Geometry of universal embedding spaces for almost complex manifolds

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Geometry of universal embedding spaces for almost complex manifolds

Gabriella Clemente

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Gabriella Clemente. Geometry of universal embedding spaces for almost complex manifolds. 2019.

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Geometry of universal embedding spaces for almost complex manifolds

Gabriella Clemente

Abstract

We study the geometry of universal embedding spaces for compact almost complex manifolds of a given dimension. These spaces are complex algebraic analogues of twistor spaces that were intro- duced by J-P. Demailly and H. Gaussier. Their original goal was the study of a conjecture made by F. Bogomolov, asserting the “transverse embeddability” of arbitrary compact complex manifolds into foliated algebraic varieties. In this work, we introduce a more general category of universal embedding spaces, and elucidate the geometric structure of the integrability locus characterizing integrable almost complex structures. Our approach can potentially be used to inves- tigate the existence (or non-existence) of topological obstructions to integrability, and to tackle Yau’s Challenge.

Contents

Introduction 2

1 Universal embedding spaces associated to

even-dimensional complex manifolds 3

1.1 Universal embedding property . . . . 5 1.2 Functorial property with respect to ´etale morphisms . . . . 8 2 The geometry ofZ_n(Y)and related bundles 9 2.1 Coordinates onZ_n(Y) . . . . 9 2.2 Sub-bundles of the Grassmannian bundle . . . . 11 2.3 A functorial group action . . . . 14

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3 Affine bundle structure 16 3.1 The quotient bundleGr^o_n,k/I^o_n,k . . . . 20 3.2 Concluding remarks . . . . 23

Appendix 23

Introduction

J-P. Demailly and H. Gaussier proved that every compact almost complex manifold (X, J_X) of dimension n admits an embedding F :X ֒→ Z that is transverse to an algebraic distribution D ⊂T_Z (cf. Theorem 1.2 [5]), and that ifJ_Xis integrable, Im( ¯∂_J_XF) is contained in a subvarietyI of the Grass- mannian bundle Gr^C(D, n) → Z that is the isotropic locus of the torsion tensor θ:D × D →T_Z/D, θ(ζ, η) = [ζ, η] modD(cf. Theorem 1.6 [5]). J-P.

Demailly proposed using this result to study

Yau’s Challenge: Determine whether there exists a compact almost complex manifold of dimension at least 3 that cannot be given a complex structure [8].

The proposed strategy is most promising when the homotopy classes of almost complex structures are well understood. This is the case for the (oriented) sphere S⁶ since any almost complex structure on S⁶ is homotopic to the well-known octonion almost complex structure.

LetJ be a hypothetical integrable almost complex structure onX that is homotopic to J_X. A (smooth) homotopy J_t such that J₀ =J_X and J₁ =J produces an isotopy of transverse toDembeddingsF_t: (X, J_t)֒→Z,giving rise, in turn, to a homotopy Im( ¯∂_J_tF_t(x)) : [0,1]×X → Gr^C(D, n), where Im( ¯∂_J₀F₀(x))⊂Gr^C(D, n) and Im( ¯∂_J₁F₁(x))⊂ I. The original proposal was to study the topology ofI relative to Gr^C(D, n) independently in hopes of detecting an obstruction to the existence of the homotopy Im( ¯∂_J_tF_t(x)) [4].

It is unclear if such an obstruction exists at all. But if it did, then in the special case of a homotopically uniqueJ_X,the conclusion would be thatX cannot be a complex manifold.

A refinement of the initial proposal can be obtained by replacing Gr^C(D, n) with a Zariski open subset Grô(D, n)⊂Gr^C(D, n) and I with its intersec- tionIôwith Grô(D, n).This is explained in section 2.2. Then, it is possible to reformulate the strategy in terms of the topology and geometry of the quotient Grô(D, n)/Iô, and to interpret said reformulation as a lineariza-

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tion of the problem of deciding when an almost complex structure is integrable. This is mentioned in the concluding remarks, and will be the topic of future work.

The main goal of this article is to study the geometry of spaces involved in the aforementioned improvements of the initial strategy. Additionally, we provide a generalization of the embedding theory that was incepted by J-P. Demailly and H. Gaussier in [5].

