The case of a fixed Riemann surface

The remainder of this chapter consists of defining (Banach) manifold structures over certain subsets of this space (although not on all ofM( ˜X, A, J,H( ˜X))) in such a way that it reflects the stratified structure ofM by the stratification by signature (where well-defined) and to define a topology on M( ˜X, A, J,H( ˜X)) compatible with the manifold topologies on these parts.

K¨ahler, S being twodimensional). This can be solved by replacing the Levi-Civita connection onXby the hermitian connection ˜∇defined byg^J andJ. It is shown in [MS04], Appendix C.7, that

∇˜_XY =∇_XY −1

2J(∇_XJ)Y.

∇˜ preserves J and the metric g^J, but it is not torsion free, its torsion being given by

T^∇˜(X, Y) =−1

4N_J(X, Y), where NJ denotes the Nijenhuis tensor of J. Also, the map

π^Hom_Hom

(j,J) : Hom(T S, VX)ˆ →Hom_(j,J)(T S, VX)ˆ η7→ 1

2(η+J◦η◦j) defines a smooth bundle morphism.

Using these structures, one can make the following definitions:

Fix, once and for all, a real number p >2. Furthermore, letk∈N. B^k,p( ˆX, A, J, H) :={u∈L^k,p( ˆX,pr₁, g^J)|[pr₂◦u] =A},

where L^k,p( ˆX,pr₂, g^J) is the Sobolev space of sections of ˆX, defined in Con-struction II.2. This is a Banach manifold, since it is a union of connected components, hence an open subset, of L^k,p( ˆX,pr₁, g^J). For a continuous path in this space via the Sobolev embedding theorem defines a continuous path of continuous functions, hence two sections in the same connected component define the same homology class. Proceeding,

E^k−1,p( ˆX, A, J, H) :={(η, u)∈L^k−1,p(Hom_(j,J)(T S, VX), hˆ ^∗⊗g^J,∇^S⊗ ∇^H, X,ˆ pr₁, g^J)|u∈B^k,p( ˆX, A, J, H)},

which, as a restriction of the Banach space bundleL^k−1,p(Hom_(j,J)(T S, VX), hˆ ^∗⊗ g^J,∇^S⊗ ∇^H,X,ˆ pr₁, g^J) from Construction II.4 and Example II.1 to an open subset, is a Banach space bundle over B^k,p( ˆX, A, J, H). The projection of this bundle will be denoted by κ^k,p:E^k−1,p( ˆX, A, J, H)→B^k,p( ˆX, A, J, H).

To define the Cauchy-Riemann operator, note that just the same way, there also is the Banach space bundle

F^k−1,p( ˆX, A, J, H) :={(η, u)∈L^k−1,p(Hom(T S, VX), hˆ ^∗⊗g^J,∇^S⊗ ∇^H, X,ˆ pr₁, g^J)|u∈B^k,p( ˆX, A, J, H)}

over B^k,p( ˆX, A, J, H) which by Lemma II.13 comes with the section D^v : B^k,p( ˆX, A, J, H)→F^k−1,p( ˆX, A, J, H). Additionally, by Lemma II.14, the bun-dle morphismπ_Hom^Hom

(j,J) from above induces a morphism of Banach space bundles fromF^k−1,p( ˆX, A, J, H) toE^k−1,p( ˆX, A, J, H), hence the composition of the sec-tion D^v with this morphism defines a section ofE^k−1,p( ˆX, A, J, H).

Finally, note thatJinduces almost complex structures on bothVXˆ and Hom_(j,J)(T S, VX),ˆ which via Lemma II.14 turn bothTB^k,p( ˆX, A, J, H) andE^k−1,p( ˆX, A, J, H) into

complex Banach space bundles over B^k,p( ˆX, A, J, H).

Definition II.21. The(nonlinear) Cauchy-Riemann operator on ˆXis the sec-tion

∂^J,H_S :B^k,p( ˆX, A, J, H)→E^k−1,p( ˆX, A, J, H) u7→

1

2(D^vu+J◦D^vu◦j), u

of the Banach space bundle

κ^k,p:E^k−1,p( ˆX, A, J, H)→B^k,p( ˆX, A, J, H).

The next result is very technical, but necessary, for later the exact form of the linearisation of the Cauchy-Riemann operator is needed. The relevant result here is Corollary II.4. Still somewhat relevant to note may be the fact that this lemma shows that the linearisation of the Cauchy-Riemann operator is a (compact) perturbation of a linear Cauchy-Riemann operator (of some Sobolev class) at all points, not just the differentiable ones.

