Ohsawa-Takegoshi Extension Theorem for Compact Kähler Manifolds and Applications

Abstract

Our main goal in this article is to prove an extension theorem for sections of the canonical bundle of a weakly pseudoconvex Kähler manifold with values in a line bundle endowed with a possibly singular metric. We also give some applications of our result.

Appendix

For the reader’s convenience, we give the proof of (26) and (27), which is a rather standard estimate (cf. [9, Prop. 12.4, Remark 12.5], [11] or [22]). Set g: = g _m , η: = η _ε , B: = B_ε, k and δ: = δ _k for simplicity. Let Y _k be a subvariety of X such that φ _k is smooth outside Y _k . Then there exists a complete Kähler metric ω₁ on X∖Y _k . Set ω _s : = ω + sω₁. Then ω _s is also a complete Kähler metric on X∖Y _k for every s > 0.

We apply the twist L ²-estimate (cf. [9, 12.A, 12.B]) for the line bundle $(L,\widetilde{h}_{k})$ on (X∖Y _k , ω _s ). Thanks to (24) and [13, Lemma 4.1], for every smooth (n, 1)-form v with compact support, we have

$$ { \vert \langle g,v\rangle _{\omega _{s}}\vert ^{2} }$$

(45)

$$ {\leq (\int _{X\setminus Y _{k}}\langle (B + \frac{2C \cdot m} {k} )^{-1}g,g\rangle dV _{\omega _{ s}}) \cdot (\|(\eta +\lambda )^{\frac{1} {2} }D^{{\prime\prime}{\ast}}v\|_{\omega _{ s}}^{2} + \frac{2C \cdot m} {k} \int _{X\setminus Y _{k}}\langle v,v\rangle dV _{\omega _{s}})}$$

Set $H_{1}:=\| \cdot \|_{L^{2}}$, where the L ²-norm $\|\cdot \|_{L^{2}}$ is defined with respect to the metrics ω _s and $(L,\widetilde{h}_{k})$. Let H₂ be a Hilbert space where the norm is defined by

$$ {\|\,f\|_{H_{2}}^{2}:= \frac{2C \cdot m} {k} \int _{X\setminus Y _{k}}\vert \,f\vert _{\widetilde{h}_{k}}^{2}dV _{\omega _{ s}}.}$$

By (45) and the Hahn-Banach theorem, we can construct a continuous linear map (cf. for example [9, 5.A])

$$ {H_{1} \oplus H_{2} \rightarrow \mathbb{C},}$$

which is an extension of the application

$$ {((\eta +\lambda )^{\frac{1} {2} }D^{{\prime\prime}{\ast}}v,v) \rightarrow \langle g,v\rangle _{\omega _{ s}}.}$$

Therefore, there exist f and h such that

$$ {\langle g,v\rangle _{\omega _{s}} =\langle f,(\eta +\lambda )^{\frac{1} {2} }D^{{\prime\prime}{\ast}}v\rangle _{\omega _{ s}} + \frac{2C \cdot m} {k} \langle h,v\rangle _{\omega _{s}}}$$

and

$$ {\|\,f\|_{\omega _{s}}^{2} + \frac{2C \cdot m} {k} \|h\|_{\omega _{s}}^{2} \leq \int _{ X}(\langle B + \frac{2C \cdot m} {k} )^{-1}g,g\rangle dV _{\omega _{ s}}}$$

Let $\beta:= 2C(\frac{m} {k} )^{\frac{1} {2} } \cdot h$ and $\gamma:= (\eta +\lambda )^{\frac{1} {2} }f$. Then

$$ {g = D^{{\prime\prime}}\gamma + (\frac{m} {k} )^{\frac{1} {2} }\beta }$$

and

$$ {\| \frac{\gamma } {(\lambda +\eta )^{\frac{1} {2} }} \|_{(X\setminus Y _{k},\omega _{s})}^{2} + \frac{1} {2C}\|\beta \|_{(X\setminus Y _{k},\omega _{s})}^{2} \leq \int _{ X\setminus Y _{k}}\langle (B + \frac{2C \cdot m} {k} )^{-1}g,g\rangle dV _{\omega _{ s}}}$$

Then (26) and (27) are proved by letting s → 0⁺.