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A strictly pseudoconvex boundary point has a local holomorphic separator

Statement

Assume the Axiom of Choice (AC). Let $D \subseteq C^{n}$ , $n \geq 1$ , be a domain and let $p \in \partial D$ be a boundary point such that $\partial D$ is of class $C^{2}$ near $p$ and strongly pseudoconvex at $p$ . In this item, $z_{j}$ denotes the canonical coordinate $z_{j - 1}$ for $1 \leq j \leq n$ , with the same relabeling for derivatives and form coefficients. There are a neighbourhood $U$ of $p$ and a $C^{2}$ function $ρ : U \to R$ with $D \cap U = {z \in U : ρ (z) < 0}, d ρ (p) \neq 0,$ and $L_{ρ} (p; v) > 0 for every v \neq 0 with \sum_{j = 1}^{n} \frac{\partial ρ}{\partial z_{j}} (p) v_{j} = 0.$

Then there are a neighbourhood $V \subseteq U$ of $p$ and a holomorphic function $h : V \to C$ such that $h (p) = 0 and Re h (z) < 0 for every z \in (D \cap V) ∖ {p} .$

In particular $h$ has no zero on $(D \cap V) ∖ {p}$ . Moreover the separator is quantitative in suitable coordinates: there are a holomorphic chart $w$ centred at $p$ , a radius $δ > 0$ and a constant $c > 0$ such that $h = w_{n}$ in this chart and $Re w_{n} \leq - c ∣ w ∣^{2}$ at every point of $D$ corresponding to $∣ w ∣ < δ$ .

Facts & Assumptions

Given: The Axiom of Choice; a domain $D \subseteq C^{n}$ ; a boundary point $p \in \partial D$ with $C^{2}$ boundary near $p$ ; a $C^{2}$ defining function $ρ$ on a neighbourhood $U$ of $p$ with $D \cap U = {ρ < 0}$ , $d ρ (p) \neq 0$ , and $L_{ρ} (p; v) > 0$ for every nonzero complex tangent vector $v$ at $p$ .

[F1]

For $u \in C^{2}$ open set and $a$ , $v$ as above, the Levi form is $L_{u} (a; v) : = \sum_{j, k} \frac{\partial^{2} u}{\partial z_{j} \partial {\overset{ˉ}{z}}_{k}} (a) v_{j} \overline{v_{k}},$ and $u$ is strictly plurisubharmonic when $L_{u} (a; v) > 0$ for all $a$ and all $v \neq 0$ (The Levi form and strict plurisubharmonicity).

[F2]

With $ρ$ a $C^{2}$ defining function of a domain on a neighbourhood of a boundary point $p$ , the complex tangent vectors at $p$ are those with $\sum_{j} ρ_{z_{j}} (p) v_{j} = 0$ , and the Levi form is evaluated on that subspace (Levi pseudoconvex domains).

[F3]

Two $C^{2}$ defining functions near the same boundary point have Levi forms on complex tangent vectors differing by a positive scalar factor; in particular the strict positivity demanded at $p$ does not depend on the choice of $ρ$ (Levi pseudoconvexity does not depend on the defining function).

[F4]

For a $C^{2}$ scalar field near $a$ , $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} ⟨ H_{f} (a) h, h ⟩ + o (∥ h ∥^{2})$ (Second-order Taylor expansion $f (a + h) = f (a) + \nabla f (a) \cdot h + \frac{1}{2} h^{T} H_{f} (a) h + o (∥ h ∥^{2})$ ).

[F5]

The Wirtinger operators in several variables satisfy $D f (a) h = \sum_{k} ((\partial_{z_{k}} f (a)) h_{k} + (\partial_{{\overset{ˉ}{z}}_{k}} f (a)) \overline{h_{k}})$ for real totally differentiable $f$ , and $ρ$ real-valued is recovered from its Wirtinger partials by this identity (Wirtinger operators in $C^{m}$ ).

[F6]

AC states that every family of nonempty sets has a choice function (The Axiom of Choice).