The statement, as well as all the other statements in the corollaries and lemmas following it, is to be read as follows: Pick measurable sections representing ξ,

∇ξ,η and∇η. Then there is a measurable sectionρof Hom(T S, u^∗VX) s. t. forˆ Z∈TzS,ρz(Z)∈V_u(z)Xˆ is given by the formula in the lemma. The equivalence class of this section in the relevant Sobolev space gives a well-defined element inE^k−1,p(Hom(T S, VX), A, J, H).ˆ

Alternatively, one can take the formula literally on differentiable sections and use a standard density argument for differentiable sections inL^k,p-sections.

Lemma II.16. Letu∈Γ^k( ˆX,pr₁, g)∩B^k,p( ˆX, A, J, H),ξ, η∈L^k,p(u^∗VX)ˆ and assume thatξis small enough that it lies in the chart aroundu. Then w. r. t. the chart forB^k,p( ˆX, A, J, H)and the trivialisation ofE^k−1,p(Hom(T S, VX), A, J, H)ˆ aroundufrom Constructions II.2 and II.4, respectively, the linearisation of∂^J,H_S atexp^⊥_u(ξ) in the direction η and evaluated on Z ∈T_zS is given by

k^s=1_s=0 exp^⊥_u(sξ(z)) −1

(D^vexp^Xˆz)_ξ(z)1

2(∇_Zη+Jξ∇_jZη) + + 1

2(∇_ηξA^u,ξ(·, Z) +J∇_ηξA^u,ξ(·, jZ))−

− 1

2J(∇_ηξJ)(A^u,ξ(ξ, Z)−J A^u,ξ(ξ, jZ)) + + H(η, ξ)∂^J,H_S exp^⊥_u(ξ)(Z)

,

where

J_ξ(z) := (D^vexp^Xˆz)⁻¹_ξ(z)◦J◦(D^vexp^Xˆz)_ξ(z) η_ξ(z) := (D^vexp^Xˆz)_ξ(z)(η(z))

A^u,ξ(ζ, Z) := (Dexp^Xˆz)ζ(D^vu(Z),∇_Zξ) +τζ(Z) H(η, ξ)(z) :=

Z ₁

0

t

k^s=1_s=t exp^⊥_u(sξ)

◦R^∇˜

(D^vexp^Xˆz)_tξ(z)η(z), (D^vexp^Xˆz)_tξ(z)ξ(z)

◦

k^s=1_s=t exp^⊥_u(sξ) −1

dt.

Proof. Let u, ξ and η be as in the statement of the Lemma and let λ ∈ R be so small that ξ +λη lies in the open subset of L^k,p(u^∗VX, h, gˆ ^J,∇) on which the chart around u is defined. For a vertical path α : [0,1] → X,ˆ i. e. pr₁◦α≡const : [0,1]→S, denote byk^s=1_s=0α(s) :V_α(0)Xˆ →V_α(1)Xˆ parallel transport w. r. t. the connection ˜∇ on X that respects J. Here, the fibre ˆX_z of ˆX over a point z ∈ S is identified with X. Note also that over each such fibre, the connection on Hom_(j,J)(T S, VX)|ˆ _Xˆ

z is given by composition with ˜∇, i. e. ( ˜∇η)(Z) = ˜∇(η(Z)) forη a section of Hom_(j,J)(T S, VX)|ˆ _Xˆ

z and Z ∈T_zS.

For this fibre is canonically identified withT_z^∗S×T X, i. e. the covariant deriva-tive in the first factor is the trivial one.

Then by definition, in the chart arounduand the trivialisation ofE^k−1,p( ˆX, A, J, H) over this chart, the derivative of (D∂^J,H_S ) at the point exp^⊥_u(ξ)∈B^k,p( ˆX, A, J, H) in the direction η and evaluated on Z∈T_zS is given by

d dλ

λ=0

k^s=1_s=0 exp^⊥_u(s(ξ+λη)) −1

∂^J,H_S exp^⊥_u(ξ+λη)(Z) =

=

k^s=1_s=0exp^⊥_u(sξ) −1 d

dλ λ=0

k^s=1_s=0 exp^⊥_u(ξ+sλη) −1

∂^J,H_S exp^⊥_u(ξ+λη)(Z) + + H(η, ξ)

k^s=1_s=0 exp^⊥_u(sξ) −1

∂^J,H_S exp^⊥_u(ξ)(Z).

