Generalized complex structures on complex 2-tori
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1 Bull. Math. Soc. Sci. Math. Roumanie Tome 5(100) No. 3, 009, Generalized complex structures on complex -tori by Vasile Brînzănescu, Neculae Dinuţă and Roxana Dinuţă To Professor S. Ianuş on the occasion of his 70th Birthday Abstract We compute the deformations, in the sense of generalized complex structures, of the standard complex structure on a complex -torus. We get a smooth complete family depending on six complex parameters and, in particular, we obtain the well-known smooth complete family of complex deformations depending on four parameters. Key Words: Generalized complex manifolds, deformations of complex structures, -tori. 000 Mathematics Subject Classification: Primary 3G05, Secondary 3G07, 53C56, 58A30, 58H15. 1 Introduction Nigel Hitchin introduced in the paper [5] new geometrical structures which unify many classical structures. He defines a generalized complex structure to be a complex structure, not on the tangent bundle T of a manifold, but on T T, unifying in this way the complex geometry and the symplectic geometry in some sense. This new geometrical structure is also, in some sense, the complex analogue of the Dirac structure introduced by Courant and Weinstein [], [3], in order to unify Poisson geometry with symplectic geometry. The study of generalized complex structures was continued by Gualtieri (see [4]). In this paper we start with the (classical) complex structure on a -torus and we compute (as in [1]) by using properties of the Lie algebroids, the family of deformations of this complex structure in the sense of generalized complex structures (see [4]). By solving the generalized Maurer-Cartan equation we get the main result of the paper, which shows that obtained family of deformations is a smooth locally complete family depending on six complex parameters. In particular, we get the family (depending on four complex parameters) of deformations of (classical) complex structures on a -torus (see, for example [6]), as well as examples of generalized complex structures of complex type, which are not (classical) complex structures on a -torus. VB was partially supported by CNCSIS grant 1189/
2 64 Vasile Brînzănescu, Neculae Dinuţă and Roxana Dinuţă Generalized complex structures on manifolds A generalized complex structure on a manifold M (see [5], [4]) is defined to be a complex structure J (J = 1) not on the tangent bundle T M, but on the sum T M T M of the tangent and cotangent bundles, which is required to be orthogonal with respect to the natural inner product on sections X + σ, Y + τ C (T M T M) defined by X + σ, Y + τ = 1 (σ(y ) + τ(x)). This is only possible if dim R M = n, which we suppose. In addition, the (+i)-eigenbundle L (T M T M) C of J is required to be involutive with respect to the Courant bracket, a skew bracket operation on smooth sections of T M T M defined by [X + σ, Y + τ] = [X, Y ] + L Xτ L Y σ 1 d(ixτ iy σ), where L X and i X denote the Lie derivative and interior product operations on forms. Since J is orthogonal with respect to,, the (+i)-eigenbundle L is a maximal isotropic subbundle of (T M T M) C of real index zero (i.e. L L = {0}). In fact, a generalized complex structure on M is completely determined by a maximal isotropic subbundle of (T M T M) C of real index zero, which is Courant involutive (see [5], [4]). For such a subbundle we have the decomposition (T M T M) C = L L, and we may use the inner product, to identify L L. Let π T : (T M T M) C T M C be the projection and let E = π T(L). Then, the type k {0, 1,..., n} of the generalized complex structure at x M is defined as the codimension of E x T x C. To deform J we will vary L in the Grassmannian of maximal isotropic. Any maximal isotropic having zero intersection with L (this is an open set containing L) can be uniquely described as the graph of a homomorphism ε : L L satisfying ( ) ε(x), Y + X, ε(y ) = 0, X, Y C (L) or equivalently, ε C ( L ). Therefore, the new isotropic is given by L ε = (1 + ε)l. As the deformed J is to remain real, we must have L ε = (1 + ε) L. The subbundle L ε has zero intersection with L ε if and only if the endomorphism on L L, described by A ε = 1 ε ε 1 is invertible; this is the case for ε in an open set around zero (see [4]). So, providing ε is small enough, J ε = A εja 1 ε is a new generalized almost complex structure. By [8], J ε is integrable if and only if ε C ( L ) satisfies the generalized Maurer-Cartan equation ( ) d Lε + 1 [ε, ε] = 0.!
