;∃ (J, ĝ) on g such that ˆF 2 = (F 2 ) ν =⇒ (J, ĝ) is balanced on g.[U] L. Ugarte: <strong>Hermitian</strong> structures on six-dimensional nilmanifolds, Transform. Groups 12 (2007), 175-202.[B] F.A. Belgun: On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), 1-40.[FG] A. Fino, G. Grantcharov: Properties of manifolds with skew-symmetric torsion and special holonomy,Adv. Math. 189 (2004), 439-450.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 12 / 28

Proof: (1) Fixed a basis {ω j } 3 j=1 for g1,0 with respect to J, thefundamental 2-form F ∈ ∧2 g ∗ of any J-<strong>Hermitian</strong> metric is given by:2 F = i(r 2 ω 1¯1 + s 2 ω 2¯2 + t 2 ω 3¯3) + u ω 1¯2 − ū ω 2¯1 + v ω 2¯3 − ¯v ω 3¯2+ z ω 1¯3 − ¯z ω 3¯1,where r 2 , s 2 , t 2 ∈ R ∗ and u, v, z ∈ C.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 13 / 28

Proof: (1) Fixed a basis {ω j } 3 j=1 for g1,0 with respect to J, thefundamental 2-form F ∈ ∧2 g ∗ of any J-<strong>Hermitian</strong> metric is given by:2 F = i(r 2 ω 1¯1 + s 2 ω 2¯2 + t 2 ω 3¯3) + u ω 1¯2 − ū ω 2¯1 + v ω 2¯3 − ¯v ω 3¯2+ z ω 1¯3 − ¯z ω 3¯1,where r 2 , s 2 , t 2 ∈ R ∗ and u, v, z ∈ C.(2) J complex-parallelizable (h 5 , J 0 ): any F is balanced.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 13 / 28

Proof: (1) Fixed a basis {ω j } 3 j=1 for g1,0 with respect to J, thefundamental 2-form F ∈ ∧2 g ∗ of any J-<strong>Hermitian</strong> metric is given by:2 F = i(r 2 ω 1¯1 + s 2 ω 2¯2 + t 2 ω 3¯3) + u ω 1¯2 − ū ω 2¯1 + v ω 2¯3 − ¯v ω 3¯2where r 2 , s 2 , t 2 ∈ R ∗ and u, v, z ∈ C.(2) J complex-parallelizable (h 5 , J 0 ): any F is balanced.+ z ω 1¯3 − ¯z ω 3¯1,(3) J nilpotent but not complex-parallelizable:{dω 1 = dω 2 = 0, dω 3 = ρ ω 12 + ω 1¯1 + b 2 ω 1¯2 + D ω 2¯2,2 F = i(ω 1¯1 + s 2 ω 2¯2 + t 2 ω 3¯3) + u ω 1¯2 − ū ω 2¯1,s 2 + D = ūb 2 i =⇒ g ∼ = h 1 , h 2 , h 3 , h 4 , h 5 , h 6L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 13 / 28

Proof: (1) Fixed a basis {ω j } 3 j=1 for g1,0 with respect to J, thefundamental 2-form F ∈ ∧2 g ∗ of any J-<strong>Hermitian</strong> metric is given by:2 F = i(r 2 ω 1¯1 + s 2 ω 2¯2 + t 2 ω 3¯3) + u ω 1¯2 − ū ω 2¯1 + v ω 2¯3 − ¯v ω 3¯2where r 2 , s 2 , t 2 ∈ R ∗ and u, v, z ∈ C.(2) J complex-parallelizable (h 5 , J 0 ): any F is balanced.+ z ω 1¯3 − ¯z ω 3¯1,(3) J nilpotent but not complex-parallelizable:{dω 1 = dω 2 = 0, dω 3 = ρ ω 12 + ω 1¯1 + b 2 ω 1¯2 + D ω 2¯2,2 F = i(ω 1¯1 + s 2 ω 2¯2 + t 2 ω 3¯3) + u ω 1¯2 − ū ω 2¯1,s 2 + D = ūb 2 i =⇒ g ∼ = h 1 , h 2 , h 3 , h 4 , h 5 , h 6{dω 1 = 0, dω 2 = ω 13 + ω 1¯3, dω 3 = ±i(ω 1¯2− ω 2¯1),(4) J non-nilp:2 F = i(r 2 ω 1¯1+ s 2 ω 2¯2+ t 2 ω 3¯3) + v ω 2¯3− ¯v ω 3¯2.g ∼ = h − 19L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 13 / 28

Deformation of balanced metricsDefinition [FY]A complex manifold M 2n satisfies the (n − 1, n)-th weak ∂ ¯∂-lemma if:∀ real ϕ n−1,n−1 | ¯∂ϕ = ∂η, there exists ψ n−2,n−1 such that ¯∂ϕ = i∂ ¯∂ψ.[FY] J-X. Fu, S-T. Yau: A note on balanced metrics under small deformations, arXiv:1105.0026v2 [math.DG]L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 14 / 28

Deformation of balanced metricsDefinition [FY]A complex manifold M 2n satisfies the (n − 1, n)-th weak ∂ ¯∂-lemma if:∀ real ϕ n−1,n−1 | ¯∂ϕ = ∂η, there exists ψ n−2,n−1 such that ¯∂ϕ = i∂ ¯∂ψ.Theorem [FY]Let M 2n complex and M λ a small deformation. If M 0 has a balancedmetric and M λ satisfies the (n − 1, n)-th weak ∂ ¯∂-lemma for smallλ ≠ 0, then ∃ a balanced structure on M λ for sufficiently small λ.[FY] J-X. Fu, S-T. Yau: A note on balanced metrics under small deformations, arXiv:1105.0026v2 [math.DG]L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 14 / 28

Deformation of balanced metricsDefinition [FY]A complex manifold M 2n satisfies the (n − 1, n)-th weak ∂ ¯∂-lemma if:∀ real ϕ n−1,n−1 | ¯∂ϕ = ∂η, there exists ψ n−2,n−1 such that ¯∂ϕ = i∂ ¯∂ψ.Theorem [FY]Let M 2n complex and M λ a small deformation. If M 0 has a balancedmetric and M λ satisfies the (n − 1, n)-th weak ∂ ¯∂-lemma for smallλ ≠ 0, then ∃ a balanced structure on M λ for sufficiently small λ.Remark: small deformation of the Iwasawa manifold (h 5 , J 0 ) does notsatisfy the (2, 3)-th weak ∂ ¯∂-lemma.Question: which nilmanifolds satisfy the (n − 1, n)-th weak ∂ ¯∂-lemma?[FY] J-X. Fu, S-T. Yau: A note on balanced metrics under small deformations, arXiv:1105.0026v2 [math.DG]L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 14 / 28

