Dolbeault cohomology and. stability of abelian complex structures. on nilmanifolds

Transcription

1 Dolbeault cohomology and stability of abelian complex structures on nilmanifolds in collaboration with Anna Fino and Yat Sun Poon HU Berlin 2006

2 M = G/Γ nilmanifold, G: real simply connected nilpotent Lie group J: invariant complex structure on M Aims : compute Dolbeault cohomology of M by invariant forms study deformations of certain types of complex structures

3 G is (k-step) nilpotent k: 0 g k 1 g k = {0} where g 0 = g, g i = [g i 1, g] (lower central series) G nilpotent with rational structure constants g has a rational structure, i.e., rational g Q : g = g Q R Malcev = a discrete subgroup Γ such that M = G/Γ is compact.

4 de Rham Cohomology Nomizu s Theorem: H k DR (M) = H k (g) where H k (g) = Ker(d k g ) d( k 1 g ) (Chevalley-Eilenberg)... k 1 g d k g d k+1 g... dα(x 1,..., x k+1 ) = i<j ( 1) i+j α([x i, x j ], x 1,... ˆx i,..., ˆx j,..., x k+1 )

5 Sketch of the proof: Set h := g k 1 last non-zero term in the lower central series. h is central ( abelian) Let H := connected Lie subgroup of G with Lie algebra h. G simply-connected & H connected = H simply-connected. Hence H = R n. HΓ/H and H Γ are discrete cocompact subgroups. = M := G/HΓ compact nilmanifold & dim M < dim M T n := H/H Γ is a torus.

6 T n = H/H Γ fibration : M = G/Γ π M = G/HΓ E p,q := Leray-Serre spectral sequence associated with the fibration: E p,q 2 = Hp dr (M, Hq dr (Tn )) = H p dr (M) q R n, E p,q Hp+q dr (M).

7 Main idea: construct a second spectral sequence Ẽ p,q = Leray-Serre spectral sequence for the complex of G-invariant forms (i.e., the Chevalley-Eilenberg complex). g subcomplex of M = Ẽ p,q E p,q & Ẽ p,q 2 = Hp (g/h) q R n, Ẽ p,q H p+q (g). dim M < dim M, induction on dim = H p dr (M) = H p (g/h) for any p. = E 2 = Ẽ 2 & E = Ẽ. i.e., H k dr (M) = H k (g) for any k.

8 Complex structures An invariant almost complex structure J on M is associated to a almost complex structure J on g J { g C = g 1,0 g 0,1 (g C ) = 1,0 (g C ) 0,1 (g C ) = g (1,0) g (0,1) J is complex if N J = 0 or dg (1,0) g (1,1) g (2,0), g (p,q) = space of (p, q)-forms on g. In general: g 1,0 is a complex subalgebra of g C

9 Complex parallelizable nilmanifolds G/Γ is complex parallelizable if G is a complex Lie group or equivalently dg (1,0) g (2,0) Abelian complex structures Definition [Barberis, Dotti, Miatello]: J is abelian if g 1,0 is abelian [JX, JY ] = [X, Y ], X, Y g or equivalently dg (1,0) g (1,1)

10 Dolbeault cohomology p,q (g C ) = g (p,q) G invariant forms of type (p, q) on M J is complex 2 = 0 = (, (g C ), ) bigraded differential algebra and d = + = H (g C ) cohom. of the Dolbeault complex of G-inv. forms on G H (M) = cohom. of Dolbeault complex ( p,q, ) of Γ-inv. forms on G

11 Problem: For which complex structure J the isomorphism holds? H, (M) = H, (gc ) Known: [Sakane]: for complex parallelizable solvmanifolds (= nilmanifolds) [Cordero, Fernandez, Gray, Ugarte]: for nilpotent ( abelian) cx structures

12 In general: the inclusion p,q (g C ) p,q induces an injective morphism H, (gc ) j H, (M) [Endowed M = G/Γ with an invariant Hermitian metric,, preserve {G-invariant forms} and {G-invariant forms} if j[ω] = 0, [ω] H p,q (gc ) = j(ω) = ϕ, ϕ {G-invariant forms} = ϕ {G-invariant forms} & 0 = ϕ, ϕ = ϕ, ϕ = ϕ = 0 = [ω] = 0 in H p,q (gc )] H (M) = cohomology of Dolbeault complex of Γ-invariant forms orthogonal to the G-invariant ones H, (M) = H, (gc ) dim H (M) = 0