Acknowledgment I thank Jean-Pierre Demailly, my PhD supervisor, for his guidance, patience and generosity. I thank the European Research Council for financial support in the form of a PhD grant from the project

“Algebraic and K¨ahler geometry” (ALKAGE, no. 670846). The PhD project emerged at the CIME School on Non-K¨ahler geometry [4].

1 Universal embedding spaces associated to even-dimensional complex manifolds

Acomplex directed manifoldis a pair (X,D) of complex manifoldX and dis- tributionD ⊂T_X,i.e.Dis a holomorphic sub-bundle of the tangent bundle ofX. Complex directed manifolds form a category whose morphisms are holomorphic mapsΨ:X→X^′ withΨ_∗(D)⊂ D^′[3]. A morphism is´etaleif it is a local isomorphism. For example, an ´etale morphism of real analytic manifolds is a real analytic map that is locally a diffeomorphism and an

´etale morphism of complex manifolds is a holomorphic map that is locally a biholomorphism. If (X, J) and (X^′, J^′) are almost complex manifolds, a map f :X →X^′ ispseudo-holomorphicprovided that it satisfies the corresponding Cauchy-Riemann equation with ¯∂_J,J^′f := ¹₂(df +J^′◦df ◦J).IfW^R is a real analytic manifold and dimR(W^R) =m,by the complexification of W^R, we mean the unique (germ of the) complex manifoldW that results from selecting an atlas onW^R and complexifying each of its real analytic transition functions near the real points. The complexification W is such that dimC(Y) =m,and it possesses a natural anti-holomorphic involution whose set of fixed points isW^R.

Let k ≥ n ≥ 1 and Y be a complex manifold of complex dimension 2k.For every y ∈ Y ,consider the complex projective manifold of flags of signature (k−n, k) inT_{Y ,y}

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F_(k−n,k)(T_{Y ,y}) ={(S^′,Σ^′)

S^′⊂Σ^′⊂T_{Y ,y} is a sequence of linear subspaces, dimC(S^′) =k−nand dimC(Σ^′) =k}

and the product manifold

F_(k−n,k)² (T_{Y ,y}) ={(S^′, S^′′,Σ^′,Σ^′′)

(S^′,Σ^′),(S^′′,Σ^′′)∈F_(k−n,k)(T_{Y ,y})}. Let

Q_y={(S^′, S^′′,Σ^′,Σ^′′)∈F_(k−n,k)² (T_{Y ,y})

Σ^′⊕Σ^′′ =T_{Y ,y}}.

Define

Z_n(Y) :=a

y∈Y

Q_y. This is a complex manifold of complex dimension

N_n,k := 2k+ 2(k²+n(k−n)),

and it bears a resemblance to twistor bundles and Grassmannians [5]. Let π_Y :Z_n(Y)→Y be the projection map defined for anyy ∈Y and q_y ∈Q_y byπ_Y(y, q_y) =y.

Define∆

n,k to be the sub-bundle ofπ^∗_Y(T_Y) such that for anyw= (y, S^′, S^′′,Σ^′,Σ^′′)∈ Z_n(Y),we have∆

n,k,w=S^′⊕Σ^′′.Now define a distributionD_n,k⊂T_Z_n_(Y₎ by D_n,k :=dπ⁻¹_Y (∆_n,k).

In the caseY =C^2k,if we let

Q:={(S^′, S^′′,Σ^′,Σ^′′)∈F_(k−n,k)² (C^2k)|Σ^′⊕Σ^′′ =C^2k},

then we simply get thatZ_n(C^2k) =C^2k×Q.Therefore, the above construction recovers the one made by J-P. Demailly and H. Gaussier, who intro- duced the complex directed manifolds (Z_n,k,D_n,k),where Z_n,k is the complex, quasi-projective manifold of all 5-tuples

{(z, S^′, S^′′,Σ^′,Σ^′′)|z∈C^2k,(S^′, S^′′,Σ^′,Σ^′′)∈Q}

andD_n,k is the co-ranknsub-bundle ofT_Z_n,k,whose fiber at anyw∈Z_n,k is D_n,k,w={(ζ, u^′, u^′′, v^′, v^′′)∈T_Z_n,k_,w

ζ∈S^′⊕Σ^′′}

(cf. Theorem 1.2 [5]). For n fixed, the above defined complex directed manifold (Z_n(Y),D_n,k) will be called here the universal embedding space associated withY ,as we will soon see its universal property.

