arXiv subjects
Publications and source records attributed to Imre Patyi.
We give an abstract definition, similar to the axioms of a Stein manifold, of a class of complex Banach manifolds in such a way that a manifold belongs to the class if and only if it is biholomorphic to a closed split complex Banach submanifold of a separable Banach space.
We show that if $X$ is a Banach space with a Schauder basis, $Ω\subset X$ is a pseudoconvex open subset, and $u:\,Ω\to(-\infty,\infty)$ is a locally bounded function, then there are a Banach space $Z$ and a holomorphic function $h:\,Ω\to Z$ with $u(x)<\|h(x)\|$ for all $x\inΩ$.
We show that any Dolbeault cohomology group $H^{p,q}(D)$, $p\ge0$, $q\ge1$, of an open subset $D$ of a closed finite codimensional complex Hilbert submanifold of $\ell_2$ is either zero or infinite dimensional. We also show that any continuous character of the algebra of holomorphic functions of a closed complex Hilbert submanifold $M$ of $\ell_2$ is induced by evaluation at a point of $M$. Lastly, we prove that any closed split infinite dimensional complex Banach submanifold of $\ell_1$ admits a nowhere critical holomorphic function.
Let $X$ be a separable Banach space and $u{:} X\to\Bbb{R}$ locally upper bounded. We show that there are a Banach space $Z$ and a holomorphic function $h{:} X\to Z$ with $u(x)<\|h(x)\|$ for $x\in X$. As a consequence we find that the sheaf cohomology group $H^q(X,\Cal{O})$ vanishes if $X$ has the bounded approximation property (i.e., $X$ is a direct summand of a Banach space with a Schauder basis), $\Cal{O}$ is the sheaf of germs of holomorphic functions on $X$, and $q\ge1$. As another consequence we prove that if $f$ is a $C^1$-smooth $\overline\partial$-closed $(0,1)$-form on the space $X=L_1[0,1]$ of summable functions, then there is a $C^1$-smooth function $u$ on $X$ with $\overline\partial u=f$ on $X$.
Let $X$ be a real Banach space with an unconditional basis (e.g., $X=\ell_2$ Hilbert space), $Ω\subset X$ open, $M\subsetΩ$ a closed split real analytic Banach submanifold of $Ω$, $E\to M$ a real analytic Banach vector bundle, and ${\Cal A}^E\to M$ the sheaf of germs of real analytic sections of $E\to M$. We show that the sheaf cohomology groups $H^q(M,{\Cal A}^E)$ vanish for all $q\ge1$, and there is a real analytic retraction $r:U\to M$ from an open set $U$ with $M\subset U\subsetΩ$ such that $r(x)=x$ for all $x\in M$. Some applications are also given, e.g., we show that any infinite dimensional real analytic Hilbert submanifold of separable affine or projective Hilbert space is real analytically parallelizable.
We show that the holomorphic ideal sheaf of a linear section of a pseudoconvex open subset $Ω$ of, say, a Hilbert space $X=\ell_2$ is acyclic. We also prove an analog of Hefer's lemma, i.e., if $f:Ω\timesΩ\to\CC$ is holomorphic and $f(x,x)=0$ for $x\inΩ$, then there is a holomorphic $g:Ω\timesΩ\to X^*$ with values in the dual space $X^*$ of $X$ such that $f(x,y)=g(x,y)(x-y)$
We show, e.g., that a holomorphic Banach vector bundle over a pseudoconvex open subset of, say, Hilbert space is holomorphically trivial if it is continuously trivial. Some applications are also given.
Among other things, we show that the ideal sheaf of a complex Hilbert submanifold of a pseudoconvex open subset of Hilbert space is acyclic over the ambient pseudoconvex open set. We also prove a vanishing theorem for a fairly general class of analytic sheaves over pseudoconvex open sets of a large class of Banach spaces. Some applications are also given.
We show that on the unit ball of a certain separable Banach space there is a smooth delbar-closed (0,1)-form which is not locally delbar-exact. Further, the Dolbeault isomorphism theorem does not generalize to arbitrary Banach spaces. Lastly, the Newlander-Nirenberg theorem does not generalize to arbitrary smooth integrable almost complex Banach manifolds.