(n-1,n)-th weak ∂ ¯∂-lemma on nilmanifoldsProposition [UV 2 ]Let M 2n = (Γ\G, J) nilmanifold, J invariant, g Lie algebra of G.(1) If (g, J) does not satisfy the weak lemma,then M does not satisfy the weak lemma.(2) If (g, J) satisfies the weak lemma and H n−2,n (M) ∼ = H n−2,n (g, J),then M satisfies the weak lemma.[UV2] L. Ugarte, R. Villacampa: <strong>Balanced</strong> <strong>Hermitian</strong> <strong>geometry</strong> on 6-dimensional nilmanifolds, arXiv:1104.5524v2 [math.DG]L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 15 / 28

(n-1,n)-th weak ∂ ¯∂-lemma on nilmanifoldsProposition [UV 2 ]Let M 2n = (Γ\G, J) nilmanifold, J invariant, g Lie algebra of G.(1) If (g, J) does not satisfy the weak lemma,then M does not satisfy the weak lemma.(2) If (g, J) satisfies the weak lemma and H n−2,n (M) ∼ = H n−2,n (g, J),then M satisfies the weak lemma.Proof: (1) Use the symmetrization process.[UV2] L. Ugarte, R. Villacampa: <strong>Balanced</strong> <strong>Hermitian</strong> <strong>geometry</strong> on 6-dimensional nilmanifolds, arXiv:1104.5524v2 [math.DG]L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 15 / 28

(n-1,n)-th weak ∂ ¯∂-lemma on nilmanifoldsProposition [UV 2 ]Let M 2n = (Γ\G, J) nilmanifold, J invariant, g Lie algebra of G.(1) If (g, J) does not satisfy the weak lemma,then M does not satisfy the weak lemma.(2) If (g, J) satisfies the weak lemma and H n−2,n (M) ∼ = H n−2,n (g, J),then M satisfies the weak lemma.Proof: (1) Use the symmetrization process.(2) Let ϕ ∈ Ω n−1,n−1 (M) real such that ¯∂ϕ = ∂η n−2,n .¯∂η =0, H n−2,n (M) ∼ = H n−2,n (g, J) =⇒ η = η ν + ¯∂(iα), α ∈ Ω n−2,n−1 (M);[UV2] L. Ugarte, R. Villacampa: <strong>Balanced</strong> <strong>Hermitian</strong> <strong>geometry</strong> on 6-dimensional nilmanifolds, arXiv:1104.5524v2 [math.DG]L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 15 / 28

(n-1,n)-th weak ∂ ¯∂-lemma on nilmanifoldsProposition [UV 2 ]Let M 2n = (Γ\G, J) nilmanifold, J invariant, g Lie algebra of G.(1) If (g, J) does not satisfy the weak lemma,then M does not satisfy the weak lemma.(2) If (g, J) satisfies the weak lemma and H n−2,n (M) ∼ = H n−2,n (g, J),then M satisfies the weak lemma.Proof: (1) Use the symmetrization process.(2) Let ϕ ∈ Ω n−1,n−1 (M) real such that ¯∂ϕ = ∂η n−2,n .¯∂η =0, H n−2,n (M) ∼ = H n−2,n (g, J) =⇒ η = η ν + ¯∂(iα), α ∈ Ω n−2,n−1 (M);¯∂ϕ = ∂η =⇒ ¯∂ϕ ν = ∂η ν = i∂ ¯∂ ˜α, ˜α ∈ ∧ n−2,n−1 (g).[UV2] L. Ugarte, R. Villacampa: <strong>Balanced</strong> <strong>Hermitian</strong> <strong>geometry</strong> on 6-dimensional nilmanifolds, arXiv:1104.5524v2 [math.DG]L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 15 / 28

(n-1,n)-th weak ∂ ¯∂-lemma on nilmanifoldsProposition [UV 2 ]Let M 2n = (Γ\G, J) nilmanifold, J invariant, g Lie algebra of G.(1) If (g, J) does not satisfy the weak lemma,then M does not satisfy the weak lemma.(2) If (g, J) satisfies the weak lemma and H n−2,n (M) ∼ = H n−2,n (g, J),then M satisfies the weak lemma.Proof: (1) Use the symmetrization process.(2) Let ϕ ∈ Ω n−1,n−1 (M) real such that ¯∂ϕ = ∂η n−2,n .¯∂η =0, H n−2,n (M) ∼ = H n−2,n (g, J) =⇒ η = η ν + ¯∂(iα), α ∈ Ω n−2,n−1 (M);¯∂ϕ = ∂η =⇒ ¯∂ϕ ν = ∂η ν = i∂ ¯∂ ˜α, ˜α ∈ ∧ n−2,n−1 (g).Therefore, ¯∂ϕ = i∂ ¯∂(˜α + α).[UV2] L. Ugarte, R. Villacampa: <strong>Balanced</strong> <strong>Hermitian</strong> <strong>geometry</strong> on 6-dimensional nilmanifolds, arXiv:1104.5524v2 [math.DG]L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 15 / 28

Weak lemma for balanced 6-nilmanifoldsProposition [UV 2 ]Let M 6 = Γ\G nilmanifold with g = h 1 , . . . , h 6 or h − 19, and J invariant onM. Then, (M, J) satisfies the weak lemma if and only if J is abelian,complex-parallelizable or of non-nilpotent type.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 16 / 28

Weak lemma for balanced 6-nilmanifoldsProposition [UV 2 ]Let M 6 = Γ\G nilmanifold with g = h 1 , . . . , h 6 or h − 19, and J invariant onM. Then, (M, J) satisfies the weak lemma if and only if J is abelian,complex-parallelizable or of non-nilpotent type.Proof: Using [CF, R], it suffices to study the weak lemma on (g, J).[CF] S. Console, A. Fino: Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001), 111-124.[R] S. Rollenske: Geometry of nilmanifolds with left-invariant complex structure and deformations in the large,Proc. London Math. Soc. 99 (2009), 425-460.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 16 / 28