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Remark 1. Z_n(Y)−−→^π^Y Y is a (holomorphic) fiber bundle with typical fiberQ, and ifY =C^2k,the bundle is trivial. The universal embedding spaceZ_n(Y)is locally diffeomorphic toZ_n(C^2k).

Proof. Given a holomorphic atlas (U_α, ψ_α) forY and p ∈U_α, the isomor- phismT_{Y ,p} ≃C^2k induces a biholomorphismQ_p ≃_q_α Q,and so (U_α, Id_U_α× q_α) is a local trivialization.

Now observe that Z_n(U_α) = π⁻¹_Y (U_α) ≃ U_α ×Q ≃ C^2k ×Q = Z_n(C^2k), where Z_n(U_α) = `

p∈U_αQ_p, where the first identification comes from the local trivialization and the second one, from the mapψ_α :U_α →C^2k. 1.1 Universal embedding property

Next we construct embeddings of compact almost complex manifolds.

Our approach is based on that used in the proof of Theorem 1.2 of [5].

Throughout, (X, J_X) is a compact almost complex manifold of dimension n≥1.

Proposition 1. Assume that there is a C^∞ real embedding of X into a real analytic 2k-dimensional manifold Y^R. Assume further that N_X/Y^R admits a complex structure J_N. Then, there is a totally real embedding F :X ֒→ Z_n(Y) that is transverse toD_n,k and that induces the almost complex structureJ_X. Proof. Define ˜J := JX ⊕JN, which is a complex structure onT_Y^R|_X ≃TX ⊕ N_X/Y^R. Consider theJ_Y-complexification of ˜J, J˜^C=J_X^C⊕J_N^C :T_Y|_X →T_Y|_X, and the sub-bundleS:={0}⊕N_X/Y^C ^RofT_Y|_X.For anyx∈X and (0, η_x)∈S_x= {0} ⊕N_X/Y^C ^R_,x,J˜^C(x)(0, η_x) = (0, J_N^C(η_x))∈S_x,implying thatS_x is ˜J^C(x)-stable so that ˜J^C(x)|S_x ∈ End(Sx). Let Σ^′_x be the +i eigenspace for ˜J^C(x) and Σ^′′_x be the −i eigenspace for ˜J^C(x). Then, the +i, respectively −i, eigenspaces for ˜J^C(x)|_S_x are S_x^′ := S_x ∩Σ^′

x and S_x^′′ := S_x∩Σ^′′

x. More explicitly, these eigenspaces are Σ^′_x = T_X,x^1,0⊕Eig(J_N^C(x), i), Σ^′′_x = T_X,x^0,1 ⊕Eig(J_N^C(x),−i), S_x^′ = {0} ⊕Eig(J_N^C(x), i), and S_x^′′ = {0} ⊕Eig(J_N^C(x),−i). Note that S_x^′ ⊂ Σ^′

x, S_x^′′ ⊂ Σ^′′

x, Σ^′

x ⊕Σ^′′

x = T_Y|_X,x, and S_x = S_x^′ ⊕S_x^′′, where dimC(Σ^′

x) = dimC(Σ^′′

x) =

12dimC(T_Y|_X,x) = k and dimC(S_x^′) = dimC(S_x^′′) = ¹₂dimC(N_X/Y^C ^R_,x) = ¹₂(2k− 2n) = k−n. Therefore, if f : X ֒→ Y^R is the given C^∞ real embedding, for any x ∈X–in reality, x belongs tof(X)–we have that (S_x^′, S_x^′′,Σ^′

x,Σ^′′

x)∈ Q_x={(S^′, S^′′,Σ^′,Σ^′′)∈F_(k−n,k)² (T_{Y ,x})|Σ^′⊕Σ^′′=T_{Y ,x}},and in this way, get an embeddingF :X ֒→ Z_n(Y),whereF(x) = (f(x), S_x^′, S_x^′′,Σ^′

x,Σ^′′

x).