13 Theorem [, Fino] The isomorphism H, (M) = H, (gc ) holds for J rational [J(g Q ) g Q ] J small deformation of a rational complex structure J 0 J abelian

14 Sketch of the proof for J abelian: Use the upper central series {g l } with g 0 = {0}, g 1 = {X g [X, g] = 0},... g l = {X g [X, g] g l 1 },... g k = g. J abelian = Jg l g l (NOTE: this is not true for the lower central series!) g 1 is central ( abelian) = exact sequence of Lie algebras 0 g 1 g g/g 1 0. abelian G 1 := connected Lie subgroup of G with Lie algebra g 1 (G 1 = R n, since it is abelian and simply-connected), G 1 := simply-connected nilpotent Lie group with Lie algebra g/g 1.

15 Given Γ G g 1 rational subalg. of g = p 0 (Γ) uniform in G 1 g 1 T 1 = G 1 /G 1 Γ g holomorphic fibration: M = G/Γ π 1 g/g 1 M 1 = G 1 /p 0 (Γ) The main tool to get cohomological information about the total space of this bundle is Borel s spectral sequence

16 Theorem [Borel] p : P B be a holomorphic fibre bundle, F : compact connected fibre, P and B connected. If either (I) F is Kähler or (I ) H u,v (F ) = b B H u,v (p 1 (b)) (scalar cohomology bundle) is trivial. = spectral sequence (E r, d r ), (r 0) with the following properties: (i) E r is 4-graded by fibre degree, base degree and type. (ii) If p + q = u + v (iii) p,q E H (P ) p,q E u,v 2 = k Hk,u k (B) H p k,q u+k (F )

17 As for the proof of Nomizu s Theorem, we use a first Borel spectral sequence E for the complex of Γ-invariant forms a second one, denoted by Ẽ, for the G-invariant forms. p,q E u,v 2 p,qẽ u,v 2 = = k k = H k,u k p k,q u+k (M 1 ) (C n ), H k,u k ((g/g 1 ) C p k,q u+k ) (C n ) M 1 nilmanifold with an abelian complex structure dim M 1 < dim M induction on dim = H k,u k = E 2 = Ẽ 2 & E = Ẽ. = H p,q (M) = H p,q (gc ). (M 1 ) = H k,u k ((g/g 1 ) C ).

18 Moreover H 0,q (M) = Hq (M, O M ) = q g (0,1) = g (0,q), where O M is the structure sheaf of M. Indeed g (0,q) = 0, for any q, since dg (1,0) g (1,1), the Dolbeault complex 0, (g C ) = g (0, ) is the zero complex.

19 Deformations of abelian complex structures on nilmanifolds 1. Show: the Dolbeault cohomology on (M, J) (J abelian) with coefficients in the structure sheaf O M and holomorphic tangent sheaf Θ M can be computed using invariant forms and invariant vectors. = invariant harmonic representatives of H k (M, Θ M ) Harmonic theory is reduced to finite dim. lin. alg. 2. Apply 1 to prove Theorem [, Fino, Poon] The Kuranishi deformations arising from an abelian J are all invariant complex structures. Indeed, main ingredients for Kuranishi deform. are harmonic repr. of H k (M, Θ M ). This generalizes [Mclaughlin, Pedersen, Poon, Salamon] where these results were proved for abelian complex structures on 2-step nilmanifolds

20 Cohomology of holomorphic tangent sheaf for abelian cx structures Theorem [, Fino, Poon] There is a natural isomorphism H j (M, Θ M ) = H j (g1,0 ) where H j (g1,0 ) is the cohomology of the following complex: V g 1,0, Ū g 0,1, set ŪV := [Ū, V ] 1,0 linear map : g 1,0 g (0,1) g 1,0 Extend to : g (0,k) g 1,0 g (0,k+1) g 1,0 by setting (ω V ) = ω V + ( 1) k ω V where ω g (0,k) and V g 1,0. complex 0 g 1,0 g (0,1) g 1,0 g (0,k 1) g 1,0 g (0,k) g 1,0 Define H k (g1,0 ) = ker ( : g (0,k) g 1,0 g (0,k+1) g 1,0) ( g (0,k 1) g 1,0)