The right hand side is calculated in Lemma II.12, with Ω^E in this case given by the curvature of ˜∇. Here and in the following, a number of evaluations of ξ and η on z∈S will be omitted, for otherwise the formulas become completely unmanageable. E. g. above, instead of

d dλ

λ=0

k^s=1_s=0 exp^⊥_u(s(ξ+λη)) −1

∂^J,H_S exp^⊥_u(ξ+λη)(Z) it should actually read

d dλ

λ=0

k^s=1_s=0exp^⊥_u(s(ξ(z) +λη(z))) −1

∂^J,H_S exp^⊥_u(ξ+λη)(Z).

The rule here is that ξ and η should be replaced by ξ(z) andη(z) unless they appear behind a differential operator such as e. g. ∇_Zξ or∂^J,H_S exp^⊥_u(ξ)(Z).

Proceeding,

d dλ

λ=0

k^s=1_s=0exp^⊥_u(ξ+sλη) −1

∂^J,H_S exp^⊥_u(ξ+λη)(Z)

= d dλ

λ=0

k^s=1_s=0exp^⊥_u(ξ+sλη) −1 1

2 D^vexp^⊥_u(ξ+λη)(Z) + + J◦D^vexp^⊥_u(ξ+λη)(jZ)

= d dλ

λ=0

k^s=1_s=0 exp^⊥_u(ξ+sλη) −11

2D^vexp^⊥_u(ξ+λη)(Z) + 1

2J ◦ d dλ

λ=0

k^s=1_s=0exp^⊥_u(ξ+sλη) −1

D^vexp^⊥_u(ξ+λη)(jZ),

because parallel transport w. r. t. the connection ˜∇ preserves J by definition.

Hence it suffices to calculate (and then use the same result withZ replaced by jZ)

d dλ

λ=0

k^s=1_s=0 exp^⊥_u(ξ+sλη) −1

D^vexp^⊥_u(ξ+λη)(Z) =

= d dλ

λ=0

k^s=1_s=0 exp^⊥_u(ξ+sλη) −1

(Dexp^Xˆz)ξ+λη(D^vu(Z),∇_Z(ξ+λη)) + + τξ+λη(Z)

= d dλ

λ=0

k^s=1_s=0 exp^⊥_u(ξ+sλη)−1

(Dexp^Xˆz)_ξ+λη(D^vu(Z),∇_Zξ) + + τ_ξ+λη(Z)

+ d dλ

λ=0

λ

k^s=1_s=0exp^⊥_u(ξ+sλη)−1

(D^vexp^Xˆz)_ξ+λη(∇_Zη)

= ˜∇_(DvexpXzˆ )_ξ(η)

(Dexp^Xˆz)·(D^vu(Z),∇_Zξ) +τ·(Z)

+ (D^vexp^Xˆz)_ξ(∇_Zη)

= ˜∇_ηξA(·, Z) + (D^vexp^Xˆz)ξ(∇_Zη)

=∇_ηξA(·, Z)−1

2J(∇_ηξJ)A(ξ, Z) + (D^vexp^Xˆz)ξ(∇_Zη).

Now it only remains to sort all the different terms, noting that J(∇_ηξJ) =

−(∇_ηξJ)J(differentiateJ²=−id), and to relateHtoHvia an easy calculation using the composition property of parallel transport to finish the proof.

Lemma II.17. Let u ∈ Γ^k( ˆX,pr₁, g^J)∩B^k,p( ˆX, A, J, H). Then w. r. t. the chart for B^k,p( ˆX, A, J, H) and the trivialisation ofE^k−1,p( ˆX, A, J, H) around u from Constructions II.2 and II.4, respectively, the linearisation of∂^J,H_S at u is

given by (for Z ∈T S)

(D∂^J,H_S )_u :TB^k,p( ˆX, A, J, H)→E^k−1,p( ˆX, A, J, H)_u ((D∂^J,H_S )u)η(Z) =∇^0,1_Z η−1

2J

(∇_ηJ)∂u(Z) +π_V^TX_X^ˆ_ˆ( ˆ∇^H_η Jˆ^H) ˜Z

=∇^0,1_Z η−π_V^TX_X^ˆ_ˆ 1

2Jˆ^H

∇ˆ^H_η Jˆ^H

(∂u(Z) + ˜Z)

