Laszlo Lempert
We introduce a class of analytic sheaves in a Banach space X, that we call cohesive sheaves. Cohesion is meant to generalize the notion of coherence from finite dimensional analysis. Accordingly, we prove the analog of Cartan's Theorems A and B for cohesive sheaves over pseudoconvex open subsets of X, provided X has an unconditional basis.
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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 consider coherent and cohesive sheaves of $\cO$--modules over open sets $Ω\subset\bC^n$. We prove that coherent sheaves, and certain other sheaves derived from them, are cohesive; and conversely, certain sheaves derived from cohesive sheaves are coherent. An important tool in all this, also proved here, is that the sheaf of Banach space valued holomorphic germs is flat.
We study Banach-valued holomorphic functions defined on open subsets of the maximal ideal space of the Banach algebra H^\infty of bounded holomorphic functions on the unit disk D\subset C with pointwise multiplication and supremum norm. In particular, we establish vanishing cohomology for sheaves of germs of such functions and, solving a Banach-valued corona problem for H^\infty, prove that the maximal ideal space of the algebra H_{\rm comp}^\infty (A) of holomorphic functions on $\Di$ with relatively compact images in a commutative unital complex Banach algebra A is homeomorphic to the direct…
We prove that every homomorphism $\mathcal{O}^E_ζ\to\mathcal{O}^F_ζ$, with $E$ and $F$ Banach spaces and $ζ\in\mathbb{C}^m$, is induced by a $\mathop{\mathrm{Hom}}(E,F)$-valued holomorphic germ, provided that $1\leq m<\infty$. A similar structure theorem is obtained for the homomorphisms of type $\mathcal{O}^E_ζ\to\mathcal{S}_ζ$, where $\mathcal{S}_ζ$ is a stalk of a coherent sheaf of positive $\mathfrak{m}_ζ$-depth. We later extend these results to sheaf homomorphisms, obtaining a condition on coherent sheaves which guarantees the sheaf to be equipped with a unique analytic structure in the s…
We study the transformation of torsion-free coherent analytic sheaves under proper modifications. More precisely, we study direct images of inverse image sheaves, and torsion-free preimages of direct image sheaves. Under some conditions, it is shown that torsion-free coherent sheaves can be realized as the direct image of locally free sheaves under modifications. Thus, it is possible to study coherent sheaves modulo torsion by reducing the problem to study vector bundles on manifolds. We apply this to reduced ideal sheaves and to the Grauert-Riemenschneider canonical sheaf of holomorphic n-for…
Relative Riemann-Hilbert and Newlander-Nirenberg Theorems for torsion-free analytic sheaves on maximal and homogeneous spaces
本文基于自由解析sheaves在最大和齐性空间上，证明了相对Riemann-Hilbert定理在考虑torsion时仍成立，并推导出Newlander-Nirenberg型定理，揭示了tame flat连接与局部系统及基本群表示的对应关系。
Quasi-coherent sheaves on complex analytic spaces
本文证明在复解析空间的解析层范畴中，拟凝聚层构成阿贝尔子范畴。
We discuss possible generalizations of Vitali convergence theorem when the source and the target are Banach analytic manifolds. These results are then applied to study behavior of holomorphic mappings between Banach analytic manifolds. Explicit examples of manifolds having Vitali properties are also provided.
In this article, we will present that the analytic adjoint ideal sheaves associated to plurisubharmonic functions are not coherent.
We characterize all entire functions that transform a weighted Banach spaces of analytic functions $\mathcal{H}^{\infty}_{μ_1}$ into another space of the same kind $\mathcal{H}^{\infty}_{μ_2}$ by superposition for very general weights $μ_1$ and $μ_2$.
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 dimensi…
We extend Langton's valuative criterion for families of coherent algebraic sheaves to a complex analytic set-up. As a consequence we derive a set of sufficient conditions for the compactness of a moduli space of semistable sheaves over a compact complex manifold. This applies also to some cases appearing in complex projective geometry not covered by previous results.
Boundary behavior of analytic functions on certain Banach spaces
本文证明了在多项式稠密且点求值连续的圆盘解析函数 Banach 空间中，若某函数模的上极限在单位圆上几乎处处无穷大，则该性质对残余生成的函数集同样成立。
We extend previous results on boundedness of sets of coherent sheaves on a compact Kähler manifold to the relative and not necessarily smooth case. This enlarged context allows us to prove properness properties of the relative Douady space as well as results related to semistability of sheaves such as the existence of relative Harder-Narasimhan filtrations.
