[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ5 (1002ﾚｽ)
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728: 2023/07/17(月)06:30 ID:GpeoaFRE(1/6) AAS
\textbf{Remark.} As was noted in [C-D], $\hat{\Omega}$ is a ramified locally pseudoconvex
Riemann domain over $\mathbb{C}^3$ whose fiber are separable by holomorphic functions
by a theorem of Siu in [S]. Hence, as a complex manifold, $\hat{\Omega}$ is embeddable into
$\mathbb{C}^N$ as a locally closed complex analytic submanifold. Since this submanifold is locally
Stein, it amounts to a counterexample of a question raised by P.A. Griffiths in 1977 in Kyoto.
However, it is still an open question whether or not the locally pseudoconvex ramified Riemann domain
constructed by Fornaess [F] is embeddable into some $\mathbb{C}^N$ as a locally closed submanifold.
省5

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729: 2023/07/17(月)08:16 ID:GpeoaFRE(2/6) AAS
\section*{A convexity property of $\hat{\Omega}$}We shall show that, although $\hat{\Omega}$ is not Stein, it has a weak convexity property with respect to the space of $L^2$ holomorphic functions. To describe this property, let us introduce the notion of $L^2$-convexity. For any Hermitian manifold $(M,g)$, we shall denote the space of $L^2$ holomorphic functions on $M$ with respect to $g$ by $A^2(M,g)$.
\begin{definition} $(M,g)$ is said to be $L^2$-convex if, for any compact subset $K\subset M$ and for every point $x$ in the completion $\overline{M}$ of $(M,g)$, there exists a neighborhood $U$ of $x$ in $\overline{M}$ such that $$\Big\{z\in M; |f(z)|\leq \sup_K|f| \;\;for \;all \;f\in A^2(M,g)\Big\}\cap U=\phi.$$ \end{definition}

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730: 2023/07/17(月)13:31 ID:GpeoaFRE(3/6) AAS
\begin{theorem}$\left(\hat{\Omega}, \displaystyle\left(\frac{du_1d\overline{u_1}}{({\rm Im }u_1)^2}+\frac{du_2d\overline{u_2}}{({\rm Im} u_2)^2} \right)|_{\hat{\Omega}}\right)$ is $L^2$-convex. \end{theorem}

Proof. Since the canonical bundle of $\hat{\Omega}$ is trivial as $A$ leaves $du_1\wedge du_2$ invariant, it is easy to see from [Dm] or [Oh-1] that Theorem 1 implies the solvability of the $\dbar$-equation with $L^2$ estimates which yields the assertion. \qed\\

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731: 2023/07/17(月)13:36 ID:GpeoaFRE(4/6) AAS
\textbf{Remark.} As for the $L^2$ $\dbar$cohomology groups $H^{p,q}_{(2)}$
of the complete K\"ahler manifold $\left(F,\displaystyle\frac{du_1d\overline{u_1}}{({\rm Im }u_1)^2}
+\frac{du_2d\overline{u_2}}{({\rm Im} u_2)^2}+\partial\dbar\log{K_F}\right)$,
it is easy to verify that $H^{p,q}_{(2)}=0$　hold if $p+q\neq 2$ and
$\dim{H^{2,0}_{(2)}}=\dim{H^{0,2}_{(2)}}=\infty$.
The author's guess is that one can show　that $\dim{H^{1,1}_{(2)}}=\infty$　similarly as in [Oh-2].
See also [Mi].

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732: 2023/07/17(月)13:38 ID:GpeoaFRE(5/6) AAS
\section*{Complete K\"ahler bundles over complete K\"ahler manifolds}
Similarly as the Serre problem, it may be asked whether or not analytic fiber bundles
with complete K\"ahler fibers with complete K\"ahler bases admit complete K\"ahler metrics.
In the circumstance of Theorem 1, the Bergman metric of $F$ is not complete,
but there happens to exist a local coordinate $(u_1,u_2)$ for which
$\displaystyle\frac{du_1d\overline{u_1}}{({\rm Im }u_1)^2}+
\frac{du_2d\overline{u_2}}{({\rm Im} u_2)^2}$ compensates the incompleteness
省4

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733: 2023/07/17(月)13:39 ID:GpeoaFRE(6/6) AAS
\begin{theorem}Let $\pi_1(X)$ be the fundamental group of a complete K\"ahler manifold $X$ and
let $\rho:\pi_1(X)\to GL(n,\mathbb{Z})$ be a homomorphism such that
$\#\rho(\pi_1(X))<\infty$ or $\rho(\pi_1(X))$ is simultaneously diagonizable.
Then the bundle $X\times_{{\rho}}(\mathbb{C}^*)^n$ has
a complete K\"ahler metric. \end{theorem}

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