Several Complex Variables and Applications IME-USP, Spring 2017

Several Complex Variables and Applications IME-USP, Spring 2017

Peter Hazard

Lecture 1 – Course Overview.

Some Problems.

Roughly speaking, we wish to understand when, given two objects in Cⁿ which are equivalent in some sense meaning there is a bijection from one to the other living in some category when is there an extension of this bijection, living in the same category, to the ambient space Cⁿ? More precisely, we formulate our problems as follows.

Problem A: Let Ω ⊂ Cⁿ be an open domain, M^m a complex manifold of dimension m≤n.

(i) Given holomorphic s0, s1: Ω → C, s⁻0¹(0) ≃ s⁻1¹(0), when does there exist a biholomorphism G: Ω→Cⁿ such that G(s⁻1¹(0)) =s⁻0¹(0)?

(ii) Given holomorphic embeddings f0, f1: M →Cⁿ, when does there exist a biholomorphism G: Ω→Cⁿ such that G◦f1 =f0?

(iii) Given M0, M1 ⊂ Ω complex submanifolds, M0 ≃ M1, when does there exist a biholomorphism G: Ω→Cⁿ such that G(M1) =M0?

Here, the equivalence between zero sets or complex submanifolds will typi- cally mean there is an (intrinsic) biholomorphism between these space. (Though we must be careful, in the case of singularities, with defining what this means.)

There are also the associated problems for isotopies. First, we must set up some terminology. Fix a positive integer k. By a C^k- 1-parameter family of mapswe meanφ: Ω×[0,1]→Cⁿwhich isC^k-smooth in the second variable.

We will use the notation φt(z) = φ(z, t) throughout. By a C^k- 1-parameter family of submanifolds, we will mean that the local parametrizations are C^k-smooth in the second variable. When k is fixed or can be chosen arbi- trarily we will also refer to such objects as 1-parameter families of maps and submanifolds.

Problem B: Let Ω and M^m be as in Problem A.

(i) Given a 1-parameter family of holomorphic st: Ω ⊆ Cⁿ → C, t ∈ [0,1], when does there exist a 1-parameter family of biholomorphisms Gt: Ω →Cⁿ such that G(s⁻_t¹(0)) =s⁻₀¹(0) for all t?

(ii) Given a 1-parameter family of holomorphic embeddings f_t: M → Cⁿ when does there exist a 1-parameter family of biholomorphismsG: Ω→ Cⁿ such that Gt◦ft=f0?

(iii) Given a 1-parameter family of complex submanifoldsMt ⊂Ω, M0 ≃Mt

for allt, when does there exist a 1-parameter family of biholomorphisms Gt: Ω →Cⁿ such that Gt(Mt) =M0, for all t?

Obviously, in Problem A, if there exists a (non-ambient) isotopy st be- tweens0 ands1 (resp. ftbetweenf0 and f1 orMtbetweenM0 andM1) then a solution to Problem B automatically gives a solution to Problem A.

Results in the Smooth Category.

Note that, in the smooth category, Problem B(iii) has a solution. This is the Isotopy Extension Theorem (see [3]):

Theorem 1 (v.1). Let M^m be a C^∞-smooth manifold, V^v ⊆ M^m a C^∞- smooth compact submanifold. Let F: V ×[0,1] → M be a C^∞-isotopy. If F(V ×[0,1])⊆M \bM then F extends to a C^∞-ambient isotopy of M. Theorem 2 (v.2). Let M be a C^∞-smooth manifold, U ⊆ M open and A⊂U compact. LetF: U×[0,1]→M be a C^∞-isotopy, such that the track Fˆ(U ×[0,1]) ⊂ M ×[0,1] is open. Then there exists an ambient isotopy G of M, with compact support, extending F in a neighbourhood of A×[0,1].

Recall that thetrack of an isotopyF: V ×[0,1]→M is the map ˆF: V × [0,1] → M ×[0,1] given by ˆF(z, t) = (Ft(z), t). Observe that the track ˆF is level-preserving. (Ex: show that every level-preserving embedding is the track of some isotopy.) We will also call the set

Vˆ = ˆF(V ×[0,1])⊂M ×[0,1] (1) the traceof V with respect to the track ˆF.

Remark 1. The requirement thatV orAbe compact can be replaced with the weaker property thatF has bounded velocity: the isotopyF: V ×[0,1]→M has bounded velocity if, for some complete Riemannian metric on M, and some positive real number B, we have

k∂_tF(z, t)k< B ∀z ∈V, ∀t∈[0,1] (2) (If the velocity is unbounded, then the corresponding time-dependent vector field defined on the trace of V under F is not Lipschitz. Consequently, we don’t necessarily have uniqueness of solutions for the corresponding ODE.)