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Letσ :Y →Y , y7→y, be an anti-holomorphic involution that realizes Y^Ras its fixed point set. For (S^′, S^′′,Σ^′,Σ^′′)∈Q_y,putS^′:=dσ|_y(S^′)⊂T_{Y ,y}, Σ^′:=dσ|_y(Σ^′)⊂T_{Y ,y},and defineS^′′ andΣ^′′ similarly. Then,σ gives rise to the anti-holomorphic involution ˜σ :Z_n(Y)→ Z_n(Y),

(y, S^′, S^′′,Σ^′,Σ^′′)7→(y, S^′′, S^′,Σ^′′,Σ^′).

The real pointsZ_n(Y)^RofZ_n(Y) are the fixed points of ˜σ,so

Z_n(Y)^R={(y, S^′, S^′′,Σ^′,Σ^′′)∈ Z_n(Y)|y∈Y^R, S^′′ =S^′,Σ^′′ =Σ^′}.

The anti-holomorphic character ofσ implies thatdσ is type-reversing and point-wise conjugate linear, soΣ^′_x=T_X,x^0,1⊕Eig(J_N^C(x),−i) =Σ^′′

x and similarly, Sx^′ =S_x^′′.Therefore,F(X)⊂ Z_n(Y)^R.

S_x^′ ⊕Σ^′′

x ={0}.HenceTZ_n(Y),F(x)=dF|_x(T_X,x)⊕D_n,k,F(x).

LetJZ_n(Y)be the given complex structure onZ_n(Y).The quotientTZ_n(Y)/D_n,k is a holomorphic vector bundle on Z_n(Y). Since D_n,k ⊂ TZ_n(Y) is a holomorphic distribution, JZ_n(Y)(D_n,k) =D_n,k. SoJZ_n(Y) descends to a complex structure on TZ_n(Y)/D_n,k. The transversality of F implies that at any x ∈ X, there is a real isomorphism ρ : T_F(X_),F(x) → T_Z_n_(Y_),F(x)/D_n,k,F(x). Then, J_F(X)^Zⁿ^(Y^),D^n,k(x) :=ρ⁻¹◦JZ_n(Y)(x)◦ρdefines an almost complex structureJ_F(X^Zⁿ^(Y₎^),D^n,k on F(X), and then since F is an embedding, the pullback section J_F :=

F^∗(J_F(X)^Zⁿ^(Y^),D^n,k) is an almost complex structure onX.Note that TZ_n(Y),F(x)/D_n,k,F(x)≃Σ^′

x/S_x^′,

and soTZ_n(Y),F(x)/D_n,k,F(x) is isomorphic to the holomorphic tangent space T_X,x^1,0, which is T_X,x endowed with the complex structure J_X^C(x). Run the above construction, with T_X,x^1,0 now playing the role of TZ_n(Y),F(x)/D_n,k,F(x), to find thatJ_F =J_X.

Lemma 1. LetMbe a real analytic manifold of dimensionk > n,letf :X ֒→M be a C^∞ real embedding, and i∆ : M ֒→ M×M be the diagonal embedding i∆(x) = (x, x).EmbedXintoM×Mviai∆◦f .Then, the normal bundleN_X/M×M has a natural complex structureJ_N.

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Proof. SinceT_M≃N_M/M×M,we obtain the following real analytic splittings N_X/M×M≃N_X/M⊕N_M/M×M|_X ≃N_X/M⊕T_M|_X ≃N_X/M⊕N_X/M⊕T_X. LetJ_N_X/M_⊕N_X/M be the tautological complex structure that is given by J_N_X/M_⊕N_X/M(ζ, η) = (−η, ζ). Put J_N := J_N_X/M_⊕N_X/M ⊕(−J_X), which defines a complex structure onN_X/M×M.

If we takeY^R=M×M as in the lemma, we then have that Σ^′

x=T_X,x^1,0⊕ {(u,−iu)|u∈N_X/M,x^C } ⊕T_X,x^0,1, Σ^′′

x =T_X,x^0,1⊕ {(u, iu)|u∈N_X/M,x^C } ⊕T_X,x^1,0, S_x^′ ={0} ⊕ {(u,−iu)|u∈N_X/M,x^C } ⊕T_X,x^0,1, and

S_x^′′={0} ⊕ {(u, iu)|u∈N_X/M,x^C } ⊕T_X,x^1,0.

