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

Given a complete K\"ahler 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 $\Delta''_k$ 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 K\"ahler families of compact complex manifolds.