MODULI SPACE OF HOLOMORPHIC

DISCS WITH BOUNDARY

WILHELM K L I N G E N B E R G

Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 72076 Tübingen, Germany Email: wilhelm@moebius.mathematik.uni-tuebingen.de

1. Introduction. Let (X, J) be a complex surface, i.e., a two complex dimensional manifold with holomorphic coordinate changes, and J the associated complex structure on X. In case that X is compact, the moduli space of rational curves in X is an important tool in classification theory. For the study of open complex surfaces it is natural to consider holomophically embedded open discs with boundary. To obtain a well posed problem for such one needs to impose a boundary condition. If it is restricted to lie in a real two-dimensional surface that is totally real - that is infinitesimally transverse to its image under the complex structure J, rather than invariant, as are holomorphic curves - then one obtains a classical Riemann-Hilbert problem [5], an elliptic boundary value problem. Here we study the corresponding moduli space and examine its dependence on the boundary condition. The study of holomorphic discs with boundary was initiated by Bishop [2] and the arguments given here follow [4], chapter 2.1. The ellipticity and the index of the

1991 Mathematics Subject Classification: Primary: 58D27 Secondary: 58G20 The paper is in final form and no version of it will be published elsewhere. problem for compact curves is studied in detail in [1], 6.22 and [6], 2.3.4. The index formula in Theorem 1c) occurs in [4] and Theorem 1d) is proven similarly as in [6]. Acknowledgement: The author is grateful to Thomasz Mrowka for enlightening discussions, and to Marie-Louise Schimmel for her expedient typing of the manuscript.

2. The analytic framework. Let 𝔐 be the infinite dimensional manifold of C^{∞} - smooth embedded totally real surfaces in X. Here a real surface M ↪ X is called totally real if T_{m}M ⊕ J(m)T_{m}M = T_{m}X for all m ∈ M or equivalently since dim_{ℝ} M = 4, T_{m}M ∩ J(m)T_{m}M = 0 ⊆ T_{m}X. Let now Δ ⊂ ℂ denote the set {z ∈ ℂ : |z| < 1}. Since W^{1,m}(Δ̄, ℝ^{N}), the Sobolev space of functions whose derivative is in L^{p}, embeds compactly into C^{0}(Δ̄, ℝ^{N}) for p > 2, we have a well-defined space W^{1,p}(Δ, ∂Δ; X, M) if X ↪ ℝ^{N} for some N ∈ ℕ. Let

𝔉 = {(f, M) ∈ W^{1,p} × 𝔐 : f : (Δ, ∂Δ) → (X, M)}.

Together with the natural map π : 𝔉 → 𝔐, this is a locally trivial fibre bundle over 𝔐, and a complete Banach manifold. For differentiable structures on 𝔐 and 𝔉 one may consult [3]. The moduli space of holomorphic discs in X with boundary in elements of M ∈ 𝔐 is now given by the points in 𝔉 that satisty the Cauchy-Riemann equation:

H = {(f, M) ∈ 𝔉 : df ∘ i = J(f) · df}.

For (f, M) ∈ 𝔉 denote by c_{1}(f^{⋆}T X, f^{⋆}T M) ∈ ℤ the first relative Chern class of the bundle (f^{⋆}T X, f^{⋆}T M) over (Δ, ∂Δ) with distinguished totally real subbundle over ∂Δ. It is also called the Maslov class and equals the first Chern class of the relative to f^{⋆}T M doubled bundle over ℂP (1). Theorem 1. Let (f, M) ∈ H and assume that c_{1}(f^{⋆}T X, f^{⋆}T M) ≥ 1. Then there exists a neighborhood 𝔘 ⊂ 𝔉 of (f, M) with:

a) H ∩ 𝔘 is an infinite dimensional submanifold of 𝔉

b) π : H ∩ 𝔘 → 𝔐 is a smooth Fredholm map

c) The index of π|H ∩ 𝔘 equals 2 + c_{1}(f^{⋆}T X, f^{⋆}T M)

d) the cokernel of π vanishes at (f, M) ∈ H, i.e.

dim π^{−1}(M_{1}) = 2 + c_{1}(f^{⋆}T X, f^{⋆}T M_{1}) for M_{1} ∈ π(𝔘) ⊂ 𝔐.