Weak lemma for balanced 6-nilmanifoldsProposition [UV 2 ]Let M 6 = Γ\G nilmanifold with g = h 1 , . . . , h 6 or h − 19, and J invariant onM. Then, (M, J) satisfies the weak lemma if and only if J is abelian,complex-parallelizable or of non-nilpotent type.Proof: Using [CF, R], it suffices to study the weak lemma on (g, J).J nilpotent: ∂ ¯∂( ∧ 1,2 (g ∗ )) = 0 ⊃ 〈ρ ω 12¯1¯2¯3〉 = ∂( ∧ 1,3 (g ∗ )) =⇒ ρ = 0[CF] S. Console, A. Fino: Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001), 111-124.[R] S. Rollenske: Geometry of nilmanifolds with left-invariant complex structure and deformations in the large,Proc. London Math. Soc. 99 (2009), 425-460.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 16 / 28

Weak lemma for balanced 6-nilmanifoldsProposition [UV 2 ]Let M 6 = Γ\G nilmanifold with g = h 1 , . . . , h 6 or h − 19, and J invariant onM. Then, (M, J) satisfies the weak lemma if and only if J is abelian,complex-parallelizable or of non-nilpotent type.Proof: Using [CF, R], it suffices to study the weak lemma on (g, J).J nilpotent: ∂ ¯∂( ∧ 1,2 (g ∗ )) = 0 ⊃ 〈ρ ω 12¯1¯2¯3〉 = ∂( ∧ 1,3 (g ∗ )) =⇒ ρ = 0J non-nilpotent: ∂ ¯∂( ∧ 1,2 (g ∗ )) = 〈ω 13¯1¯2¯3〉 = ∂( ∧ 1,3 (g ∗ ))[CF] S. Console, A. Fino: Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001), 111-124.[R] S. Rollenske: Geometry of nilmanifolds with left-invariant complex structure and deformations in the large,Proc. London Math. Soc. 99 (2009), 425-460.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 16 / 28

Weak lemma for balanced 6-nilmanifoldsProposition [UV 2 ]Let M 6 = Γ\G nilmanifold with g = h 1 , . . . , h 6 or h − 19, and J invariant onM. Then, (M, J) satisfies the weak lemma if and only if J is abelian,complex-parallelizable or of non-nilpotent type.Proof: Using [CF, R], it suffices to study the weak lemma on (g, J).J nilpotent: ∂ ¯∂( ∧ 1,2 (g ∗ )) = 0 ⊃ 〈ρ ω 12¯1¯2¯3〉 = ∂( ∧ 1,3 (g ∗ )) =⇒ ρ = 0J non-nilpotent: ∂ ¯∂( ∧ 1,2 (g ∗ )) = 〈ω 13¯1¯2¯3〉 = ∂( ∧ 1,3 (g ∗ ))Example: h 5 = (0, 0, 0, 0, 13 − 24, 14 + 23). For λ ∈ [0, 1),I λ : dµ 1 = dµ 2 = 0,dµ 3 = λ µ 12 + µ 1¯2[CF] S. Console, A. Fino: Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001), 111-124.[R] S. Rollenske: Geometry of nilmanifolds with left-invariant complex structure and deformations in the large,Proc. London Math. Soc. 99 (2009), 425-460.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 16 / 28

Weak lemma for balanced 6-nilmanifoldsProposition [UV 2 ]Let M 6 = Γ\G nilmanifold with g = h 1 , . . . , h 6 or h − 19, and J invariant onM. Then, (M, J) satisfies the weak lemma if and only if J is abelian,complex-parallelizable or of non-nilpotent type.Proof: Using [CF, R], it suffices to study the weak lemma on (g, J).J nilpotent: ∂ ¯∂( ∧ 1,2 (g ∗ )) = 0 ⊃ 〈ρ ω 12¯1¯2¯3〉 = ∂( ∧ 1,3 (g ∗ )) =⇒ ρ = 0J non-nilpotent: ∂ ¯∂( ∧ 1,2 (g ∗ )) = 〈ω 13¯1¯2¯3〉 = ∂( ∧ 1,3 (g ∗ ))Example: h 5 = (0, 0, 0, 0, 13 − 24, 14 + 23). For λ ∈ [0, 1),I λ : dµ 1 = dµ 2 = 0,dµ 3 = λ µ 12 + µ 1¯2<strong>Balanced</strong> I λ -<strong>Hermitian</strong> metric: g λ =diag(1, 1,√1+λ1−λ , √1+λ1−λ[CF] S. Console, A. Fino: Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001), 111-124.[R] S. Rollenske: Geometry of nilmanifolds with left-invariant complex structure and deformations in the large,Proc. London Math. Soc. 99 (2009), 425-460., 1+ λ, 1+λ)L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 16 / 28

Holonomy of the Bismut connectionGiven (J, g) <strong>Hermitian</strong> on M 2n , Hol(∇ B ) ⊂ U(n).L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 17 / 28

Holonomy of the Bismut connectionGiven (J, g) <strong>Hermitian</strong> on M 2n , Hol(∇ B ) ⊂ U(n).Def.: An SU(n)-structure on M 2n is a pair (F, Ψ) such that F is thefundamental 2-form of an almost <strong>Hermitian</strong> structure (J, g) andΨ is a (n,0)-form such that F ∧ Ψ = 0, F n = c n Ψ ∧ Ψ ≠ 0.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 17 / 28

Holonomy of the Bismut connectionGiven (J, g) <strong>Hermitian</strong> on M 2n , Hol(∇ B ) ⊂ U(n).Def.: An SU(n)-structure on M 2n is a pair (F, Ψ) such that F is thefundamental 2-form of an almost <strong>Hermitian</strong> structure (J, g) andΨ is a (n,0)-form such that F ∧ Ψ = 0, F n = c n Ψ ∧ Ψ ≠ 0.If, in addition, ∇ B Ψ = 0 then Hol(∇ B ) ⊂ SU(n).L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 17 / 28

Holonomy of the Bismut connectionGiven (J, g) <strong>Hermitian</strong> on M 2n , Hol(∇ B ) ⊂ U(n).Def.: An SU(n)-structure on M 2n is a pair (F, Ψ) such that F is thefundamental 2-form of an almost <strong>Hermitian</strong> structure (J, g) andΨ is a (n,0)-form such that F ∧ Ψ = 0, F n = c n Ψ ∧ Ψ ≠ 0.If, in addition, ∇ B Ψ = 0 then Hol(∇ B ) ⊂ SU(n).Proposition [FPS]: An invariant <strong>Hermitian</strong> structure on a nilmanifoldM 2n = Γ\G is balanced if and only if Hol(∇ B ) ⊂ SU(n).∇ B (ω 1 ∧ · · · ∧ ω n ) = 0 for any basis {ω k } n k=1of g1,0[FPS] A. Fino, M. Parton, S. Salamon: Families of strong KT structures in six dimensions, Comment. Math. Helv. 79 (2004)L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 17 / 28