On any pure $n$-dimensional, possibly non-reduced, analytic space $X$ we introduce the sheaves $\mathscr{E}_X^{p,q}$ of smooth $(p,q)$-forms and certain extensions $\mathscr{A}_X^{p,q}$ of them such that the corresponding Dolbeault complex is exact, i.e., the $\bar\partial$-equation is locally solvable in $\mathscr{A}_X$. The sheaves $\mathscr{A}_X^{p,q}$ are modules over the smooth forms, in particular, they are fine sheaves. We also introduce certain sheaves $\mathscr{B}_X^{n-p,n-q}$ of currents on $X$ that are dual to $\mathscr{A}_X^{p,q}$ in the sense of Serre duality. More precisely, we s…
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 prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite dimensional) analytic family of complex curves and its domain of existence is a pinched domain filled in by this analytic …
Let X be a complex space and F a coherent O_X-module. A F-(co)framed} sheaf on X is a pair (E,f) with a coherent O_X-module E and a morphism of coherent sheaves f : F -> E (resp. f : E -> F). Two such pairs (E,f) and (E',f') are said to be isomorphic if there exists an isomorphism of sheaves g : E -> E' such that gof = f' (resp. f'og = f). A pair (E,f) is called simple if its only automorphism is the identity on E. In this note we prove a representability theorem in a relative framework, which implies in particular that there is a moduli space of simple F-(co)framed sheaves on a given compact …
On an injectivity theorem for log-canonical pairs with analytic adjoint ideal sheaves
本文利用lc-测度的残值函数改进了Matsumura关于紧Kähler流形上plt对的内射性定理，并借助解析伴随理想层提出了Fujino猜想广义版本的解决方案，同时给出了2维lc对的证明。
Smith et al. recently gave the sufficient and necessary conditions for the boundedness of Volterra type operators on Banach spaces of bounded analytic functions when the symbol functions are univalent. In this paper, we give the complete characterizations of the conditions for the boundedness and compactness of Volterra type operators $T_g$ and their companion operators $S_g$ between weighted Banach spaces of analytic functions, which essentially generalize their works.
In the smooth case, we prove quasi-flasqueness for the sheaves of all subelliptic multipliers as well as at each of the steps of the Kohn algorithm on a pseudoconvex domain in $\C^n.$ We use techniques by Jean-Claude Tougeron to show that if the domain has a real-analytic defining function, the modified Kohn algorithm involving generating ideals and taking real radicals only in the ring of real-analytic germs yields quasi-coherent sheaves. This sharpens a result obtained by J. J. Kohn in 1979.
We give a rather simple proof of the generalization of Kuhlmann's quasi-proper direct image theorem to the case of a map with values in a Banach analytic set. The proof uses a generalization of the Remmert-Stein's theorem to this context. AMS Classification. 32 H 02-32 K 05-32 C 25.
We present a generalization of Takegoshi's relative version of the Grauert-Riemenschneider vanishing theorem. Under some natural assumptions, we extend Takegoshi's vanishing theorem to the case of Nakano semi-positive coherent analytic sheaves on singular complex spaces. We also obtain some new results about proper modifications of torsion-free coherent analytic sheaves.
In this note, for the multiplier ideal sheaves with weights $\log\sum_{i}|z_{i}|^{a_{i}}$, we present the sufficient and necessary condition of the existence of decreasing equisingular approximations with analytic singularities.