3. An Elliptic Boundary Value Operator. To study the space H it is convenient to view it as the zero set of a section of a vector bundle 𝔊 over 𝔉. The fibre at (f, M) ∈ 𝔉 of 𝔊 is given by Γ(Δ, Homℂ̄(T Δ, f^{⋆}T X)), the vector space of sections of the bundle over Δ whose fibre in turn at δ ∈ Δ is given by the (i, −J)-linear maps Λ from (T_{δ}Δ, i) ≈ (ℂ, i) to (T_{f(δ)}X, J(f(δ))) : Λ(iv) + J(f(δ))Λv = 0 for all v ∈ T_{δ}Δ. We denote by 𝔊 → 𝔉 the resulting bundle and the section ¯∂: 𝔉 → 𝔊 we are looking at is given by

¯∂(f, M) = \frac{1}{2}(df + J(f) ∘ df ∘ i).

Then we have H = (¯∂)^{−1}0_{𝔊} ⊂ 𝔉, if 0_{𝔊} denotes the zero section of 𝔊 → 𝔉. We next consider the linearization of the section ¯∂, using a connection ∇ on the bundle 𝔊. The result is a linear operator

∇¯∂: T_{f}𝔉 ≈ Γ(Δ, f^{⋆}T X) → Γ(Δ, Hom(T Δ, f^{⋆}T X)).

For a not necessarily integrable complex structure J, the value of ∇¯∂ at a section s ∈ T_{f}𝔉 is given by the following expression.

(1)  ∇_{s} ¯∂ =\frac{1}{2}[f^{⋆}(∇)_{(·)}s + J(f) ∘ f^{⋆}(∇)_{(·)}s ∘ i]

+ \frac{1}{2}[Tor^{f^{⋆}∇}(s, df(·)) + J(f) · Tor^{f^{⋆}∇}(s, df ∘ i(·))]

+ \frac{1}{2}(f^{⋆}(∇)_{s}(f^{⋆}J) ∘ df ∘ i

One checks that this is an element of Γ(Δ, Homℂ̄(T Δ, f^{⋆}T X)) for f ∈ ker ¯∂. To verify this expression, let φ : Δ×(−1, 1) → X; (δ, t) → φ(δ, t) be a curve in 𝔉 tangent to s ∈ T_{f}𝔉 : φ(·, 0) = f, ∂/∂t|_{t=0} φ = s, and v a section of T Δ, which can be trivially extended to a section of T (Δ × (−1, 1)), as can the vector field ∂/∂t ∈ T (−1, 1). We are now in a position to compute (∇_{s}d)(f) · v. It equals

∇_{∂/∂t}^{φ^{⋆}T X} dφ ∘ v |_{t=0}.

On the other hand, the torsion of ∇ on φ^{⋆}T is defined to be

Tor (dφ · ∂/∂t, dφ · v) = ∇_{∂/∂t}^{φ^{⋆}T X} dφ · v − ∇_{v}^{φ^{⋆}T X}dφ · ∂/∂t − dφ [∂/∂t, v] .