Remark 2. In Theorem 2, the idea is that we start with a smooth isotopy on an arbitrary set A in M. However, this makes no sense, unless A is a sub- manifold. This explains why the neighbourhood U is required. (However, one could consider a homotopy endowed with some form of jet, although un- der suitable hypotheses this would be equivalent to having a smooth isotopy defined on some neighbourhood.)

Sketch of Proof of Theorem 1. Let n = m−v, the codimension of V in M. Let ˆV denote the trace of V under ˆF. Then the codimension of ˆV in ˆM = M ×[0,1] is also n. The track ˆF determines a vector field on the trace ˆV of the form

X( ˆˆ F(z, t)) = (X(Ft(z), t),1) ∀(z, t)∈V ×[0,1] (3) Observe that Xt(z) =X(Ft(z), t) is a time-dependent vector field (partially defined) onM. For some positiveǫsufficiently small, take anǫ-tubular neigh- bourhood N of ˆV in ˆM. Denote the corresponding tubular neighbourhood map by

τǫ: ˆV ×Bⁿ(ǫ)→ Nǫ (4) whereBⁿ(ǫ) denotes theǫ-ball inRⁿ. By the tubular neighbourhood theorem, this is a diffeomorphism ifǫ is sufficiently small. Via this diffeomorphism we transfer all objects to act on ˆV ×Bⁿ(ǫ). Let Y0 = (τǫ)^∗( ˆX), T = (τǫ)^∗(∂t) and η = (τǫ)^∗(dt). Observe that we have a splitting of the form

Y0(z, t,0) = Y0^⊥(z, t,0) +Y0^k(z, t,0) ∀(z, t)∈Vˆ (5) where η(Y0^⊥) = 0 and η(Y0^k) = η(Y0). First, let us (trivially) extend Y0 to Vˆ×Bⁿ(ǫ). Take a finite good open cover of ˆV by charts{(U_α, ϕ_α)}. Observe that, since ˆV ×Bⁿ(ǫ) is a trivial bundle, these correspond to a cover by local trivialisations {(U_α^′, ϕ^′_α)}={(Uα×Bⁿ(ǫ), ϕα×id)}of the bundle ˆV ×Bⁿ(ǫ).

Take a partition of unity {φ_α}subordinate to the cover {U_α}. Define Y0(z, t, s) =X

α

φα(z, t)Y0(z, t,0) ∀(z, t)∈V ,ˆ ∀s ∈Bⁿ(ǫ) (6) (What we mean by this slightly ambiguous notation is that, in each local triv- ialisation, we define Y0 by (parallel) translating Y0(z, t,0) along each fibre.) Observe that we can also extend the splitting

Y0(z, t, s) =Y0^⊥(z, t, s) +Y0^k(z, t, s) ∀(z, t)∈V ,ˆ ∀s ∈Bⁿ(ǫ) (7) whereη(Y0^⊥) = 0 andη(Y0^k) =η(Y0). Secondly, we restrictY0to a neighbour- hood of the zero section ˆV × {0} so that η(Y0) > ¹₂ in this neighbourhood.

More precisely, since η(Y0(z, t,0)) = 1, there exists a positive ǫ1 such that η(Y0(z, t, s))> ¹₂ whenever|s|< ǫ1. We modifyY0 on this neighbourhood as follows: Let β: [0, ǫ1] → R be a smooth bump function so that β ≡ 1 in a neighbourhood of 0 and β ≡0 in a neighbourhood of ǫ1. Define

Y1(z, t, s) =β(s)Y0^⊥(z, t, s) +Y0^k(z, t, s) ∀(z, t)∈V ,ˆ ∀s∈Bⁿ(ǫ1) (8) (Again, this notation is shorthand: we actually consider these expression in local trivialisations and using the partition of unity to glue.) Then in a neighbourhood of the boundary of Bⁿ(ǫ1), Y1 is parallel to T. Moreover, Y1^k =Y0^k is nowhere vanishing in ˆV×Bⁿ(ǫ1). Therefore there exists a smooth function κ: ˆV ×Bⁿ(ǫ1)→R₊ so that

Y2(z, t, s) = κ(z, t, s)Y1(z, t, s) ∀(z, t)∈V ,ˆ ∀s∈Bⁿ(ǫ1) (9) satisfies η(Y2) ≡ 1. (Once more, this notation is shorthand.) Thus if we define ˆX2 =τ∗(Y2) then ˆX2 is a smooth vector field on a neighbourhood of ˆV in ˆM which (i) extends ˆX smoothly, (ii) coincides with∂_tin a neighbourhood of the boundary of this neighbourhood. Hence we may extend ˆX2 smoothly to a vector field on ˆM, which we also denote by ˆX2, by defining it equal to

∂_t outside of this neighbourhood.