Holonomy of the Bismut connectionGiven (J, g) <strong>Hermitian</strong> on M 2n , Hol(∇ B ) ⊂ U(n).Def.: An SU(n)-structure on M 2n is a pair (F, Ψ) such that F is thefundamental 2-form of an almost <strong>Hermitian</strong> structure (J, g) andΨ is a (n,0)-form such that F ∧ Ψ = 0, F n = c n Ψ ∧ Ψ ≠ 0.If, in addition, ∇ B Ψ = 0 then Hol(∇ B ) ⊂ SU(n).Proposition [FPS]: An invariant <strong>Hermitian</strong> structure on a nilmanifoldM 2n = Γ\G is balanced if and only if Hol(∇ B ) ⊂ SU(n).∇ B (ω 1 ∧ · · · ∧ ω n ) = 0 for any basis {ω k } n k=1of g1,0Question: which balanced nilmanifolds have Hol(∇ B ) = SU(n)?[FPS] A. Fino, M. Parton, S. Salamon: Families of strong KT structures in six dimensions, Comment. Math. Helv. 79 (2004)L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 17 / 28

Proposition [UV 2 ]Let (J,g) be an invariant balanced <strong>Hermitian</strong> structure on a nilmanifoldM 6 = Γ\G. Then, Hol(∇ B ) = SU(3) if and only if J is non-abelian.Moreover, if J is abelian then Hol(∇ B ) ⊆ SU(2), with “=” iff g ∼ = h 5 .L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 18 / 28

Proposition [UV 2 ]Let (J,g) be an invariant balanced <strong>Hermitian</strong> structure on a nilmanifoldM 6 = Γ\G. Then, Hol(∇ B ) = SU(3) if and only if J is non-abelian.Moreover, if J is abelian then Hol(∇ B ) ⊆ SU(2), with “=” iff g ∼ = h 5 .Remarks: (1) Hol(∇ B ) only depends on the complex structure J.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 18 / 28

Proposition [UV 2 ]Let (J,g) be an invariant balanced <strong>Hermitian</strong> structure on a nilmanifoldM 6 = Γ\G. Then, Hol(∇ B ) = SU(3) if and only if J is non-abelian.Moreover, if J is abelian then Hol(∇ B ) ⊆ SU(2), with “=” iff g ∼ = h 5 .Remarks: (1) Hol(∇ B ) only depends on the complex structure J.(2) For (h 5 , I λ , g λ ): Hol(∇ B 0 ) = SU(2) and Hol(∇B λ) = SU(3) for λ ≠ 0.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 18 / 28

Proposition [UV 2 ]Let (J,g) be an invariant balanced <strong>Hermitian</strong> structure on a nilmanifoldM 6 = Γ\G. Then, Hol(∇ B ) = SU(3) if and only if J is non-abelian.Moreover, if J is abelian then Hol(∇ B ) ⊆ SU(2), with “=” iff g ∼ = h 5 .Remarks: (1) Hol(∇ B ) only depends on the complex structure J.(2) For (h 5 , I λ , g λ ): Hol(∇ B 0 ) = SU(2) and Hol(∇B λ) = SU(3) for λ ≠ 0.(3) The result does not hold on solvmanifolds. For example:g = (0, 0, −13 − 24, −14 + 23, 15 + 26, 16 − 25)Abelian complex structure: Je 1 = −e 2 , Je 3 = −e 4 , Je 5 = −e 6<strong>Balanced</strong> structure: F = e 12 + e 34 + e 56Ψ = (e 1 + i e 2 ) ∧ (e 3 + i e 4 ) ∧ (e 5 + i e 6 ) satisfies ∇ B Ψ = 0and Hol(∇ B ) = SU(3).L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 18 / 28

Proof: adapted basis: Je 1 = −e 2 , Je 3 = −e 4 , Je 5 = −e 6 ,F = e 12 + e 34 + e 56 , Ψ = (e 1 + i e 2 ) ∧ (e 3 + i e 4 ) ∧ (e 5 + i e 6 )L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 19 / 28

Proof: adapted basis: Je 1 = −e 2 , Je 3 = −e 4 , Je 5 = −e 6 ,F = e 12 + e 34 + e 56 , Ψ = (e 1 + i e 2 ) ∧ (e 3 + i e 4 ) ∧ (e 5 + i e 6 )• J nilpotent but not complex-parallelizable:⎧⎪⎨de 1 = de 2 = de 3 = de 4 = 0,⎪⎩de 5 =s t (ρ + b2 )e 13 −s t (ρ − b2 )e 24 ,de 6 = − 2 t (e 12 − e 34 ) +s t (ρ − b2 )e 14 +s t (ρ + b2 )e 23 ,⎧(de = 2sY 1 |u|b (e − e ) − tu 1 |u| Y 2 b2 (e 13 + e 24 ) + su 1 ρ (e 13 − e 24 ) + su 2 (ρ − b 2 )e 14 + (ρ + b 2 )e 23)] ,⎪⎩ de 6 = 2sY[(2s 2 −u 2 b 2 )|u| (e 12 − e 34 )+ tu 2 |u| Y 2 b2 (e 13 + e 24 )− su 2 ρ (e 13 − e 24 )+ su 1((ρ−b 2 )e 14 + (ρ+b 2 )e 23)]⎪⎨de 1 = de 2 = de 3 = de 4 = 0,[u 2 12 34 √ρ ∈ {0, 1}, b ∈ R, s, t ∈ R ∗ , s 2 > |u| 2 > 0, Y = 2 s 2 − |u| 2 /|u|t;L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 19 / 28

Proof: adapted basis: Je 1 = −e 2 , Je 3 = −e 4 , Je 5 = −e 6 ,F = e 12 + e 34 + e 56 , Ψ = (e 1 + i e 2 ) ∧ (e 3 + i e 4 ) ∧ (e 5 + i e 6 )• J nilpotent but not complex-parallelizable:⎧⎪⎨⎪⎩de 1 = de 2 = de 3 = de 4 = 0,de 5 = t s (ρ + b2 )e 13 − t s (ρ − b2 )e 24 ,de 6 = − 2 t (e 12 − e 34 ) + t s (ρ − b2 )e 14 + t s (ρ + b2 )e 23 ,⎧de 1 = de 2 = de 3 = de 4 = 0,⎪⎨(de 5 = 2sY[u 1 |u|b 2 (e 12 − e 34 ) − tu 1 |u| Y 2 b2 (e 13 + e 24 ) + su 1 ρ (e 13 − e 24 ) + su 2 (ρ − b 2 )e 14 + (ρ + b 2 )e 23)] ,⎪⎩ de 6 = 2sY[(2s 2 −u 2 b 2 )|u| (e 12 − e 34 )+ tu 2 |u| Y 2 b2 (e 13 + e 24 )− su 2 ρ (e 13 − e 24 )+ su 1((ρ−b 2 )e 14 + (ρ+b 2 )e 23)]√ρ ∈ {0, 1}, b ∈ R, s, t ∈ R ∗ , s 2 > |u| 2 > 0, Y = 2 s 2 − |u| 2 /|u|t;ρ = 0 ⇒⎧⎨∇ B (e 12 + e 34 ) = 0, ∇ B ((e 1 + i e 2 ) ∧ (e 3 + i e 4 )) = 0,⎩∇ B e 5 = 0, ∇ B e 6 = 0.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 19 / 28