3 Generalized complex structures 65 3 Deformations of generalized complex structures on -tori Let N = C /Λ be a complex -torus, where C denotes the space of two complex variables (z, w) and Λ C is an integral lattice of rank 4. We shall identify C with R 4, the space of four real variables (x, y, u, v) by z = x + iy, w = u + iv. From the point of view of differential structure, a complex -torus is a parallelizable manifold, i.e. the tangent bundle T N is globally generated by invariant vector fields {X, Y, U, V } with all Poisson brackets zero. The complex structure endomorphism J is acting on T N by JX = Y, JY = X, JU = V, JV = U. Let T = 1 (X iy ), W = 1 (U iv ). Then, the tangent bundle T N is globally generated by {T, W, T, W } and the cotangent bundle T N is globally generated by the dual basis of 1-forms {ω, ρ, ω, ρ}. We have JT = it, JW = iw, J T = i T, J W = i W. It follows that the subbundle T 0,1 and T 1,0 of the tangent bundle T N are globally generated by { T, W }, respectively {T, W } and the dual bundle T 1,0 is globally generated by {ω, ρ}. The standard complex structure on a -torus N can be seen as a generalized complex structure given by the subbundle L (T N T N) C of the form L = { T, W, ω, ρ} e = (T 0,1 T 1,0) C, which is maximal isotropic and Courant involutive (see [4]). Using the inner product we can identify L with L by the isomorphism: θ : L e L, θ(t) = 1 ω, θ(w) = 1 ρ, θ( ω) = 1 T, θ( ρ) = 1 W. In the following we shall study the deformations of this generalized complex structure on a -torus N. In order to obtain the deformations of the generalized complex structure we must consider the linear maps ε : L L, which verify the condition (*) or, equivalently, we can consider the linear maps ε = θ ε : L L. The map ε is given by the following matrix: t 3 t 11 t 1 eε = 1 t 3 0 t 1 t B C, tij t 11 t 1 0 t 14 A t 1 t t 14 0 and eε C ( L ). Now, as in [1], we shall solve the generalized Maurer-Cartan equation (**), where [ ε, ε] is the Schouten bracket and the derivative d L : C ( L ) C ( 3 L ), for a Lie algebroid is given in this case by the formula d L ε(x 0, X 1, X ) = a(x 0) ε(x 1, X ) a(x 1) ε(x 0, X ) + a(x ) ε(x 0, X 1) ε([x 0, X 1], X ) + ε([x 0, X ], X 1) ε([x 1, X ], X 0);
