## A solution of an $$L^{2}$$ extension problem with an optimal estimate and applications.(English)Zbl 1348.32008

The authors give the proof of the generalization of the Ohsawa-Takegoshi extension theorem announced in [C. R. Math. Acad. Sci. Paris, 350 (2012), 753-756]. A slightly less general version, sufficient for several applications, is given below. Let $$c_A (t)$$ denote a smooth function defined on $$(-A, +\infty )$$ and such that $$c_A (t) e^{-t}$$ is decreasing. Let $$M$$ be either a Stein manifold or a complex projective manifold, and $$S$$ a closed complex subvariety. Let $$S'$$ be a set containing $$S$$ and negligible for $$L^2$$ functions on $$M$$. Further, suppose $$\Psi$$ is almost plurisubharmonic on $$M$$, smooth on $$M\setminus S'$$, with $$S$$ contained in the $$-\infty$$ set of $$\Psi$$, with singularities prescribed in the following way: if $$S$$ is $$k$$-dimensional around a given regular point and in a local chart $$S$$ is given by $$z_{k+1} = ... = z_n =0$$, then in this chart $$\Psi$$ has the same singularities as $$(n-k)\log \sum _{k+1}^n |z_j |^2$$. Consider also a holomorphic vector bundle $$E$$ on $$M$$ of rank $$r$$, and $$h$$ a smooth metric on $$E$$ with the property that $$h e^{-\Psi}$$ is semipositive in the sense of Nakano on $$M\setminus S'$$. Then, for a holomorphic section $$f$$ of $$K_M$$ tensor $$E$$ restricted to $$S$$, there exists an extension $$F$$ to the whole tensor product on $$M$$ with the estimate
$\int _M c_A (\psi )|F|^2 _h dV_M \leq \int _A ^{\infty }c_A (t)e^{-t} dt \sum _{k=1}^{n} \frac{\pi ^k}{k!} \int _{S_{n-k}}|f|_h ^2 dV_M [\Psi ],\tag{1}$ provided the right hand side is finite. Here $$dV_M [\Psi ]$$ is defined as a minimal element of the set of positive measures $$\mu$$ satisfying $\int _{S_k} fd\mu \geq \limsup _{t\to \infty }\frac{2(n-k)}{\sigma _{2n-2k-1} }\int _M fe^{-\Psi }\chi _{\{ -1-t<\Psi <-t \} } dV_M ,$ for any continuous, compactly supported $$f$$, where $$\chi$$ denotes the characteristic function and $$\sigma _k$$ the area of the unit $$k$$-dimensional sphere. The important point is that the estimate (1) is no longer true if the right hand side is multiplied by a constant smaller than 1.
The authors give several applications of this result. One of them is the characterisation of the sets for which the equality holds in the Suita conjecture [N. Suita, Arch. Ration. Mech. Anal. 46, 212–217 (1972; Zbl 0245.30014)]. Others include a related conjecture of A. Yamada, a question of T. Ohsawa on boundedness of the extension operators [Contemp. Math. 332, 235–239 (2003; Zbl 1049.32010)], the log-plurisubharmonicity of the Bergman kernel, and the optimal constant in various versions of the $$L^2$$ extension theorem.

### MSC:

 32L05 Holomorphic bundles and generalizations 32S05 Local complex singularities

### Citations:

Zbl 0245.30014; Zbl 1049.32010
Full Text:

### References:

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