Here we can see why it is nicer to take−J_X in the definition of J_N,as oth- erwiseΣ^′_xandΣ^′′_x would have repeated direct sum factors ofT_X,x^1,0 andT_X,x^0,1. Corollary 1. Ifn≥1andk≥2n,any compact almost complexn-dimensional manifold (X, J_X) admits a universal embedding F : (X, J_X) ֒→ (Z_n(Y),D_n,k), wheredim^R(Y^R) = 2k.

Proof. Take any real analytic manifold M of dimension k and put Y^R = M×M.Letψ:X²ⁿ֒→R⁴ⁿbe a Whitney embedding andφ:R⁴ⁿ→Bbe a diffeomorphism ofR⁴ⁿwith some open ballB⊂Y^R.Then,φ◦ψ:X ֒→Y^R is a real analytic embedding. The proposition buildsF:X ֒→(Z_n(Y),D_n,k) out of the embeddingφ◦ψ.

We call an embedding F as in Corollary 1 a universal embedding and denote it byF: (X, J_X)֒→(Z_n(Y),D_n,k).

Remark 2. Consider the 6-dimensional sphere S⁶ equipped with the octo- nion almost complex structureJ^O.Proposition 1 manufactures a universal em- bedding F : (S⁶, J^O) ֒→ (Z₃(C⁸),D_3,4) from the inclusion mapping i : S⁶ ֒→ Im(O)⊂R⁸,whereIm(O)≃R⁷are the imaginary octonions.

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1.2 Functorial property with respect to ´etale morphisms

For each n∈N,let C-Manⁿ_Et be the category of complex manifolds of dimension nwhose morphisms are ´etale morphisms. For eachk ≥n≥1,we have functorsZ_n:C-Man^2k_Et →C-Man^N_Et^n,k,given byZ_n(Y) =Z_n(Y),and for any ´etale morphism f :Y →Y^′,Z_n(f) :Z_n(Y)→ Z_n(Y^′) is the map that is defined at eachw= (y, S^′, S^′′,Σ^′,Σ^′′)∈ Z_n(Y) by

Z_n(f)(w) =

f (y), df|_y(S^′), df |_y(S^′′), df|_y(Σ^′), df |_y(Σ^′′) ,

wheredf|_y(S^′) := Im(df(y)|_S^′), df|_y(S^′′) := Im(df(y)|_S^′′),and so forth. Through- out, we will writeZ_n(f) instead ofZ_n(f).

For a fixedn,the universal embedding space associated withY is a complex directed manifold (Z_n(Y),D_n,k) such that every compact almost complex n-dimensional manifold (X, J_X) admits a totally real, transverse to D_n,k, J_X-inducing embedding F : X ֒→ Z_n(Y), and it satisfies the following universal property. For any n-dimensional compact almost complex manifold (X^′, J_X^′),anyC^∞ real embeddingsg :X ֒→Y^Randg^′:X^′֒→Y^′^R, any pseudo-holomorphic ´etale map ψ_X : (X, J_X)→ (X^′, J_X^′), and any ´etale morphismψ_Y :Y^R→Y^′^Rthat fit into a commutative diagram

(X, J_X) Y^R

(X^′, J_X^′) Y^′^R,

g

ψ_X ψ_Y

g^′

there is a corresponding functorially defined morphismZ_n(ψ_Y) : (Z_n(Y),D_n,k)→ (Z_n(Y^′),D^′_n,k) of complex directed manifolds making the diagram

(X, J_X) Z_n(Y)

(X^′, J_X^′) Z_n(Y^′),

F

ψ_X ∃Z_n(ψ_Y)

F^′

commute. By the Nash-Tognoli Theorem, we may assume that X and X^′ are smooth real algebraic varieties and thatg :X ֒→Y^Randg^′:X^′֒→Y^′^R are algebraic. Then, our construction gives thatF andF^′are real algebraic as well.