Choice use. AC is the ambient hypothesis stated in the lemma. The normalization, the multiplication by the positive function $g$ , the choice of the polynomial $q$ and the final pullback are all explicit formulas, so neither [F6] nor any weaker selection principle is consumed by the construction itself; the cited suppliers are used as stated.

Proof

technique · direct

1.1F2F5F6given

With $U$ and $ρ$ as given, the only rephrasing needed is the description of the complex tangent space: by [F2] the complex tangent vectors at $p$ form the kernel of the $C$ -linear form $ℓ (v) : = \sum_{a} ρ_{z_{a}} (p) v_{a}$ , and $ℓ \neq 0$ because $d ρ (p) \neq 0$ (if all Wirtinger partials of $ρ$ vanished at $p$ , then $D ρ (p) = 0$ by [F5]); relabel the indices so that $ρ_{z_{n}} (p) \neq 0$ , so that $\ker ℓ$ has complex dimension $n - 1$ and the hypothesis of the statement says that $\sum_{a, b} ρ_{z_{a} {\overset{ˉ}{z}}_{b}} (p) v_{a} \overline{v_{b}} > 0$ for every $0 \neq v \in \ker ℓ$ . The Axiom of Choice [F6] is the ambient hypothesis of the statement, and this step selects nothing.

2.1F4F5step 1.1algebra

Expansion of $ρ$ at $p$ in complex notation: applying the second-order Taylor expansion [F4] to the $C^{2}$ function $ρ$ and rewriting its linear and quadratic terms with the differential identity of [F5] (for the linear term $D ρ (p) h = \sum_{a} (ρ_{z_{a}} (p) h_{a} + ρ_{{\overset{ˉ}{z}}_{a}} (p) \overline{h_{a}}) = 2 Re ℓ (h)$ , because $ρ$ is real; for the quadratic term, substituting the real coordinates $ξ_{a} = (h_{a} + \overline{h_{a}}) / 2$ and $η_{a} = (h_{a} - \overline{h_{a}}) / (2 i)$ into the real Hessian form and collecting the $h_{a} h_{b}$ , $h_{a} \overline{h_{b}}$ , $\overline{h_{a}} \overline{h_{b}}$ terms), one obtains with $ℓ (h) : = \sum_{a} ρ_{z_{a}} (p) h_{a}$ , $A (h) : = \sum_{a, b} ρ_{z_{a} z_{b}} (p) h_{a} h_{b}$ and $Q_{0} (h) : = \sum_{a, b} ρ_{z_{a} {\overset{ˉ}{z}}_{b}} (p) h_{a} \overline{h_{b}}$ the expansion $ρ (p + h) = 2 Re ℓ (h) + Re A (h) + Q_{0} (h) + o (∣ h ∣^{2})$ as $h \to 0$ .

3.1F1F2F3step 1.1step 2.1algebra

First normalization: define the holomorphic affine map $Φ$ by $Φ (ζ)_{a} : = p_{a} + ζ_{a}$ for $1 \leq a < n$ and $Φ (ζ)_{n} : = p_{n} + c (ζ_{n} - 2 \sum_{1 \leq a < n} ρ_{z_{a}} (p) ζ_{a})$ with $c : = (2 ρ_{z_{n}} (p))^{- 1}$ , so that $ℓ (Φ (ζ) - p) = ζ_{n} / 2$ and the Jacobian of $Φ$ is triangular with diagonal entries $1$ and $c \neq 0$ ; shrinking $U$ makes $Φ$ a biholomorphism onto a neighbourhood of $0$ , and $ρ_{1} : = ρ \circ Φ$ is a $C^{2}$ defining function of $D_{1} : = Φ^{- 1} (D \cap U)$ near $0$ with expansion $ρ_{1} (ζ) = Re ζ_{n} + Q_{1} (ζ) + Re A_{1} (ζ) + o (∣ ζ ∣^{2})$ from step 2.1, where $Q_{1} (ζ) = \sum_{a, b} ρ_{1, ζ_{a} {\overset{ˉ}{ζ}}_{b}} (0) ζ_{a} \overline{ζ_{b}}$ and $A_{1} (ζ) = \sum_{a, b} ρ_{1, ζ_{a} ζ_{b}} (0) ζ_{a} ζ_{b}$ . The hypothesis survives: $ℓ_{1} (w) : = \sum_{a} ρ_{1, ζ_{a}} (0) w_{a}$ equals $ℓ (L w) = w_{n} / 2$ for the linear part $L$ of $Φ$ , and for $v \neq 0$ with $ℓ_{1} (v) = 0$ the chain rule gives $L v \in \ker ℓ ∖ {0}$ together with $L_{ρ_{1}} (0; v) = L_{ρ} (p; L v) > 0$ by step 1.1.