Proof: adapted basis: Je 1 = −e 2 , Je 3 = −e 4 , Je 5 = −e 6 ,F = e 12 + e 34 + e 56 , Ψ = (e 1 + i e 2 ) ∧ (e 3 + i e 4 ) ∧ (e 5 + i e 6 )• J nilpotent but not complex-parallelizable:⎧⎪⎨⎪⎩de 1 = de 2 = de 3 = de 4 = 0,de 5 = t s (ρ + b2 )e 13 − t s (ρ − b2 )e 24 ,de 6 = − 2 t (e 12 − e 34 ) + t s (ρ − b2 )e 14 + t s (ρ + b2 )e 23 ,⎧de 1 = de 2 = de 3 = de 4 = 0,⎪⎨(de 5 = 2sY[u 1 |u|b 2 (e 12 − e 34 ) − tu 1 |u| Y 2 b2 (e 13 + e 24 ) + su 1 ρ (e 13 − e 24 ) + su 2 (ρ − b 2 )e 14 + (ρ + b 2 )e 23)] ,⎪⎩ de 6 = 2sY[(2s 2 −u 2 b 2 )|u| (e 12 − e 34 )+ tu 2 |u| Y 2 b2 (e 13 + e 24 )− su 2 ρ (e 13 − e 24 )+ su 1((ρ−b 2 )e 14 + (ρ+b 2 )e 23)]√ρ ∈ {0, 1}, b ∈ R, s, t ∈ R ∗ , s 2 > |u| 2 > 0, Y = 2 s 2 − |u| 2 /|u|t;ρ = 0 ⇒⎧⎨∇ B (e 12 + e 34 ) = 0, ∇ B ((e 1 + i e 2 ) ∧ (e 3 + i e 4 )) = 0,⎩∇ B e 5 = 0, ∇ B e 6 = 0.If, in addition, b = 0 ⇒ ∇ B (e 12 ) = 0, ∇ B (e 34 ) = 0.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 19 / 28

Proof: adapted basis: Je 1 = −e 2 , Je 3 = −e 4 , Je 5 = −e 6 ,F = e 12 + e 34 + e 56 , Ψ = (e 1 + i e 2 ) ∧ (e 3 + i e 4 ) ∧ (e 5 + i e 6 )• J nilpotent but not complex-parallelizable:⎧⎪⎨⎪⎩de 1 = de 2 = de 3 = de 4 = 0,de 5 = t s (ρ + b2 )e 13 − t s (ρ − b2 )e 24 ,de 6 = − 2 t (e 12 − e 34 ) + t s (ρ − b2 )e 14 + t s (ρ + b2 )e 23 ,⎧de 1 = de 2 = de 3 = de 4 = 0,⎪⎨(de 5 = 2sY[u 1 |u|b 2 (e 12 − e 34 ) − tu 1 |u| Y 2 b2 (e 13 + e 24 ) + su 1 ρ (e 13 − e 24 ) + su 2 (ρ − b 2 )e 14 + (ρ + b 2 )e 23)] ,⎪⎩ de 6 = 2sY[(2s 2 −u 2 b 2 )|u| (e 12 − e 34 )+ tu 2 |u| Y 2 b2 (e 13 + e 24 )− su 2 ρ (e 13 − e 24 )+ su 1((ρ−b 2 )e 14 + (ρ+b 2 )e 23)]√ρ ∈ {0, 1}, b ∈ R, s, t ∈ R ∗ , s 2 > |u| 2 > 0, Y = 2 s 2 − |u| 2 /|u|t;ρ = 0 ⇒⎧⎨∇ B (e 12 + e 34 ) = 0, ∇ B ((e 1 + i e 2 ) ∧ (e 3 + i e 4 )) = 0,⎩∇ B e 5 = 0, ∇ B e 6 = 0.If, in addition, b = 0 ⇒ ∇ B (e 12 ) = 0, ∇ B (e 34 ) = 0.• J complex-parallelizable or non-nilpotent: always Hol(∇ B ) = SU(3).L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 19 / 28

Applications to heterotic string theoryHeterotic string compactifications with fluxes:10-dimensional manifold R 1,3 × M 6 , M compact, endowed with:L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 20 / 28

Applications to heterotic string theoryHeterotic string compactifications with fluxes:10-dimensional manifold R 1,3 × M 6 , M compact, endowed with:Three-form field strength H (flux);Dilaton function φ;Non-zero curvature 2-form F A of instanton type(Donaldson-Uhlenbeck-Yau instanton A);Curvature 2-form R of some metric connection ∇;String tension α ′ > 0.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 20 / 28

Consider H = T , the torsion 3-form of ∇ B .Strominger system [St]: compact SU(3)-manifold satisfyingGravitino and dilatino equations: Hol(∇ B ) ⊂ SU(3),J integrable and Lee form θ = 2dφ;[St] A. Strominger: Superstrings with torsion, Nuclear Phys. B 274 (1986), 254-284.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 21 / 28

Consider H = T , the torsion 3-form of ∇ B .Strominger system [St]: compact SU(3)-manifold satisfyingGravitino and dilatino equations: Hol(∇ B ) ⊂ SU(3),J integrable and Lee form θ = 2dφ;Gaugino equation: Donaldson-Uhlenbeck-Yau’sSU(3)-instanton A:(F A ) i j (Je k, Je l ) = (F A ) i j (e ∑k, e l ),k (F A ) i j (e k, Je k ) = 0;[St] A. Strominger: Superstrings with torsion, Nuclear Phys. B 274 (1986), 254-284.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 21 / 28