The expression ∇_{s}(J(f) ∘ df ∘ i) splits up into (∇_{s}J(f)df ∘ i + J(f)(∇_{s}df) ∘ i and collecting terms, one arrives at (1). To keep track of the boundary we need to extend the operator as follows. One needs to understand the tangent space of 𝔐, the manifold of totally real surfaces in X. If we fix such a surface M ∈ 𝔐, the tangent space to 𝔐 at M is given by Γ(M, T X|_{M}/T M). The full linearization of our operator is thus

∇¯∂ ⊕ b : Γ(f^{⋆}T X) → Homℂ̄(T Δ, f^{⋆}T M) ⊕ Γ(∂Δ; f|_{∂Δ}^{⋆}T X|_{M}/T M),

where b : Γ(f^{⋆}T X) → Γ(∂Δ; f|_{∂Δ}^{⋆}T X|_{M}/T M) is given by b(s) = s|_{∂Δ}/T M. Regarding this operator, we have the following. Proposition 2. The linearized operator ∇¯∂ ⊕ b is an elliptic boundary value operator. Proof. There are two parts to this dicussion. First we shall show that the symbol of our differential operator ∇¯∂ is invertible. Then the boundary condition is shown to be elliptic. Let δ ∈ Δ, v ∈ T_{δ}^{⋆}Δ \ 0, then the symbol is a linear map

σ_{δ}(v) : f^{⋆}T M|_{δ} → Homℂ̄(T_{δ}Δ, f^{⋆}T M|_{δ})

Since the domain and range of σ are of the same dimension to check invertibility it suffices to show that σ_{δ}(v)e ≠ 0 for all e ∈ f^{⋆}T M|_{δ}\{0}. But the value of σ_{δ}(v) · e is

(2)  \frac{1}{2} (v ⊗ e + v · i ⊗ J(f(δ)) · e)

which is nonzero. To check the boundary conditions we fix δ ∈ ∂Δ and also require the following data. Near ∂Δ we fix a collar neighborhood C = [0, ϵ] × ∂Δ. Let ξ denote a unit covector along ∂Δ in the neighborhood and fix n = ξ ∘ i a conormal vector. Using this notation we consider the ordinary differential equation

(3)  σ (ξ + n (−i\frac{∂}{∂t})) e(t) = 0

for e(t) a curve in f^{⋆}T X|_{δ} ⊗ ℂ. Let B ⊂ f^{⋆}T X|_{δ} ⊗ ℂ denote the space of initial conditions at t = 0 giving rise to solutions of equation (3) which are bounded for t > 0. Let the map R : f^{⋆}T X|_{δ} ⊗ ℂ → f^{⋆}T X/f^{⋆}T M|_{δ} ⊗ ℂ denote the quotient map. To check ellipticity it suffices to show that the induced map R : B → f^{⋆}T X/f^{⋆}T M|_{δ} ⊗ ℂ is an isomorphism. To this end we write equation (2) more explicitly. Using equation (1) we have

n ⊗ −i\frac{∂e(t)}{∂t} + n ∘ i ⊗ J · (−i\frac{∂e(t)}{∂t}) + ξ ⊗ e(t) + ξ ∘ i ⊗ Je(t) = 0.

Let e(t) = e_{+}(t) + e_{−}(t) denote the decomposition along the ±i eigenspaces of J. The previous equation then splits into the follwing pair of equations.

±ie_{±} − i\frac{∂e_{±}}{∂t} = 0.

The solutions to these equations are

e_{±}(t) = e_{±}(0)e^{±t}.

Thus B is the −i eigenspace of J. Thus B is naturally isomorphic to f^{⋆}T X|_{δ} by the map sending f ∈ f^{⋆}T X|_{δ} to \frac{1}{2}(f + iJf) ∈ B. Since M is totally real there is a natural isomorphism of f^{⋆}T X/f^{⋆}T M|_{δ} ⊗ ℂ with f^{⋆}T X|_{δ} induced from the map sending ξ + iη ∈ f^{⋆}T X|_{δ} ⊗ ℂ to ξ + Jη ∈ f^{⋆}T X|_{δ}. Since the corresponding composition is the identity it follows directly that R is an isomorphism. q.e.d.