4.1F1step 3.1algebra

Multiplication by a positive function: for $t > 0$ put $g_{t} (ζ) : = 1 + t Re ζ_{n}$ and $ρ_{t} : = g_{t} ρ_{1}$ , a $C^{2}$ defining function of the same domain near $0$ with $g_{t} (0) = 1$ and $d ρ_{t} (0) = d ρ_{1} (0) \neq 0$ . Writing $ℓ_{1} (ζ) = ζ_{n} / 2$ and $L_{t} : = t ℓ_{1}$ and using the identity $2 Re (X) 2 Re (Y) = 2 Re (X \overline{Y}) + 2 Re (X Y)$ , multiplication of the expansion of step 3.1 by $g_{t} = 1 + 2 Re L_{t}$ gives $ρ_{t} (ζ) = 2 Re [ℓ_{1} (ζ) + \frac{1}{2} A_{1} (ζ) + L_{t} (ζ) ℓ_{1} (ζ)] + [Q_{1} (ζ) + 2 Re (L_{t} (ζ) \overline{ℓ_{1} (ζ)})] + o (∣ ζ ∣^{2})$ , the $O (∣ ζ ∣^{3})$ terms of $2 Re (L_{t}) (Re A_{1} + Q_{1})$ having been absorbed into $o (∣ ζ ∣^{2})$ ; thus the holomorphic quadratic part of $ρ_{t}$ is $B_{t} : = ζ_{n} / 2 + A_{1} / 2 + t ζ_{n}^{2} / 4$ and its Hermitian quadratic part is the Hermitian form $H_{t} (v) : = Q_{1} (v) + (t / 2) ∣ v_{n} ∣^{2}$ in the variable $v$ .

5.1step 1.1step 4.1algebra

$H_{t}$ is positive definite for all large $t$ : the hypothesis of step 1.1 says $Q_{1} (v) > 0$ for $0 \neq v$ with $v_{n} = 0$ , so on the hyperplane $E : = {v_{n} = 0}$ there is $δ > 0$ with $Q_{1} (v) \geq δ ∣ v ∣^{2}$ , while the Hermitian form $Q_{1}$ satisfies $∣ Q_{1} (v, w) ∣ \leq C ∣ v ∣ ∣ w ∣$ for $v, w \in E$ with a constant $C$ independent of $t$ . Decomposing $v = v^{'} + λ e_{n}$ with $v^{'} \in E$ and $λ = v_{n}$ gives $H_{t} (v) = Q_{1} (v^{'}) + 2 Re Q_{1} (v^{'}, λ e_{n}) + ∣ λ ∣^{2} Q_{1} (e_{n}) + (t / 2) ∣ λ ∣^{2} \geq δ ∣ v^{'} ∣^{2} - 2 C ∣ v^{'} ∣ ∣ λ ∣ + (Q_{1} (e_{n}) + t / 2) ∣ λ ∣^{2}$ , and $2 C ∣ v^{'} ∣ ∣ λ ∣ \leq (δ / 2) ∣ v^{'} ∣^{2} + (2 C^{2} / δ) ∣ λ ∣^{2}$ because $(\sqrt{δ / 2} ∣ v^{'} ∣ - \sqrt{2 / δ} C ∣ λ ∣)^{2} \geq 0$ . Choosing $t > 0$ with $Q_{1} (e_{n}) + t / 2 \geq 1 + 2 C^{2} / δ$ therefore gives $H_{t} (v) \geq (δ / 2) ∣ v^{'} ∣^{2} + ∣ λ ∣^{2} \geq c ∣ v ∣^{2}$ for all $v$ , where $c : = \min (δ / 2, 1) > 0$ and $∣ v ∣^{2} = ∣ v^{'} ∣^{2} + ∣ λ ∣^{2}$ ; fix such a $t$ and write $ρ_{t}$ and $H$ for $ρ_{t}$ and $H_{t}$ .