4 66 Vasile Brînzănescu, Neculae Dinuţă and Roxana Dinuţă the anchor map a : C (L) C (T N) is the projection on the tangent bundle T N and X 0, X 1, X C (L). We shall use the notation: where u i, α i, β i C (N), i = 1,, 3, 4. We have X 0 = u 1 T + u W + u3ω + u 4ρ, X 1 = α 1 T + α W + α3ω + α 4ρ, X = β 1 T + β W + β3ω + β 4ρ, ε(x 1, X ) = 1 (β1(t3α t11α3 t1α4) + β( t3α1 t1α3 tα4)+ +β 3(t 11α 1 + t 1α + t 14α 4) + β 4(t 1α 1 + t α t 14α 3)). Since a(x 0) = u 1 T + u W, by direct computation as in [1], we get: Lemma 3.1 a(x 0) ε(x 1, X ) = 1 (t11u1( T(α 1)β 3 + T(β 3)α 1 T(α 3)β 1 T(β )α 3)+ +t 1u 1( T(α )β 3 + T(β 3)α T(α 3)β T(β )α 3)+ +t 1u 1( T(α 1)β 4 + T(β 4)α 1 T(α 4)β 1 T(β 1)α 4)+ +t u 1( T(α )β 4 + T(β 4)α T(α 4)β T(β )α 4)+ +t 14u 1( T(α 4)β 3 + T(β 3)α 4 T(α 3)β 4 T(β 4)α 3)+ +t 3u 1( T(α )β 1 + T(β 1)α T(α 1)β T(β )α 1)+ +t 11u ( W(α 1)β 3 + W(β 3)α 1 W(α 3)β 1 W(β 1)α 3)+ +t 1u ( W(α )β 3 + W(β 3)α W(α 3)β W(β )α 3)+ +t 1u ( W(α 1)β 4 + W(β 4)α 1 W(α 4)β 1 W(β 1)α 4)+ +t u ( W(α )β 4 + W(β 4)α W(α 4)β W(β )α 4)+ +t 14u ( W(α 4)β 3 + W(β 3)α 4 W(α 3)β 4 W(β 4)α 3)+ +t 3u ( W(α )β 1 + W(β 1)α W(α 1)β W(β )α 1)). Analogous formulae can be written for the terms a(x 1) ε(x 0, X ) and a(x ) ε(x 0, X 1). Lemma 3. For the Courant bracket we have: [X 0, X 1] = (X(α 1) Y (u 1)) T + (X(α ) Y (u )) W+ +(X(α 3) Y (u 3))ω + (X(α 4) Y (u 4))ρ. where X 0 = X + σ, X 1 = Y + τ, X = u 1 T + u W, Y = α1 T + α W C (T N)
5 Generalized complex structures 67 and σ = u 3ω + u 4ρ, τ = α 3ω + α 4ρ C (T N). Proof: By direct computation we have L Xτ = X(α 3)ω + X(α 4)ρ, L Y σ = Y (α 3)ω + Y (u 4)ρ, and i Xτ = 0, i Y σ = 0. By using the definition of the Courant bracket the result follows. We have analogous formulae for [X 0, X ] and [X 1, X ]. Now, a tedious but direct computation gives: Lemma 3.3 ε([x 0, X ], X 1) = 1 (t11( α1u1 T(β 3) α 1u W(β3) + α 1β 1 T(u3)+ +α 1β W(u3) + α 3u 1 T(β1)+ +α 3u W(β1) α 3β 1 T(u1) α 3β W(u1)) + t 1( α u 1 T(β3) α u W(β3) + α β 1 T(u3) + α β W(u3)+ +α 3u 1 T(β) + α 3u W(β) α 3β 1 T(u) α 3β W(u)) + t 1( α 1u 1 T(β4) α 1u W(β4) + α 1β 1 T(u4)+ +α 1β W(u4) + α 4u 1 T(β1) + α 4u W(β1) α 4β 1 T(u1) α 4β W(u1)) + t ( α u 1 T(β4) α u W(β4) + α β 1 T(u4)+ +α β W(u4) + α 4u 1 T(β) + α 4u W(β) α 4β 1 T(u) α 4β W(u)) + t 14(α 3u 1 T(β4) + α 3u W(β4) α 3β 1 T(u4) α 3β W(u4) α 4u 1 T(β3) α 4u W(β3) + α 4β 1 T(u3)+ +α 4β W(u3) + t 3(α 1u 1 T(β)+ +α 1u W(β) α 1β 1 T(u) α 1β W(u) α u 1 T(β1) α u W(β1) + α β 1 T(u1) + α β W(u1))). There are analogous formulae for the terms ε([x 0, X 1], X ) and ε([x 1, X ], X 0). By using all the above formulae, we get Theorem 3.4 For the differential d L we have: (d L ε)(x 0, X 1, X ) = 0, X 0, X 1, X C (L).
