[Paper Review] $L^2$ Extension of $\bar\partial$-closed forms from a hypersurface

Jeffery D. McNeal, Dror Varolin|arXiv (Cornell University)|Feb 27, 2015

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper establishes $L^2$ extension theorems for $\bar\partial$-closed $(0,q)$-forms with values in a holomorphic line bundle with smooth Hermitian metric, from a smooth hypersurface $Z$ in a Stein manifold $X$. Using the Kohn solution to the $\bar\partial$-Neumann problem, the authors prove that under curvature-type positivity conditions on $\varphi - \lambda_Z$ and $\varphi - (1+\delta)\lambda_Z$, any $L^2$-bounded, $\bar\partial$-closed form on $Z$ admits a smooth, $\bar\partial$-closed extension to $X$ with controlled $L^2$-norm, extending Berndtsson's result to non-compact settings with a new, classical proof.

ABSTRACT

We establish $L^2$ extension theorems for $\bar \partial$-closed $(0,q)$-forms with values in a holomorphic line bundle with smooth Hermitian metric, from a smooth hypersurface on a Stein manifold. Our result extends (and gives a new, perhaps more classical, proof of) a theorem of Berndtsson on compact Kähler manifolds, which itself is a sharpening of the theorem of Koziarz. The proof makes use of the Kohn solution, which is the solution of an (interior) elliptic problem, to handle the well-known regularity issues. As such, our methods require the line bundle to be equipped with a smooth metric.

Motivation & Objective

• To extend $L^2$-bounded, $\bar\partial$-closed $(0,q)$-forms from a smooth hypersurface $Z$ to the ambient Stein manifold $X$ with controlled $L^2$-norm.
• To provide a new, classical proof of Berndtsson's $L^2$ extension theorem on compact Kähler manifolds by adapting it to non-compact Stein manifolds.
• To handle the regularity issues of $\bar\partial$-closed forms on $X$ by employing the Kohn solution to the $\bar\partial$-Neumann problem.
• To clarify the distinction between intrinsic and ambient restrictions of forms on $Z$, and to establish extension theorems for both cases under smooth metric assumptions.
• To prove that transverse forms (those annihilated by restriction to $Z$) can be extended with arbitrarily small $L^2$-norm, enabling decomposition-based extension strategies.

Proposed method

• The proof relies on the Kohn solution to the $\bar\partial$-Neumann problem, which resolves the regularity issues inherent in solving $\bar\partial$ equations for non-elliptic $q > 0$.
• The authors use a decomposition of ambient forms into intrinsic and transverse parts via the orthogonal projection $P$ induced by the Kähler metric $\omega$.
• For the intrinsic part, the extension is achieved via Theorem 2, which provides a universal constant $C$ such that the $L^2$-norm of the extension is bounded by $C/\delta$ times the $L^2$-norm on $Z$.
• For the transverse part, the method constructs a form $u = \bar\partial \hat{\alpha}$ using a cutoff function $\rho_\varepsilon$ and a smooth extension $\tilde{\alpha}$ of the form $\alpha$ such that $\xi = \alpha \wedge d\bar{f}_Z$, ensuring $u|_Z = \xi$ and $\|u\|_{L^2}$ can be made arbitrarily small.
• The curvature conditions $\sqrt{-1}(\partial\bar\partial(\varphi - \lambda_Z) + \mathrm{Ricci}(\omega)) \wedge \omega^q \geq 0$ and $\sqrt{-1}(\partial\bar\partial(\varphi - (1+\delta)\lambda_Z) + \mathrm{Ricci}(\omega)) \wedge \omega^q \geq 0$ are used to ensure the positivity required for the $L^2$ estimates.
• The proof of Theorem 4 uses a local model $X = B \times \mathbb{D}$ to isolate the transverse component and construct a form with small $L^2$-norm that matches $\xi$ on $Z$.

Experimental results

Research questions

• RQ1 Can $L^2$-bounded, $\bar\partial$-closed $(0,q)$-forms on a smooth hypersurface $Z$ be extended to $\bar\partial$-closed forms on a Stein manifold $X$ with controlled $L^2$-norm?
• RQ2 How do the intrinsic and ambient notions of restriction differ, and what conditions ensure compatibility with $\bar\partial$-closedness?
• RQ3 Can the $L^2$ extension problem be solved for transverse forms (those with zero intrinsic restriction) with arbitrarily small $L^2$-norm?
• RQ4 What curvature conditions on the metrics $\varphi$ and $\lambda_Z$ are sufficient to guarantee such extensions with universal constants?
• RQ5 Is there a new, classical proof of Berndtsson’s $L^2$ extension theorem that avoids the use of current-theoretic methods?

Key findings

• Theorem 1 establishes an ambient $L^2$ extension for $\bar\partial$-closed forms on $Z$ under the curvature conditions $\sqrt{-1}(\partial\bar\partial(\varphi - \lambda_Z) + \mathrm{Ricci}(\omega)) \wedge \omega^q \geq 0$ and $\sqrt{-1}(\partial\bar\partial(\varphi - (1+\delta)\lambda_Z) + \mathrm{Ricci}(\omega)) \wedge \omega^q \geq 0$, with the $L^2$-norm of the extension bounded by $\frac{C}{\delta}$ times the $L^2$-norm on $Z$.
• Theorem 2 provides an intrinsic $L^2$ extension result, showing that any $\bar\partial$-closed form $\eta$ on $Z$ with finite $L^2$-norm relative to $|df_Z|^{-2}e^{-\lambda_Z}$ admits a smooth, $\bar\partial$-closed extension $u$ to $X$ with $\|u\|_{L^2}^2 \leq \frac{C}{\delta} \|\eta\|_{L^2}^2$.
• The constant $C$ in the extension estimate is universal and independent of the data, including $\delta$, $\varphi$, $\lambda_Z$, and $\omega$.
• For transverse forms $\xi$ (those with $\iota^*\xi = 0$), Proposition 6.2 shows that there exists a $\bar\partial$-closed extension $u$ with $\|u\|_{L^2}^2 < \varepsilon$ for any $\varepsilon > 0$, enabling decomposition-based extension strategies.
• The proof of Theorem 4 reduces to Theorem 3 by decomposing any ambient form $\xi$ into its intrinsic and transverse parts, with the intrinsic part handled by Theorem 2 and the transverse part extended with arbitrarily small norm.
• The method successfully extends Berndtsson’s result from compact Kähler manifolds to Stein manifolds by using the Kohn solution instead of current-theoretic techniques, providing a more classical and regular approach.