21 Sketch of the proof: Use the upper central series {g l } with J abelian = Jg l g l. g 0 = {0}, g 1 = {X g [X, g] = 0},... g l = {X g [X, g] g l 1 },... g k 1 = {X g [X, g] g k 2 }, g k = g. t l := g l /g l 1, l 1 is abelian (g 1 = g 1 /g 0 abelian) g / g k 1 is abelian exact sequences of Lie algebras 0 t l g/g l 1 g/g l 0 abelian with for l = k 1 the base g/g k 1 abelian

22 Everything goes well (inductively) with uniform subgroups g l /g l 1 T l g/g l 1 holomorphic fibration: M l 1 π l g/g l M l For the last fibration π k 1 : M k 2 M k 1 fibre & base = torus. The idea is to start with the last fibration and go on inductively. At any step the basis of the fibre bundle is not a torus but it has good properties (it is the total space of previous step s bundle).

23 O Ml 1 (Θ Ml 1 ): structure sheaf (tangent sheaf) of M l 1 (total space). For j 1, the direct image sheaves with respect to π l are R j π l O Ml 1 = t (0,j) l O Ml, fibre base R j π l π l Θ Ml = t (0,j) l fibre Θ Ml base H j (M l, O Ml ) = (g/g l ) (0,j) (Already proved, as a special case of H p,q (M l) = H p,q ((g/g l) C )) [One can also prove using the Leray spectral sequence] H j (M l, Θ Ml ) = H j ((g/g l) (1,0) )

24 For the proof, we use induction, beginning with l = k 1 and finishing at l = 0. Note that M k 1 is a torus ( base & fibre tori) For the general inductive step, consider π l+1 : M l M l+1 Leray spectral seq. of π l+1 Θ M l+1 = E p,q 2 = H p (M l+1, R q π l+1 π l+1 Θ Ml+1 ) = = q (t l+1 ) (0,1) H p (M l+1, Θ Ml+1 ) = q (t l+1 ) (0,1) H p ((g/g l+1) 1,0 ), by induction = The Leray spectral sequence degenerates at E 2 [d 2 is generated by ] = H j (M l, πl+1 Θ M l+1 ) = E p,q 2 = q (t l+1 ) (0,1) H p ((g/g l+1) 1,0 ) p+q=j p+q=j fibre base

25 To get H j (M l, Θ Ml ) = H j ((g/g l) (1,0) ) the main tool is the exact sequence 0 (t l+1 ) 1,0 O Ml Θ Ml π l+1 Θ M l+1 0, the induced long exact sequence... (t l+1 ) 1,0 H j (M l, O Ml ) H j (M l, Θ Ml ) H j (M l, π l+1 Θ M l+1 ) δ j (tl+1 ) 1,0 H j+1 (M l, O Ml )... and diagram chasing H j (M l, Θ Ml ) = H j ((g/g l) (1,0) ) for l = 0 yields the Theorem.

26 Kuranishi deformations Harmonic theory is reduced to finite dim. lin. alg. : There are invariant harmonic representatives for H k (M, Θ M ) [Namely, these are im k 1 (:= {im k 1 } in ker k )] Let µ g (0,k) g 1,0 = µ w. r. L 2 -norm on the compact manifold M = µ w. r. Hermitian inner prod. on the finite-dim. g (0,k) g 1,0 Consider the Schouten-Nijenhuis bracket {, } : H 1 (X, Θ X ) H 1 (X, Θ X ) H 2 (X, Θ X ) : Let ω V and ω V be vector-valued (0,1)-forms representing elements in H 1 (X, Θ X ). Then {ω V, ω V } = ω L V ω V + ω L V ω V + ω ω [V, V ]

27 Kuranishi s recursive formula: {β 1,..., β N }: ON basis of the harmonic representatives of H 1 (M, Θ M ). t = (t 1,..., t N ) C N, let µ(t) = t 1 β t N β N and set φ 1 = µ. φ r is defined inductively for r 2: = adjoint operator to w. r. to an Hermitian metric on M = + = Laplacian. φ r (t) := 1 2 r 1 s=1 G = Green s operator G{φ s (t), φ rs (t)} = 1 2 r 1 s=1 G {φ s (t), φ r s (t)},

28 Consider the formal sum Φ(t) = r 1 φ r {γ 1,..., γ P }: ON basis for {harmonic (0, 2)-forms with values in Θ M }. Define f k (t) := {Φ(t), Φ(t)}, γ k (L 2 -inner product ). Kuranishi theory: ɛ > 0 such that Kur := {t C N : t < ɛ, f 1 (t) = 0,..., f P (t) = 0} forms a locally complete family of deformations of M.