6 68 Vasile Brînzănescu, Neculae Dinuţă and Roxana Dinuţă As in [1], by similar computation we get: Theorem 3.5 For the Schouten bracket we have: [ ε, ε] = 0. From the two theorems, we get the following: Corollary 3.6 The solutions of the generalized Maurer-Cartan equation are given by ε = 4(t 3 ω ρ t 11 ω T t 1 ρ T t 1 ω W t ρ W + t 14T W) Now, we need the following result: Lemma 3.7 The image of the differential d L : C (L ) C ( L ) is zero. The proof is similar to the proof of Proposition 4.8 in [1]. By all the above results we get the main result: Theorem 3.8 The deformations in the sense of generalized complex structures of the standard complex structure on a -torus N are given by ε = t 3 ω ρ t 11 ω T t 1 ρ T t 1 ω W t ρ W + t 14T W, where (t 3, t 11, t 1, t 1, t, t 14) C 6. In particular, taking the parameters t 3 = 0 and t 14 = 0 we get the classical deformations of complex structures (see [6]). We have the following: Corollary 3.9 The family of deformations of generalized complex structures on a complex -torus N, given by eε = t 3 ω ρ t 11 ω T t 1 ρ T t 1 ω W t ρ W + t 14T W, with (t 3, t 11, t 1, t 1, t, t 14) U C 6, where U is an open neighborhood of 0 C 6, is a smooth locally complete family. Proof: By Theorem 3.5 we have [eε, eε] = 0. From the definition of the obstruction map φ by Theorem 5.4, [4] (see also [7]) we get φ = 0. Then, applying again Theorem 5.4 in [4] it follows that the above family of deformations is a smooth locally complete family in a open neighborhood U of 0 C 6.
7 Generalized complex structures 69 We have the following result: Proposition 3.10 Let N be a complex -torus. The type of the generalized complex structure given by the subbundle L ε = (1+ε)L (T N T N) C is k ε = 0 (symplectic type) or k ε = (complex type). Proof: The type k ε of a generalized complex structure is the codimension in any fibre of the projection of L ε on T N C by the canonical map Since we get: π T : (T N T N) C T N C. L = { T, W, ω, ρ}, (1 + ε)( T) = T + t 11T + t 1W t 3 ρ (1 + ε)( W) = W + t 1T + t W t 3 ω (1 + ε)(ω) = t 14W + ω t 11 ω t 1 ρ (1 + ε)(ρ) = t 14T + ρ t 1 ω t ρ. It follows that the projection of L ε on T N C is globally generated by { T + t 11T + t 1W, W + t 1T + t W, t 14W, t 14T }. If t 14 0, then the type k ε = 0 (symplectic type) and, if t 14 = 0,then k ε = (complex type). Remark If, in the case t 14 = 0, we have also t 3 = 0, we get classical deformations of complex structures. If, in the case t 14 = 0, we have t 3 0, we get examples of generalized complex structures of complex type, which are not classical complex structures. References [1] V. Brînzănescu, O.A. Turcu, Generalized complex structures on Kodaira surfaces, arxiv: [] T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319(1990), [3] T. Courant, A. Weinstein, Beyond Poisson structures, In: Action hamiltoniennes de groupes, Troisième théorème de Lie (Lyon,1986), volume 7 of Travaux en Cours, 39-49, Hermann,Paris [4] M.Gualtieri, Generalized complex geometry, D. Phil. Thesis, St. John s College, University of Oxford, 003. [5] N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (3) (003), [6] K. Kodaira, Complex Manifolds and Deformation of Complex Structures, Springer-Verlag, [7] M. Kuranishi, New proof for the existence of locally complete families of complex structures, In: Proc. Conf. Complex Analysis (Mineapolis, 1964), , Springer [8] Z.-J. Liu, A. Weinstein, P. Xu, Manin triples for Lie bialgebroids, J. Diff. Geom. 45,(1997),
8 70 Vasile Brînzănescu, Neculae Dinuţă and Roxana Dinuţă Received: Vasile Brînzănescu Simion Stoilow Institute of Mathematics of the Romanian Academy P.O. Box 1-764, Bucharest, Romania and University of Piteşti str. Târgu din Vale, nr Piteşti, Romania Neculae Dinuţă and Roxana Dinuţă University of Piteşti str. Târgu din Vale, nr Piteşti, Romania
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