=∇^0,1_Z η−K_JˆH(η, ∂u(Z) + ˜Z)−

− 1

8π^T_V^X_X^ˆ_ˆNJˆ^H(η, ∂u(Z) + ˜Z), where

∇^0,1_Z η := 1

2(∇_Zη+J∇_jZη),

∂u(Z) := 1

2(D^vu(Z)−J D^vu(jZ)), KJˆ^H is the symmetric part of the bundle morphism

TXˆ ⊗TXˆ →VX,ˆ (η, ξ)7→π^T_V^X_X^ˆ_ˆ 1

2Jˆ^H

∇ˆ^H_η Jˆ^H

ξ

and where Z˜ denotes the horizontal lift of Z ∈T S toX,ˆ ∇ˆ^H denotes the Levi-Civita connection on Xˆ and Jˆ^H denotes the almost complex structure on Xˆ defined by J, j and the connection given byH as in Definition II.19.

Proof. This is the special case of Lemma II.16 forξ= 0. One checks easily that in this case J_ξ =J,η_ξ=η and H(η, ξ) = 0 as well as (k^s=1_s=0 exp^⊥_u(sξ(z)))⁻¹ = (Dexp^X_u(z)^ˆz )_ξ(z) = id. Also, by elementary properties of the differential of the (full) exponential map together with Lemma II.5, A^u,0(0, Z) =D^vu(Z). This accounts for the first and second summand in the formula. Again by elementary properties of the differential of the exponential map together with Lemma II.6,

∇_ηA^u,0(0, Z) =−π^TXˆ

VXˆ

∇ˆ^H_η Z, where ˜˜ Z denotes the horizontal lift. So 1

2(∇_ηA^u,0(·, Z) +J∇_ηA^u,0(·, jZ) = 1 2(−π^TXˆ

VXˆ∇ˆ^H_η Z˜−J π^T_V^X_X^ˆ_ˆ∇ˆ^H_η jZ)f

=−π^TXˆ

VXˆ

1

2( ˆ∇^H_η Z˜+ ˆJ^H∇ˆ^H_η jZ).f Now

Jˆ^H∇ˆ^H_η jZf = ˆ∇^H_η ( ˆJ^HjZf

| {z }

=jjZ=−g Z˜

)−( ˆ∇^H_η Jˆ^H)jZf

=−∇ˆ^H_η Z˜−( ˆ∇^H_η Jˆ^H) ˆJ^HZ.˜

So 1

2(∇_ηA^u,0(·, Z) +J∇_ηA^u,0(·, jZ)) = 1

2π^T_V^X_X^ˆ_ˆ( ˆ∇^H_η Jˆ^H) ˆJ^HZ˜

=−1

2π_V^TX_X^ˆ_ˆJˆ^HZ˜( ˆ∇^H_η Jˆ^H)

=−1

2J π_V^TX_X^ˆ_ˆ( ˆ∇^H_η Jˆ^H) ˜Z, which accounts for the remaining term in the formula.

The last equality follows from the decomposition of a linear morphism TXˆ ⊗ TXˆ →VXˆ into its symmetric and antisymmetric part together with the right formula in line (C.7.5) of Lemma C.7.1, p. 566, in [MS04].

This result seems to differ by the term involving ˜Z from the corresponding formula in [MS04], Section 8.3, p. 257 f., esp. Remark 8.3.8. Although that is not a real argument for why the formula above is correct, Corollary II.4 and Lemma II.18 at least show that it produces the consequences one (or at least the author) would hope for.

Corollary II.3.

∂^J,H_S :B^k,p( ˆX, A, J, H)→E^k−1,p( ˆX, A, J, H) is a Fredholm operator of index

dim_C(X) ˆχ+ 2c1(A).

Proof. By Lemma II.16, the differential ofD∂^J,H_S at a point exp^⊥_u(ξ) inB^k,p( ˆX, A, J, H) withudifferentiable andξaL^k,p-section ofu^∗VX, in the trivialisations aroundˆ u, is given by the operator defined, on η∈L^k,p(u^∗VX), byˆ

TzS 3Z 7→

k^s=1_s=0exp^⊥_u(sξ(z)) −1

(D^vexp^Xˆz)_ξ(z)1

2(∇_Zη+Jξ∇_jZη) + + 1

2(∇_ηξA^u,ξ(·, Z) +J∇_ηξA^u,ξ(·, jZ))−

− 1

2J(∇_ηξJ)(A^u,ξ(ξ, Z)−J A^u,ξ(ξ, jZ)) + + H(η, ξ)∂^J,H_S exp^⊥_u(ξ)(Z)

. Claim. The expressionz7→ k^s=1_s=0exp^⊥_u(sξ(z))−1

◦(D^vexp^Xˆz)_ξ(z)defines an el-ement Ψ ofL^k,p(Hom(u^∗VX, uˆ ^∗VX)), with image in the bundle isomorphisms.ˆ Proof. That for fixedz∈S this defines an isomorphism of V_u(z)Xˆ is clear from the standing assumption on kξkin the chart for B^k,p( ˆX, A, J, H) aroundu.