Consider H = T , the torsion 3-form of ∇ B .Strominger system [St]: compact SU(3)-manifold satisfyingGravitino and dilatino equations: Hol(∇ B ) ⊂ SU(3),J integrable and Lee form θ = 2dφ;Gaugino equation: Donaldson-Uhlenbeck-Yau’sSU(3)-instanton A:(F A ) i j (Je k, Je l ) = (F A ) i j (e ∑k, e l ),k (F A ) i j (e k, Je k ) = 0;Anomaly cancellation condition:dT = α′4 (tr R ∧ R − tr F A ∧ F A ), α ′ > 0,R is the curvature of a metric connection ∇ over M.[St] A. Strominger: Superstrings with torsion, Nuclear Phys. B 274 (1986), 254-284.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 21 / 28

Consider H = T , the torsion 3-form of ∇ B .Strominger system [St]: compact SU(3)-manifold satisfyingGravitino and dilatino equations: Hol(∇ B ) ⊂ SU(3),J integrable and Lee form θ = 2dφ;Gaugino equation: Donaldson-Uhlenbeck-Yau’sSU(3)-instanton A:(F A ) i j (Je k, Je l ) = (F A ) i j (e ∑k, e l ),k (F A ) i j (e k, Je k ) = 0;Anomaly cancellation condition:dT = α′4 (tr R ∧ R − tr F A ∧ F A ), α ′ > 0,R is the curvature of a metric connection ∇ over M.Existence of solutions: [FY][FY] J-X. Fu, S-T. Yau: The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation,J. Differential Geom. 78 (2008), 369-428.[St] A. Strominger: Superstrings with torsion, Nuclear Phys. B 274 (1986), 254-284.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 21 / 28

Explicit solutions with constant dilaton φ [FIUV]:compact SU(3) manifold satisfyingGravitino and dilatino equations with constant dilaton:Hol(∇ B ) ⊂ SU(3) and <strong>Balanced</strong> <strong>Hermitian</strong> structure;Gaugino equation: SU(3)-instanton A:(F A ) i j (Je k, Je l ) = (F A ) i j (e ∑k, e l ),k (F A ) i j (e k, Je k ) = 0;Anomaly cancellation condition:dT = α′4 (tr R ∧ R − tr F A ∧ F A ), α ′ > 0.Two cases: R = R B and R = R C .[FIUV] M. Fernández, S. Ivanov, L. Ugarte, R. Villacampa: Non-Kaehler heterotic-string compactifications with non-zerofluxes and constant dilaton, Commun. Math. Phys. 288 (2009), 677-697.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 22 / 28

Explicit solutions with constant dilaton φ [FIUV]:compact SU(3) manifold satisfyingGravitino and dilatino equations with constant dilaton:Hol(∇ B ) ⊂ SU(3) and <strong>Balanced</strong> <strong>Hermitian</strong> structure;Gaugino equation: SU(3)-instanton A:(F A ) i j (Je k, Je l ) = (F A ) i j (e ∑k, e l ),k (F A ) i j (e k, Je k ) = 0;Anomaly cancellation condition:dT = α′4 (tr R ∧ R − tr F A ∧ F A ), α ′ > 0.Two cases: R = R B and R = R C .Question: given a complex structure J on a nilmanifold M 6 ,can we ensure existence of solution?[FIUV] M. Fernández, S. Ivanov, L. Ugarte, R. Villacampa: Non-Kaehler heterotic-string compactifications with non-zerofluxes and constant dilaton, Commun. Math. Phys. 288 (2009), 677-697.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 22 / 28

Solutions with respect to ∇ BTheoremLet M 6 nilmanifold with invariant BH structure (J, g). If J isabelian then for any g there is a non-flat instanton solving theStrominger system with respect to ∇ B .L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 23 / 28

Solutions with respect to ∇ BTheoremLet M 6 nilmanifold with invariant BH structure (J, g). If J isabelian then for any g there is a non-flat instanton solving theStrominger system with respect to ∇ B .Proof: two families of balanced structures for abelian J:(I)⎧⎪⎨⎪⎩de 1 = de 2 = de 3 = de 4 = 0,de 5 = t s b2 (e 13 + e 24 ),de 6 = − 2 t (e 12 − e 34 ) − t s b2 (e 14 − e 23 ),(II)⎧⎪⎨⎪⎩de 1 = de 2 = de 3 = de 4 = 0,[]de 5 = 2sY u 1 |u|b 2 (e 12 − e 34 ) − tu 1 |u| Y 2 b2 (e 13 + e 24 ) − su 2 b 2 (e 14 − e 23 ) ,[]de 6 = 2sY (2s 2 −u 2 b 2 )|u| (e 12 − e 34 )+ tu 2 |u| Y 2 b2 (e 13 + e 24 )− su 1 b 2 (e 14 − e 23 ) ,√where b 2 ∈ {0, 1}, s, t ∈ R ∗ , s 2 > |u| 2 > 0, Y = 2 s 2 −|u| 2.|u|tL. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 23 / 28

Instanton A: for each τ ∈ R, let A be the SU(3)-connection:(σ A ) 1 2 = −(σA ) 2 1 = −(σA ) 3 4 = (σA ) 4 3 = τ(e5 + e 6 );then any (F A ) i j∈ su(3) because is a linear combination ofe 12 − e 34 , e 13 + e 24 , e 14 − e 23 .L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 24 / 28

Instanton A: for each τ ∈ R, let A be the SU(3)-connection:(σ A ) 1 2 = −(σA ) 2 1 = −(σA ) 3 4 = (σA ) 4 3 = τ(e5 + e 6 );then any (F A ) i j∈ su(3) because is a linear combination ofe 12 − e 34 , e 13 + e 24 , e 14 − e 23 .For (I) and (II): tr F A ∧ F A = τ 2 C 1 (b, s, t, u) e 1234 , C 1 < 0L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 24 / 28

Instanton A: for each τ ∈ R, let A be the SU(3)-connection:(σ A ) 1 2 = −(σA ) 2 1 = −(σA ) 3 4 = (σA ) 4 3 = τ(e5 + e 6 );then any (F A ) i j∈ su(3) because is a linear combination ofe 12 − e 34 , e 13 + e 24 , e 14 − e 23 .For (I) and (II): tr F A ∧ F A = τ 2 C 1 (b, s, t, u) e 1234 , C 1 < 0tr R B ∧ R B = C 2 (b, s, t, u) e 1234 , C 2 < 0L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 24 / 28

