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Removal of Singularites for Stein Manifolds

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Removal of Singularites for Stein Manifolds Removal of Singularities for Stein Manifolds Undergraduate Honors Thesis in Mathematics Luis Kumanduri Abstract We adapt the technique of removal of singularities to the holomorphic setting and prove a general flexibility result for holomorphic vector bundles over Stein manifolds. If D : V ! W is an elliptic differential operator between holomorphic vector bundles over a Stein Manifold, then a q-tuple (θ1; : : : ; θq) of holomorphic sections generating W may be deformed to an exact holomorphic q-tuple (Dφ1; : : : ; Dφq) generating W . We also prove a parametric version of this theorem with holomorphic dependence on a Stein parameter X and obtain a 1-parametric h-principle. The parametric h- principle works relative to closed complex analytic subsets A of X. As corollaries we will obtain h-principles for holomorphic immersions and free maps of Stein Manifolds. Contents 1 Introduction 2 2 Jets and the h-principle 4 2.1 Jet Bundles and Transversality . .4 2.2 Differential Relations and the h-principle . .6 3 Geometry of Several Complex Variables 9 3.1 Basics of Several Complex Variables . .9 3.2 Stein Manifolds . 12 3.3 Cartan's Theorems . 16 4 Holomorphic Transversality 20 5 Main Results 24 5.1 Proof of Main Theorem . 24 5.2 Parametric h-principle . 29 5.3 Applications . 31 1 1 Introduction Problems in smooth geometry are often subject to a partial differential inequality or rela- tion that satisfies an h-principle. For any differential relation, there is a notion of a formal solution where the derivatives are replaced with algebraic relations. In situations where formal solutions can be deformed into actual solutions, we say that a problem satisfies an h-principle. More precisely, we can study the inclusion of the space of genuine solutions into the space of formal solutions and ask what properties this map has, e.g. if it induces an isomorphism on π0 or if it is a homotopy equivalence. For example, an immersion is a smooth map f with injective df. We can consider a formal immersion to be a pair (f; g) where f is a smooth map and g is a tangent bundle monomorphism covering f, not necessarily equal to df. Under some mild conditions it turns out that formal immersions can be deformed into actual immersions, which is to say the problem of finding immersions satisfies an h-principle. There are several powerful methods, introduced by Gromov and Eliashberg, for proving h-principles. The one we will focus on in this thesis is known as the technique of removal of singularities (introduced in [3]). It can be used to prove the h-principle for immersions, among others. The goal of this thesis is to adapt this technique to a holomorphic setting and use it to prove a variety of h-principles there. In holomorphic geometry there is much more rigidity than seen in the smooth case. Let M be a compact complex manifold. Then all holomorphic maps M ! Cq are constants by the maximum principle. In particular, M admits no immersions to complex Euclidean space. Even so, it is natural to ask if there are any examples of complex manifolds which admit holomorphic flexibility properties similar to those seen in smooth geometry. It turns out the answer is yes, there is an important class of complex manifolds known as Stein manifolds for which much of the theory carries over, which we will discuss in section 3. The main result of this thesis is the following Theorem 1.1. Let M be a Stein manifold, and let V and W be holomorphic vector bundles over M. Let D : A(V ) !A(W ) be an elliptic differential operator of order s from the holomorphic sections of V to the holomorphic sections of W . Let θ1; : : : ; θq be holomorphic sections of W that span the fibers at every point so that q is greater than the rank of W . Then the q-tuple (θ1; : : : ; θq) is homotopic through q-tuples generating W to a q-tuple of exact holomorphic sections (Dφ1; : : : ; Dφq) with the same spanning property. We will also prove the following parametric version of the theorem in section 5.2. Theorem 1.2. Let Sec(Vq(W )) denote the holomorphic sections of q-tuples spanning W . Let X be a Stein manifold, and let Θ(x) = (θ1; : : : ; θq)(x) be a q-tuple of sections spanning W with holomorphic dependence on a parameter x 2 X. Then there is a homotopy H : X × I ! Sec(Vq(W )) such that H(x; 0) = Θ(x), each q-tuple H(x; 1) is exact, and each family H(·; t) has a holomorphic dependence on X. Furthermore, if A ⊂ X is a closed complex analytic subset of X such that Θ is exact on A, then H may be chosen to be fixed on A, 2 These theorems are very general, but give many geometrically interesting examples of h-principles as corollaries. Let V = M × C, where sections are just holomorphic functions, and let W = T ∗M be the complex cotangent bundle and D be the exterior derivative d. Theorem 1.1 then states that if q > dim(M), a q-tuple of holomorphic 1-forms generat- ing the cotangent bundle can be deformed into a q-tuple of exact holomorphic 1-forms q (df1; : : : ; dfq). By taking the map (f1; : : : ; fq), we obtain an immersion M ! C . The parametric version of this theorem gives a holomorphic analog of the classical Smale-Hirsch theorem. Another example is the h-principle for holomorphic free maps. A map f : M ! Cq is said to be free if all of the first and second complex derivatives of f are linearly independent. Consider the bundle W = J 2(M; C)=C, where we take second order jets and forget the constant term. Given a map g : M ! C, there is an operator D that takes j2g and forgets the value of g. Writing f = (f1; : : : ; fq), a free map is exactly one for which the sections Df1; : : : ; Dfq generate W . Thus the main theorem applies, and gives an h-principle for holomorphic free maps of Stein manifolds. A smooth analog of Theorem 1.1 was stated and proven in [3]. A similar result to Theorem 1.1 was stated in [4] with only a brief sketch of the proof and no technical details, we give the first proof here as well as the first statement and proof of the parametric h-principle in this setting. We will present necessary background material as well as tools needed in the proof in sections 2 and 3 of this thesis. In section 4, we discuss the holomorphic jet transversality theorem, and we adapt the transversality theorem to the specfic situations we need to prove the main theorem. In section 5.1 we give a detailed proof of Theorem 1.1, in 5.2 we will prove Theorem 1.2, as well as a proof of the 1-parametric h-principle, and in 5.3 we will discuss applications to holomorphic immersions and free maps of Stein manifolds into Cq. 3 2 Jets and the h-principle This section contains background material and language needed to discuss and prove h- principles. It will also include some examples and results, without proofs, that are precur- sors of the main results in the holomorphic setting. A good and accessible reference for the material in this section is the book [6] of Eliashberg and Mishachev. A more advanced reference is the monograph [9]. 2.1 Jet Bundles and Transversality 2.1.1 Jet Bundles We carry out the construction of jet bundles in the case of smooth fibrations. Later on we will need these in the holomorphic setting, but the construction goes through in exactly the same manner and we will not repeat it. Given a smooth map f : Rn ! Rq, its s-jet n captures all derivatives up to order s. Let f = (f1; : : : ; fq) and let α = (α1; : : : ; αn) 2 N P s range over all tuples such that 0 ≤ jαj = αi ≤ s. Then j f(x) is the tuple of all partial derivatives jαj α @ fi @ fi = (x) α1 αn @x1 ··· @xn We can consider the manifold J s(Rn; Rq) = Rn ×Rc(n;s;q) to be the bundle of s-jets over Rn, where c(n; s; q) is the number of all partial derivatives up to order s. We can compute s+n c(n; s; q) = q n by a simple combinatorial argument. Notice that this is the dimension of the space P s(n; q) of polynomials from Rn ! Rq, and there is a parametrization n s ∼ s n q s R × P (n; q) = J R ; R ):(x; f) ! (x; j f(x)) In order to define the jet space of an arbitrary smooth fiber bundle, we will first want to give an invariant definition of s-jets. We say that two smooth functions f and g are s-tangent at x if jsf(x) = jsg(x). By the chain rule in local coordinates, s-tangency is independent of the choice of coordinates. The jet space J s(Rn; Rq) is then the set of s- tangency classes of maps Rn ! Rq. Local coordinates on Rn and Rq give local coordinate charts on J s(Rn; Rq) which give it a natural topology and smooth structure so that the natural map J s(Rn; Rq) ! Rn is a smooth fibration. If X ! M is any smooth fiber bundle, we can define the s-jet space X(s) ! M in the same way by defining s-tangency of local sections M ! X in local coordinates.
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