Forξ= 0, Ψ is clearly the identity. Using Lemma II.8 and Lemma II.12 and the line of argument used in Subsection II.2.3, one then shows that, in the notation used there, _∂ρ^∂∇ⁱΨ, for ξ, ρ ∈ Γ^k(u^∗X) andˆ i = 0, . . . , k, can be bounded in

L^p-norm by a multiple of the L^k,p-norms of ξ and ρ. Hence the L^k,p-norm of

∂

∂ρΨ can be bounded by a multiple of the L^k,p-norms of ξ and ρ as well. The claim then follows by the density argument via the mean value theorem used before.

By the Sobolev multiplication theorem (remember thatkp >2), such a section defines an isomorphism L^k−1,p(u^∗VX)ˆ → L^k−1,p(u^∗VX). One can hence dis-ˆ regard this part of the first summand. The second part of the first summand, η 7→ ¹₂(∇_·η+J_ξ∇_j·η), defines a linear Cauchy-Riemann operator of class L^k,p, by the following claim:

Claim. If ξ is of class L^k,p, then so is J_ξ.

Proof. The proof follows along the same lines of argument as the previous one.

All the remaining summands factor through the compact inclusion of L^k,p in C⁰ (by the Sobolev embedding theorem, see [MS04], Theorem B.1.11, p. 517), hence the above operator is a compact perturbation of a linear Cauchy-Riemann operator of class L^k,p. By the Riemann-Roch theorem, see [MS04], Theorem C.1.10, p. 545, this is a Fredholm operator of the given index.

Corollary II.4. Let u∈B^k,p( ˆX, A, J, H), η∈T_uB^k,p( ˆX, A, J, H). If ∂^J,H_S u= 0, then

(((D∂^J,H_S )u)(J η)−J((D∂^J,H_S )u)η)(Z) =π_V^TX_X^ˆ_ˆN_JˆH(η, Du(Z)),

where Du:T S →TXˆ is the usual differential. In particular, if N_JˆH(η, v) = 0 for all η∈VX|ˆ _imu and v∈imDu, then

(D∂^J,H_S )_u :T_uB^k,p( ˆX, A, J, H)→E^k−1,p( ˆX, A, J, H)_u is a complex linear operator.

Proof. First, assume thatηis of classC^k. Then by definition,∇_Zη =π^TXˆ

VXˆ∇ˆ^H_Du(Z)η (in casek= 1, the right hand side of this formula does not make any literal sense for sections of class L^k,p, whereas the left hand side does by definition of the L^k,p-spaces), where one considersηas a vertical vector field on ˆXon the image of u. Furthermore, becauseηis a vertical vector field,J η= ˆJ^Hηandπ^TXˆ

VXˆJˆ^H =J.

Also, by definition of ∂^J,H_S u and ∂u, Du(Z) = ∂^J,H_S u(Z) +∂u(Z) + ˜Z, in par-ticular Du(Z) =∂u(Z) + ˜Z if∂^J,H_S u= 0. With this, by the second formula for

(D∂^J,H_S )u from Lemma II.17,

(((D∂^J,H_S )u)(J η)−J((D∂^J,H_S )u)η)(Z) =π^T_V^X_X^ˆ_ˆ(((D∂^J,H_S )u)( ˆJ^Hη)−Jˆ^H((D∂^J,H_S )u)η)(Z)

=π^TXˆ

VXˆ

∇ˆ^H_Du(Z)( ˆJ^Hη)−1 2

Jˆ^H

∇ˆ^H_ˆ

J^HηJˆ^H

Du(Z)−

− Jˆ^H∇ˆ^H_Du(Z)η−1 2

∇ˆ^H_η Jˆ^H

Du(Z)

=π^T_V^X_X^ˆ_ˆ

∇ˆ^H_Du(Z)Jˆ^H η−

∇ˆ^H_η Jˆ^H

Du(Z) by the Leibniz rule and the left formula in line (C.7.5) of Lemma C.7.1, p. 566, in [MS04]. The claim for η of class C^k now follows from the right formula in line (C.7.5) of Lemma C.7.1, p. 566, in [MS04].