Instanton A: for each τ ∈ R, let A be the SU(3)-connection:(σ A ) 1 2 = −(σA ) 2 1 = −(σA ) 3 4 = (σA ) 4 3 = τ(e5 + e 6 );then any (F A ) i j∈ su(3) because is a linear combination ofe 12 − e 34 , e 13 + e 24 , e 14 − e 23 .For (I) and (II): tr F A ∧ F A = τ 2 C 1 (b, s, t, u) e 1234 , C 1 < 0tr R B ∧ R B = C 2 (b, s, t, u) e 1234 , C 2 < 0α ′4 (tr RB ∧ R B − tr F A ∧ F A ) = α′4 (C 2 − τ 2 C 1 ) e 1234L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 24 / 28

Instanton A: for each τ ∈ R, let A be the SU(3)-connection:(σ A ) 1 2 = −(σA ) 2 1 = −(σA ) 3 4 = (σA ) 4 3 = τ(e5 + e 6 );then any (F A ) i j∈ su(3) because is a linear combination ofe 12 − e 34 , e 13 + e 24 , e 14 − e 23 .For (I) and (II): tr F A ∧ F A = τ 2 C 1 (b, s, t, u) e 1234 , C 1 < 0tr R B ∧ R B = C 2 (b, s, t, u) e 1234 , C 2 < 0α ′4 (tr RB ∧ R B − tr F A ∧ F A ) = α′4 (C 2 − τ 2 C 1 ) e 1234dT = C 3 (b, s, t, u) e 1234 , C 3 < 0L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 24 / 28

Instanton A: for each τ ∈ R, let A be the SU(3)-connection:(σ A ) 1 2 = −(σA ) 2 1 = −(σA ) 3 4 = (σA ) 4 3 = τ(e5 + e 6 );then any (F A ) i j∈ su(3) because is a linear combination ofe 12 − e 34 , e 13 + e 24 , e 14 − e 23 .For (I) and (II): tr F A ∧ F A = τ 2 C 1 (b, s, t, u) e 1234 , C 1 < 0tr R B ∧ R B = C 2 (b, s, t, u) e 1234 , C 2 < 0α ′4 (tr RB ∧ R B − tr F A ∧ F A ) = α′4 (C 2 − τ 2 C 1 ) e 1234dT = C 3 (b, s, t, u) e 1234 , C 3 < 0τ small enough ⇒ C 2 − τ 2 C 1 < 0 and α ′ = 4 C 3C 2 −τ 2 C 1> 0L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 24 / 28

Iwasawa manifold: on M 5 any abelian J has a compatiblebalanced metric g, and for any such g there is a non-flatinstanton solving the Strominger system with respect to ∇ B .L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 25 / 28

Iwasawa manifold: on M 5 any abelian J has a compatiblebalanced metric g, and for any such g there is a non-flatinstanton solving the Strominger system with respect to ∇ B .Generalized Heisenberg manifold: on M 3 there are two (abelian)complex structuresJ ± :dω 1 = dω 2 = 0, dω 3 = ω 1¯1 ± ω 2¯2Only J − has balanced metrics g, and for any such g thesolutions of the Strominger system also solve the heteroticequations of motion.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 25 / 28

Iwasawa manifold: on M 5 any abelian J has a compatiblebalanced metric g, and for any such g there is a non-flatinstanton solving the Strominger system with respect to ∇ B .Generalized Heisenberg manifold: on M 3 there are two (abelian)complex structuresJ ± :dω 1 = dω 2 = 0, dω 3 = ω 1¯1 ± ω 2¯2Only J − has balanced metrics g, and for any such g thesolutions of the Strominger system also solve the heteroticequations of motion.Theorem [I]Given a solution of the Strominger system with respect to ∇,it solves the equations of motion ⇐⇒ ∇ is of instanton type.[I] S. Ivanov: Heterotic supersymmetry, anomaly cancellation and equations of motion, Phys. Lett. B 685 (2010), 190-196.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 25 / 28

Solutions with respect to ∇ CTheoremLet M nilmanifold corresponding to h − 19. For any J there is a solution ofthe Strominger system with respect to ∇ C with non-flat instanton.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 26 / 28

Solutions with respect to ∇ CTheoremLet M nilmanifold corresponding to h − 19. For any J there is a solution ofthe Strominger system with respect to ∇ C with non-flat instanton.Proof: Any J ∼ = J ± and for J ± consider with respect to an adaptedbasis {e 1 , . . . , e 6 }, the family of balanced structures F :⎧⎨⎩de 1 = 0, de 2 = 0, de 3 = 2sre 15 , de 4 = 2sre 25 ,de 5 = 0, de 6 = ± 2 rs (e13 + e 24 ), r, s ∈ R ∗ .L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 26 / 28

Solutions with respect to ∇ CTheoremLet M nilmanifold corresponding to h − 19. For any J there is a solution ofthe Strominger system with respect to ∇ C with non-flat instanton.Proof: Any J ∼ = J ± and for J ± consider with respect to an adaptedbasis {e 1 , . . . , e 6 }, the family of balanced structures F :⎧⎨⎩de 1 = 0, de 2 = 0, de 3 = 2sre 15 , de 4 = 2sre 25 ,de 5 = 0, de 6 = ± 2 rs (e13 + e 24 ), r, s ∈ R ∗ .Instanton A: for each τ ∈ R, let A be the SU(3)-connection(σ A ) 2 3 = (σA ) 2 5 = (σA ) 4 5 = 1 2 (σA ) 5 6 = −τ e6 , σ j i= −σ i j ,(σ A ) i j= τ e 6 , for 1 ≤ i < j ≤ 6, (i, j) ≠ (2, 3), (2, 5), (4, 5), (5, 6).Then, any (F A ) i jis a multiple of e 13 + e 24 ∈ su(3).L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 26 / 28

tr F A ∧ F A = −144 τ 2r 2 s 2 e 1234L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 27 / 28

tr F A ∧ F A = −144 τ 2r 2 s 2 e 1234tr R C ∧ R C = − 16r 4 e 1234 − 16r 4 e 1256L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 27 / 28

tr F A ∧ F A = −144 τ 2r 2 s 2 e 1234tr R C ∧ R C = − 16r 4 e 1234 − 16r 4 e 1256α ′4 (tr RC ∧ R C − tr F A ∧ F A ) = − 4(s2 −9τ 2 r 2 )α ′e 1234 − 4α′ e 1256r 4 s 2 r 4L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 27 / 28

tr F A ∧ F A = −144 τ 2r 2 s 2 e 1234tr R C ∧ R C = − 16r 4 e 1234 − 16r 4 e 1256α ′4 (tr RC ∧ R C − tr F A ∧ F A ) = − 4(s2 −9τ 2 r 2 )α ′e 1234 − 4α′ e 1256r 4 s 2 r 4dT = − 8r 2 s 2 e 1234 − 8s2r 2 e 1256L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 27 / 28