In the first part of this paper, we study the properties of some particular plurisubharmonic functions, namely the toric ones. The main result of this part is a precise description of their multiplier ideal sheaves, which generalizes the algebraic case studied by Howald. In the second part, almost entirely independent of the first one, we generalize the notion of the adjoint ideal sheaf used in algebraic geometry to the analytic setting. This enables us to give an analogue of Howald's theorem for adjoint ideals attached to monomial ideals. Finally, using the local Ohsawa-Takegoshi-Manivel theo…
The goal of this note is to present some recent results of our research concerning multiplier ideal sheaves on complex spaces and singularities of plurisubharmonic functions. We firstly introduce multiplier ideal sheaves on complex spaces (\emph{not} necessarily normal) via Ohsawa's extension measure, as a special case of which, it turns out to be the so-called Mather-Jacobian multiplier ideals in the algebro-geometric setting. As applications, we obtain a reasonable generalization of (algebraic) adjoint ideal sheaves to the analytic setting and establish some extension theorems on Kähler mani…
Analytic multiplier ideals of toric plurisubharmonic functions on a toric complex space
本文用Newton凸体刻画正规仿射环面复空间上环面多次调和函数的解析乘子理想层，证明其由有限个亚纯单项式生成，并推广了An、Howald、Blickle、Rashkovskii和Guenancia的既有结果。
M. J. Beltrán-Meneua et al. \cite{beltran1} and E. Jordá and A. Rodríguez-Arenas \cite{jorda} characterized the (uniformly) mean ergodic composition operators on $H^\infty(\mathbb{D})$ and $H^\infty_ν(\mathbb{D})$, respectively. In this paper, by using the interpolating sequences, we give other necessary and sufficient conditions for the (uniformly) mean ergodicity of composition operators on these spaces.
A $2$-dimensional real Banach space with constant of analyticity less than one
本文证明了在实2维Banach空间$\ell_1^2$上存在解析函数，其某点幂级数的均匀收敛半径严格小于$1-\\|a\\|$，揭示了实解析函数中解析半径与均匀收敛半径的根本区别，并首次为解析常数提供了非平凡上界。
A new definition of analytic adjoint ideal sheaves via the residue functions of log-canonical measures I
本文提出仅含整洁解析奇点的拟多重次调和函数的解析伴随理想层新定义，证明其满足留数短正合列、具备对数分解不变性，可刻画σ-lc中心并集，建立其与代数伴随理想层的关联，还证明了不依赖Ohsawa--Takegoshi技术的局部L²延拓定理。
A solution to Fujita's freeness conjecture via an extension theorem with analytic adjoint ideal sheaves
本文利用伴随理想层对乘子理想层不可积轨迹的精细控制，并结合基于内射性定理技术的扩张定理，直接处理 lc 奇点及其极小 lc 中心，从而在 Angehrn-Siu 思路下解决了藤田有效自由性猜想。
In the paper compact multiplier operators on Banach spaces of analytic functions on the unit disk with the range in Banach sequence lattices are studied. If the domain space $X$ is such that $H_\infty\hookrightarrow X\hookrightarrow H_1$, necessary and sufficient conditions for compactness are presented. Moreover, the calculation of the Hausdorff measure of noncompactness for diagonal operators between Banach sequence lattices is applied to obtaining the characterization of compact multipliers in case the domain space $X$ satisfies $H_\infty\hookrightarrow X\hookrightarrow H_2$.
Local constants and Bohr's phenomenon for Banach spaces of analytic polynomials
本研究构建通用框架分析多元解析多项式Banach空间的投影常数等基本常数的渐近最优行为，推导一般Banach序列格的Bohr半径估计，并在有限维Lorentz序列空间中得到宽参数范围的最优结果。
Analytic functions defined on a tube domain $T^{C}\subset \mathbb{C}^{n}$ and taking values in a Banach space $X$ which are known to have $X$-valued distributional boundary values are shown to be in the Hardy space $H^{p}(T^{C},X)$ if the boundary value is in the vector valued Lebesgue space $L^{p}(\mathbb{R}^{n},X)$, where $1\leq p \leq \infty$ and $C$ is a regular open convex cone. Poisson integral transform representations of elements of $H^{p}(T^{C}, X)$ are also obtained for certain classes of Banach spaces, including reflexive Banach spaces.
Gateaux differentiability in the Banach space of meromorphic functions
该研究利用Birkhoff-James正交性技术刻画了亚纯函数巴拿赫空间中的Gateaux可微性，提出扩展正交覆盖集（EOCS）概念以改进解析函数可微性相关结果，并推导了亚纯函数的相关性质。
The purpose of this survey is to present analytic versions of the injectivity theorem and their applications. The proof of our injectivity theorems is based on a combination of the L^2-method for the dbar-equation and the theory of harmonic integrals. As applications, we obtain Nadel type vanishing theorems and extension theorems for pluri-canonical sections of log pairs. Moreover, we give some results on semi-ampleness related to the abundance conjecture in birational geometry (the minimal model program).
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