L19: Fredholm theory. where E u = u T X and J u = Formally, J-holomorphic curves are just 1
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1 L19: Fredholm theory We want to understand how to make moduli spaces of J-holomorphic curves, and once we have them, how to make them smooth and compute their dimensions. Fix (X, ω, J) and, for definiteness, take Σ = C P 1 ; then J : C (P 1, X) Ω 0,1 (E) where E u = u T X and J u = 2 1 (du + J du j). Formally, J-holomorphic curves are just 1 J (0), the zero-set of a section J of a vector bundle E C (P 1, X). We would like to show smoothness by the implicit function theorem. This is possible in infinite dimensions if we work with Banach manifolds (modelled on Banach spaces); but T u (C (P 1, X)) = Γ(u T X) is a Fréchet space rather than a Banach space, it has no natural complete norm. Definition: for f : R n R m, the Sobolev norm f L k,p = k j=0 ( j )1 f p L p
2 Ex: if k = 0 we omit it: f L p = ( f p dµ) 1/p. Recall that given a norm on a vector space, there is a unique completion (e.g. defined as a space of Cauchy sequences), which is Banach. By definition L k,p (R n, R m ) is just the completion of the space of smooth functions of finite L k,p -norm. There is a generalisation from functions to sections of bundles over manifolds or to maps of manifolds. E.g. given E M and metrics on M, a Hermitian metric on E, and a connexion d A on E, we set L k,p (E) = v k j=0 p d 1 j A v L p < where φ L p = ( M φ p dµ M ) 1/p. One can check the resulting spaces are independent of the various choices provided M is compact; similarly, we get L k,p (M, N), mappings of local regularity L k,p in any local chart. Note: All our Banach spaces are separable.
3 Warning: for local regularity to be independent of choice of chart may require some care. E.g. if a map R 2 R 2n is in L k,p, and we compose with diffeomorphisms of R 2 and R 2n, we stay in L k,p ; but if we use a transition map which is C k on R 2 and C k on R 2n, we only stay in L k,p if kp > 2. Since we will be working with completions, some of our maps will be C k objects, and it only makes sense to choose C k charts, so this is a real issue which we ll hence elide. Let Eu k,p = L k,p (Λ 0,1 T Σ u T M); then j J : L k,p (Σ, M) E k 1,p is a section of a Banach bundle over a Banach manifold. Note the complex structures j on Σ and J on M are fixed. Write D u = π d( J ) u for the projection to the vertical tangent directions of the derivative at u (i.e. project to the fibre of the vector bundle E). This satisfies the following fundamental elliptic inequality. Theorem: ξ L k,p C ( D u ξ L k 1,p + ξ L 1 )
4 To see the power of this, recall a key fact from analysis: on a 2n-manifold L k,p ct C r if 1 p < k r 2n That inclusion is continuous (bounded!) is the Sobolev embedding theorem: that inclusion is a compact operator is the Rellich theorem. Definition: a continuous linear operator P : U V of (separable) Banach spaces is Fredholm if ker P and coker P are finite dimensional, and P has closed range. In this case, the index ind(p ) = dim(ker P ) dim(coker P ). A classical fact says that the index defines a map π 0 (F red(u, V )) Z so is invariant under continuous deformation. The Sard-Smale theorem says that if P : E F is a smooth map of Banach manifolds, and if s P 1 (y) the linearisation dp s is Fredholm and surjective, then P 1 (y) is a smooth finite dimensional manifold, T s (P 1 (y)) = ker(dp s ), of dimension ind(p ). We apply this to J.
5 Proposition: The linearisation D u is Fredholm. Proof: Fix k > p > 2. If D u ξ = 0 the elliptic inequality says ξ L k,p is bounded for all k, p, so Sobolev embedding says ξ is C r for all r, hence smooth. Let B = {ξ ker D u ξ L k,p 1}. Since the inclusion L k,p C 3 is compact, B lies in a compact subset of C 3, hence its closure B in C 3 is compact. Now D u : L k,p C 0 is certainly continuous, so D u ξ = 0 for ξ B D u ξ = 0 ξ B. Thus B ker D u L k,p, and so the unit ball in ker D u is compact as a topological space; but the only Banach spaces for which this is true are finite-dimensional. The proof for the cokernel is similar, but uses the (elliptic inequality for the) formal adjoint D u instead of D u. We omit the argument for the cokernel but make a definition: J is regular if D u is onto for every u 1 J (0).
6 Corollary: Fix (X, ω, J) and A H 2 (X; Z). Consider holomorphic spheres representing the class A. If J is regular, M J (X; A) = 1 J (0) is smooth of (real) dimension 2n + 2 c 1 (X), A. Proof: only the value for the index is new. But in the space of Fredholm operators, we can always join D u to an honest complex linear operator. Indeed, given a solution u : C R 2n to the local equation s u + J(u) t u = 0 and a vector field ξ : C R 2n along u, we can differentiate the LHS in direction ξ: D u ξ = η ds J(u)η dt with 2η = s ξ+j(u) t ξ+ ξ J(u) t u. We therefore see that D u ξ = J ξ + R for a zero-th order perturbation term R; perturbations don t change the index, so ind(d u ) = ind( J ) is the index of a C -linear operator in the holomorphic bundle u T M Σ. The result now follows from Riemann-Roch; for genus g curves we d get n(2 2g) + 2 c 1, A.
7 Three caveats accompany these local analyses: (i) We haven t yet shown that regular J exist; (ii) Even if regular J exist, the thing that s smooth is the moduli space of L k,p -curves which are holomorphic, but really we re interested in smooth solutions to the equation; (iii) and worst of all, nothing so far says any of these spaces M J (X; A) are nonempty. We re developing a deformation theory for J-curves, and existence questions are a separate issue! The second of these concerns is a general analytical issue, resolved by elliptic bootstrapping : the proof is too analytical for here. Proposition: if J u = 0 and J C r then u C r, in particular if J is smooth then u is smooth. The upshot is that though we widen our framework to Banach spaces, so we can use Sard- Smale, in the end the objects we construct really will be smooth: strangely, to show spaces of smooth J-curves are smooth, one has to enter the world of L k,p -curves.
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