6.1step 4.1step 5.1algebra

Killing the holomorphic quadratic part: let $q (ζ) : = - A_{1} (ζ) - (t / 2) ζ_{n}^{2}$ and define the holomorphic polynomial map $Φ_{2} (w) : = w + q (w) e_{n}$ , which fixes the first $n - 1$ coordinates and sends $w_{n}$ to $w_{n} + q (w)$ ; since $D Φ_{2} (0) = I$ it is a local biholomorphism fixing $0$ , and with $\hat{ρ} : = ρ_{t} \circ Φ_{2}$ one computes for $z = Φ_{2} (w)$ that $ℓ_{1} (w + q (w) e_{n}) = w_{n} / 2 + q (w) / 2$ and $A_{1} (w + q (w) e_{n}) = A_{1} (w) + O (∣ w ∣^{3})$ , hence $2 Re B_{t} (Φ_{2} (w)) = Re w_{n} + Re [q (w) + A_{1} (w) + (t / 2) w_{n}^{2}] + O (∣ w ∣^{3}) = Re w_{n} + O (∣ w ∣^{3})$ , while $H (Φ_{2} (w)) = H (w) + O (∣ w ∣^{3})$ because $H$ is quadratic; therefore $\hat{ρ} (w) = Re w_{n} + H (w) + o (∣ w ∣^{2})$ , and $\hat{ρ}$ is a $C^{2}$ defining function near $0$ of the image of $D$ under the change of coordinates.

7.1step 5.1step 6.1algebra∎

Conclusion: since $\hat{ρ} (w) - Re w_{n} - H (w) = o (∣ w ∣^{2})$ , after shrinking the ball to a radius $δ > 0$ on which $Φ_{2}$ is biholomorphic and $∣ \hat{ρ} (w) - Re w_{n} - H (w) ∣ \leq (c / 2) ∣ w ∣^{2}$ , every $w$ with $∣ w ∣ < δ$ , $w \neq 0$ and $\hat{ρ} (w) < 0$ satisfies $Re w_{n} = \hat{ρ} (w) - H (w) - (\hat{ρ} (w) - Re w_{n} - H (w)) \leq 0 - c ∣ w ∣^{2} + (c / 2) ∣ w ∣^{2} = - (c / 2) ∣ w ∣^{2} < 0$ ; define $V : = Φ (Φ_{2} (B (0, δ))) \subseteq U$ and $h (z) : = w_{n} (z)$ where $w = Φ_{2}^{- 1} (Φ^{- 1} (z))$ is the inverse chart, so that $h$ is holomorphic on $V$ , $h (p) = 0$ , and $Re h < 0$ on $(D \cap V) ∖ {p}$ because those points correspond exactly to the parameters $∣ w ∣ < δ$ , $w \neq 0$ , $\hat{ρ} (w) < 0$ , and the displayed inequality is the quantitative bound $Re w_{n} \leq - (c / 2) ∣ w ∣^{2}$ in the chart $w$ with the constant $c / 2 > 0$ of step 5.1.

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Sources

Harold P. Boas, Lecture Notes on Several Complex Variables (standard reference, not scraped)
Jean-Pierre Demailly, Complex Analytic and Differential Geometry (standard reference, not scraped)