4. Calculation of the Index. Let E → Δ be a complex vector bundle and R → ∂Δ a totally real subbundle of E|∂Δ. We begin by showing that there is a well defined integer valued relative first Chern class also called the Maslov index of (E, R). By taking determinants we may reduce to the case of a complex line bundle. If the (necessarily totally) real line subbundle is orientable, it provides a preferred choice of nonvanishing section of E along the boundary of Δ and hence there is a well defined relative Chern class. If the totally real line subbundle is not orientable we take the complex tensor square of E. The induced totally real line subbundle is now orientable and we can argue as before to define twice the relative Chern class. One may also use the short exact sheaf sequence

O → ℤ → O(Δ, ∂Δ; E, R)⟶^{exp}O^{⋆}(Δ, ∂Δ; E, R) → O

of holomorphic sections of E which take values in the subbundle R on ∂Δ and define the Chern class by the connecting homomorphism in homology

H^{1}(Δ, ∂Δ; O^{⋆}(E), ℝ)⟶^{c_{1}}H^{2}(Δ, ∂Δ; ℤ),

the latter being isomorphic to ℤ via Poincaré duality. Note that we have here ℝ instead of the real bundle R. A third way that one may go is to double the vector bundle (E, R) → (Δ, ∂Δ) with respect to R to a bundle E_{1} → ℂP (1) over the rational sphere by identifying points on the boundary of the two copies of Δ and glueing the fibres along the oriented prescribed subbundle R. By Schwarz reflection principle, holomorphic sections of (E, R) may be continued to sections of E_{1}. This operation defines an involution on the space O(E_{1}) and the eigenspaces to the eigenvalues ±1 are subspaces of equal dimension, span the entire spaces, and H^{0}(Δ, ∂Δ; O(E), R) is isomorphic to either. The same holds for O(E_{1}^{⋆}) and H^{1}(Δ, ∂Δ; O(E), ℝ). In this situation the theorem of Riemann-Roch states that

h^{0}(Δ, ∂Δ; O(E), R) − h^{1}(Δ, ∂Δ; O(E), R) = r(1 − g) + c_{1}(E, R).

Here h^{0}, h^{1} are the real dimensions of the real vector spaces of holomorphic sections of E with boundary values in R and corresponding holomorphic one-forms, and r is the rank over ℂ of the vector bundle E, and g the genus of ℂP (1), namely zero. The left hand side of the above formula is the real index of the usual ¯∂-operator on the relative bundle (E, R). Note that in (1) the zero order terms vanish if we choose the connection ∇ to be torsion free and since for an integrable complex structure, f^{⋆}(∇)f^{⋆}J vanishes. We conclude in this case that ∇¯∂ is identical to the usual ¯∂-operator and proceed to show that the cokernel of the latter vanishes in the vector bundle E_{1} → ℂP^{1} described above as the doubling of (f^{⋆}T X, f^{⋆}T M). This is the case precisely if the (0, 1)-Dolbeault cohomology group H0,1 ; ¯∂(ℂP (1), E_{1}) vanishes. Now E_{1} splits into the direct sum of the tangent line bundle L_{1} and the normal line bundle L_{2}. By Dolbeault isomorphy and Kodaira-Serre duality

H0,1 ; ¯∂(ℂP (1), L_{j}) ≅ H^{1}ℂP (1), O(L_{j})) ≅ H^{0}(ℂP (1), O(L_{j}^{⋆} ⊗ K)).

Here K is the canonical bundle of ℂP (1). Now the last group is trivial if the first Chern class of the bundle L_{j}^{⋆} ⊗ K is negative. Since c_{1}(L_{j}^{⋆} ⊗ K) = −c_{1}(L_{ı}) − 2, the condition is c_{1}(L_{ı}) ≥ −1. We have c_{1}(L_{1}) = 2, so the condition is satisfied. For L_{2} this condition is satisfied if c_{1}(E_{1}) ≡ c_{1}(L_{1}) + c_{1}(L_{2}) ≡ 2 + c_{1}(L_{2}) ≥ 1. This is the condition mentioned in Theorem 1. We conclude that ∇¯∂ is regular at (f, M) ∈ H, i.e. part a) and part d) of Theorem 1. This concludes the proof of Theorem 1.

References

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