tr F A ∧ F A = −144 τ 2r 2 s 2 e 1234tr R C ∧ R C = − 16r 4 e 1234 − 16r 4 e 1256α ′4 (tr RC ∧ R C − tr F A ∧ F A ) = − 4(s2 −9τ 2 r 2 )α ′e 1234 − 4α′ e 1256r 4 s 2 r 4Therefore, α ′ = 2r 2 s 2 > 0 and τ =dT = − 8r 2 s 2 e 1234 − 8s2r 2 e 1256√s 4 −19r 2 s 2L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 27 / 28

tr F A ∧ F A = −144 τ 2r 2 s 2 e 1234tr R C ∧ R C = − 16r 4 e 1234 − 16r 4 e 1256α ′4 (tr RC ∧ R C − tr F A ∧ F A ) = − 4(s2 −9τ 2 r 2 )α ′e 1234 − 4α′ e 1256r 4 s 2 r 4Therefore, α ′ = 2r 2 s 2 > 0 and τ =dT = − 8r 2 s 2 e 1234 − 8s2r 2 e 1256√s 4 −19r 2 s 2 (⇒ s 2 ≥ 1).⊲ instanton A is non-flat if and only if s 2 > 1.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 27 / 28

CorollaryLet M nilmanifold corresponding to h − 19. For any J there is (F, Ψ, A),with A non-flat, solving the anomaly cancellation conditions for both∇ B and ∇ C .L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 28 / 28

CorollaryLet M nilmanifold corresponding to h − 19. For any J there is (F, Ψ, A),with A non-flat, solving the anomaly cancellation conditions for both∇ B and ∇ C .Proof: tr R B ∧ R B = 16(s4 −4)r 4 s 4e 1234 −16 s4r 4 e 1256L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 28 / 28

CorollaryLet M nilmanifold corresponding to h − 19. For any J there is (F, Ψ, A),with A non-flat, solving the anomaly cancellation conditions for both∇ B and ∇ C .Proof: tr R B ∧ R B = 16(s4 −4)r 4 s 4e 1234 −16 s4r 4 e 1256α ′4 (tr RB ∧ R B − tr F A ∧ F A ) = (4 s4 −16+36 r 2 s 2 τ 2 )α ′e 1234 − 4 s4 α ′e 1256r 4 s 4 r 4L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 28 / 28

CorollaryLet M nilmanifold corresponding to h − 19. For any J there is (F, Ψ, A),with A non-flat, solving the anomaly cancellation conditions for both∇ B and ∇ C .Proof: tr R B ∧ R B = 16(s4 −4)r 4 s 4e 1234 −16 s4r 4 e 1256α ′4 (tr RB ∧ R B − tr F A ∧ F A ) = (4 s4 −16+36 r 2 s 2 τ 2 )α ′e 1234 − 4 s4 α ′e 1256r 4 s 4 r 4dT = − 8r 2 s 2 e 1234 − 8s2r 2 e 1256L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 28 / 28

CorollaryLet M nilmanifold corresponding to h − 19. For any J there is (F, Ψ, A),with A non-flat, solving the anomaly cancellation conditions for both∇ B and ∇ C .Proof: tr R B ∧ R B = 16(s4 −4)r 4 s 4e 1234 −16 s4r 4 e 1256α ′4 (tr RB ∧ R B − tr F A ∧ F A ) = (4 s4 −16+36 r 2 s 2 τ 2 )α ′e 1234 − 4 s4 α ′e 1256r 4 s 4 r 4Therefore, α ′ = 2r 2s 2 > 0 and ˜τ =dT = − 8 e 1234 − 8s2r 2 s 2√2(2−s 4 )9r 2 s 2r 2 e 1256L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 28 / 28

CorollaryLet M nilmanifold corresponding to h − 19. For any J there is (F, Ψ, A),with A non-flat, solving the anomaly cancellation conditions for both∇ B and ∇ C .Proof: tr R B ∧ R B = 16(s4 −4)r 4 s 4e 1234 −16 s4r 4 e 1256α ′4 (tr RB ∧ R B − tr F A ∧ F A ) = (4 s4 −16+36 r 2 s 2 τ 2 )α ′e 1234 − 4 s4 α ′e 1256r 4 s 4 r 4Therefore, α ′ = 2r 2s 2 > 0 and ˜τ =dT = − 8 e 1234 − 8s2r 2 s 2r 2 e 1256√2(2−s 4 )9r 2 s 2 (⇒ s 2 ≤ √ 2).L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 28 / 28

CorollaryLet M nilmanifold corresponding to h − 19. For any J there is (F, Ψ, A),with A non-flat, solving the anomaly cancellation conditions for both∇ B and ∇ C .Proof: tr R B ∧ R B = 16(s4 −4)r 4 s 4e 1234 −16 s4r 4 e 1256α ′4 (tr RB ∧ R B − tr F A ∧ F A ) = (4 s4 −16+36 r 2 s 2 τ 2 )α ′e 1234 − 4 s4 α ′e 1256r 4 s 4 r 4Therefore, α ′ = 2r 2s 2 > 0 and ˜τ =dT = − 8 e 1234 − 8s2r 2 s 2r 2 e 1256√2(2−s 4 )9r 2 s 2 (⇒ s 2 ≤ √ 2).⊲ For s 2 ∈ (1, √ 2), ∃ non-flat A τ and A˜τ solving the systems.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 28 / 28

CorollaryLet M nilmanifold corresponding to h − 19. For any J there is (F, Ψ, A),with A non-flat, solving the anomaly cancellation conditions for both∇ B and ∇ C .Proof: tr R B ∧ R B = 16(s4 −4)r 4 s 4e 1234 −16 s4r 4 e 1256α ′4 (tr RB ∧ R B − tr F A ∧ F A ) = (4 s4 −16+36 r 2 s 2 τ 2 )α ′e 1234 − 4 s4 α ′e 1256r 4 s 4 r 4Therefore, α ′ = 2r 2s 2 > 0 and ˜τ =dT = − 8 e 1234 − 8s2r 2 s 2r 2 e 1256√2(2−s 4 )9r 2 s 2 (⇒ s 2 ≤ √ 2).⊲ For s 2 ∈ (1, √ 2), ∃ non-flat A τ and A˜τ solving the systems.√⊲ If s 2 5=3then τ = ˜τ and the instanton solves both systems.L. Ugarte (Univ. Zaragoza) <strong>Balanced</strong> <strong>Hermitian</strong> nilmanifolds Golden Sands, Bulgaria 28 / 28