29 t Kur, Φ = Φ(t) defines a family complex structures J Φ whose (1,0)-forms are ω Φ(ω), ω g (1,0) and whose (0,1)-vectors are X + Φ( X), X g 0,1. integrability condition for Φ: ω g (1,0) and X, Ȳ g 0,1, (d(ω Φ(ω))) ( X + Φ( X), Ȳ + Φ(Ȳ )) = 0. Relation with the Maurer-Cartan equation: (d(ω Φ(ω))) ( X + Φ( X), Ȳ + Φ(Ȳ )) = ω ( (Φ + 1 ) 2 {Φ, Φ})( X, Ȳ ). Every term in the power series Φ(t) = r 1 φ r lies in g (0,1) g 1,0 = Theorem [, Fino, Poon] J be an abelian invariant complex structure on M = G/Γ = the deformations arising from J parameterized by Kur are all invariant complex structures

30 Deformations remaining abelian Aim: find under which conditions J Φ remains abelian Recall: J is abelian dg (1,0) g (1,1) ω g (1,0) & X, Y g 1,0, d(ω Φ(ω))(X + Φ(X), Y + Φ(Y )) = 0. If one extends the Schouten-Nijenhuis bracket {, } to the exterior algebra by anti-derivative and make use of the assumption that J is abelian, a short computation shows that d(ω Φ(ω))(X + Φ(X), Y + Φ(Y )) = { Φ, ω Φ(ω)}(X, Y ). = Theorem [, Fino, Poon] Φ defines an abelian deformation if and only if it is integrable and (A) { Φ, ω Φ(ω)} = 0 ω g (1,0).

31 Infinitesimally, suppose Φ = tµ + t 2 φ 2 + t 3 φ µ g (0,1) g (1,0) Φ = tµ + t 2 φ {Φ, Φ} quadratic } deg 1 term in integr. cond. µ = 0 deg 1 terms in equation (A) {µ, ω} = 0 ω g (0,1) = if φ j are constructed through the Kuranishi recursive formula, then φ j = 0 for all j 2. Conversely, if µ is -closed and satisfies the above condition, it is integrable to an abelian complex structure. = Theorem [, Fino, Poon] A parameter µ H 1 (M, Θ M ) defines an integrable infinitesimal abelian deformation if and only if and µ = 0 & {µ, ω} = 0 ω g (0,1).

32 Deformations & Dolbeault cohomology J t (t B): deformation of a complex structure J 0 consisting of invariant complex structures (at least locally) Theorem [, Fino, ] Suppose H, (M, J 0) = H, (gc ) for a complex struct. J 0. Then this isomorphism still holds for any small deformation J t (t B) of J 0 consisting of invariant complex structures.

33 Sketch of the proof: For J 0 we know that dim H, (M, J 0 ) = 0. By Kodaira-Spencer theory dim H, (M, J t ) is an upper semicontinuous function, there exists a sufficiently small neighbourhood U 0 of J 0 such that J t U 0 dim H, Thus H, (M, J t) = H, (gc ) in U 0 B. (M, J t ) dim H, (M, J 0 ) = 0. Application:. given an abelian complex structure J, for any small deformation J t (t U 0 Kur), we have H, (M, J t) = H, (gc ).

34 Stability results: If we deform a complex structure J of some type (abelian, nilpotent), the deformed structure J t can still be invariant but will not be of the same type in general. stable complex structure: it is of the same type when deformed. The deformation of abelian complex structures are not stable, beginning from real dimension six [Mclaughlin, Pedersen, Poon, Salamon]. In real dimension six, the deformation of abelian complex structures is stable within the class of nilpotent complex structures [Ugarte]. Examples show this phenomenon does not occur in higher dimension [, Fino, Poon]. neither nilpotent complex structures are stable. rational complex structures are not stable under deformations, [, Fino].