Finally, dt( ˆX2) =η(Y2) = 1, which implies that ˆX2 has the form (X2,1), for some time-dependent vector fieldX2 onM. Therefore, upon integrating, the corresponding flow is level-preserving, and thus, as was noted above, is therefore the track of some isotopy on M.

Results in the Holomorphic Category.

The above suggests a general approach to solving Problems A and B in the smooth category. Yet, we are interested in the (more difficult) holomorphic case. In this case we will find that there are additional obstructions. However, by weakening the requirements so that we only consider approximations we may still get some positive results. A large part of the course will consist in understanding results of the following type.

Theorem 3 (Forstneriˇc-Rosay [2]). 1. Let M0, M1 ⊂Cⁿ be compact real- analytic submanifolds (with or without boundary). AssumeM0 is totally real and polynomially convex.

Then M0 and M1 are Cⁿ-equivalent if and only if they are isotopic in Cⁿ through a family of totally real, polynomially convex submanifolds Mt.

2. Let M^m be a compact real-analytic manifold, m≤n. Let f0, f1: M → Cⁿ be real analytic embeddings. Assume that f0(M) is totally real and polynomially convex.

Then f0 and f1 are Cⁿ-equivalent if and only if there is an isotopy of real-analytic embeddingsf_t: M →Cⁿsuch that, for allt∈[0,1], f_t(M) is totally real and polynomially convex in Cⁿ.

Submanifolds M0, M1 ⊂ Cⁿ are Cⁿ-equivalent if there exists a sequence fj ∈ Aut(Cⁿ) and a neighbourhood Ω of M0 is Cⁿ, such that fj → f uni- formly on compact subsets of Ω, where f: Ω → f(Ω) is a biholomorphism onto its image with the property that f(M0) =M1.

Embeddings f0, f1: M →Cⁿ are Cⁿ-equivalent if there exists a sequence F_j ∈ Aut(Cⁿ) and a neighbourhood Ω of f0(M) in Cⁿ, such that F_j → F uniformly on compact subsets of Ω, whereF: Ω →F(Ω) is a biholomorphism onto its image with the property that F ◦f0 =f1.

A submanifold M ⊆Cⁿ is totally realif for allz ∈M, the tangent space TzM contains no complex lines.

A compact set K ⊆ Ω is polynomially convex in Ω if for all z0 ∈ Ω\K there exists a holomorphic polynomial P such that |P(z0)|>sup_K|P(z)|.

Remark 3. Thus, one possible approach to solving Problem B is to use the same strategy used in the proof of the Isotopy Extension Theorem. However, if we restrict attention to the case when the ambient space is Cⁿ,

(i) We must consider the problem on the extension of vector fields which arecomplete, i.e. which have solutions for all timet∈[0,1]. A stronger concept is the following:

LetXbe a holomorphic vector field onCⁿwhereX = (X1, X2, . . . , Xn), Xj entire for all j = 1,2, . . . , n. X is complete if and only if, for all z ∈Cⁿ, the initial value problem

dR

dt =X(R(t)), R(0) =z (10)

can be integrated to a solution R defined for all time−∞< t <∞.

(ii) Boundedness of velocity won’t hold generally in the holomorphic case in Cⁿ, by Liouville’s theorem (extended to the multi-variable case).

(iii) Holomorphic partitions of unity do not exist.

PeH -2017/04/06- (last updated: 2017/04/25)

References

[1] L.V. Ahlfors. Complex Analysis (Second Edition). International Series in Pure and Applied Mathematics, McGraw-Hill, (1966).

[2] F. Forstneriˇc and J.-P. Rosay. Approximation of biholomorphic map- pings by automorphisms of Cⁿ. Invent. Math., 112, (1993), 323–349.

[3] M. Hirsch. Differential Topology. Graduate Texts in Mathematics, 33, Springer-Verlag, (1976).

[4] L. H¨ormander. An Introduction to Complex Analysis in Several Vari- ables. North-Holland Mathematical Library, vol. 7, North Holland Pub- lishing Company, (1973).

[5] V.S. Vladmimirov. Methods of the Theory of Functions of Many Com- plex Variables. The M.I.T. Press, (1966).

[6] H. Whitney. Complex Analytic Varieties. Addison-Wesley series in mathematics, Addison-Wesley Pub. Co., (1972).