The general case (ηof classL^k,p) then follows by the standard density argument.

The following lemma should motivate the appearance of the almost complex structure ˆJ^H in the lemma and corollary above.

Lemma II.18. In the notation of the above construction, for a section u ∈ Γ^k( ˆX, h, g^J,∇^H)∩B^k,p( ˆX, A, J, H), π^TXˆ

HXˆ◦∂^Jˆ

H

S u= 0and π^TXˆ

VXˆ◦∂^Jˆ

H

S u=∂^J,H_S u, where Jˆ^H is the almost complex structure on Xˆ as in Construction II.5 and

∂^Jˆ

H

S is the standard Cauchy-Riemann operator on functions between the almost complex manifolds S andX. In particular,ˆ u satisfies∂^J,H_S u= 0 iff u:S→Xˆ is a(j,Jˆ^H)-holomorphic map.

Proof. By definition of ˆJ^H,

∂^Jˆ

H

S u= 1

2(Du+ ˆJ^H ◦Du◦j)

= 1

2(π_V^TX_X^ˆ_ˆ ◦Du+J◦π_V^TX_X^ˆ_ˆ ◦Du◦j+ + π^TXˆ

HXˆ ◦Du+ (π∗|_HXˆ)⁻¹◦j◦π∗◦(π^TXˆ

HXˆ)∗◦Du◦j)

=∂^J,H_S u+1

2((π∗|_HXˆ)⁻¹+ (π∗|_HXˆ)⁻¹◦j◦j) because by definition of a connectionπ∗◦(π^TXˆ

HXˆ)∗ =π∗ andπ∗◦Du= id as well asπ^TXˆ

HXˆ ◦Du= (π∗|_HXˆ)⁻¹

=∂^J,H_S u+ 0.

Lemma II.19. Let v ∈ B^k,p( ˆX, A, J, H) with ∂^J,H_S v = 0. Then v is smooth, i. e. v∈Γ( ˆX,pr₁, g^J).

Proof. In a chart around an element u ∈ Γ^k( ˆX, h, g,∇), v is given by v = exp_u(ξ) for some ξ ∈ L^k,p(u^∗VX). By the formula given in Lemma II.13, forˆ Z ∈TzS,

∂^J,H_S v(Z) = 1

2 Dexp^X_ξ^ˆz(D^vu(Z),∇_Zξ) +τ_ξ(Z) + + J Dexp^X_ξ^ˆz(D^vu(jZ),∇_jZξ) +J τξ(jZ)

, hence ∂^J,H_S v(Z) = 0 ⇔

1

2(D^vexp^Xˆz(∇_Zξ) +J D^vexp^Xˆz(∇_jZξ)) =

−1

2(τ_ξ(Z) +J τ_ξ(jZ) + D^hexp^Xˆz

ξ(D^vu(Z)) +J D^hexp^Xˆz

ξ(D^vu(jZ))).

Composing from the left with (D^vexp^Xˆz)⁻¹_ξ(z)and definingJ_ξ:= (D^vexp^Xˆz)⁻¹_ξ(z)◦ J◦(D^vexp^Xˆz)_ξ(z) as before, yields

1

2(∇_Zξ+J_ξ◦ ∇_jZξ) =−1

2(D^vexp^Xˆz)⁻¹_ξ(z) τ_ξ(Z) +J τ_ξ(jZ)c; + + D^hexp^Xˆz

ξ(D^vu(Z)) +J D^hexp^Xˆz

ξ(D^vu(jZ)) . By the second claim in the proof of Corollary II.3,J_ξis an almost complex struc-ture on the vector bundleu^∗VXˆ of the same class asξ (here, a prioriL^k,p) and the right hand side of the above equation defines a section of Hom(T S, u^∗VX)ˆ also of the same class (again a priori L^k,p) asξ. This is shown using the same proof as in that of smoothness of the transition functions in Subsection II.2.3.

After going to local charts of this bundle, one can apply the bootstrapping pro-cedure from Appendix B.4 in [MS04], esp. Lemma B.4.6 and Proposition B.4.9, to show the Lemma.

Im Dokument Universal moduli spaces in Gromov-Witten theory (Seite 69-78)