A Non-Integrable Ohsawa-Takegoshi-Type $L^2$ Extension Theorem

Abstract

Given a complete Kähler manifold $(X,\omega )$ with finite second Betti number, a smooth complex hypersurface $Y\subset X$ and a smooth real $d$-closed $(1,1)$-form $\alpha$ on $X$ with arbitrary, possibly non-rational, De Rham cohomology class $\{\alpha \}$ satisfying a certain assumption, we obtain extensions to $X$, with control of their $L^2$-norms, of smooth sections of the canonical bundle of $Y$ twisted by the restriction to $Y$ of any $C^{\infty }$ complex line bundle $L_k$ in a sequence of asymptotically holomorphic line bundles whose first Chern classes approximate the positive integer multiples $k\{\alpha \}$ of the original class. Besides a known non-integrable $(0,1)$-connection ${\bar{\partial }}_k$ on $L_k$, the proof uses two twisted Laplace-type elliptic differential operators that are introduced and investigated, leading to Bochner-Kodaira-Nakano-type (in-)equalities, a spectral gap result and an a priori $L^2$-estimate. The main difference from the classical Ohsawa-Takegoshi extension theorem is that the objects need not be holomorphic, but only asymptotically holomorphic as $k\to \infty$. The possibility that ${\bar{\partial }}_k$ does not square to $0$ accounts for its lack of commutation with the Laplacian ${\Delta }_k^{\prime \prime }$ it induces. We hope this study is a possible first step in a future attack on Siu's conjecture predicting the invariance of the plurigenera in Kähler families of compact complex manifolds.

53 pages