Subanalytic sets and complex analytic geometry
Y. Peterzil, S. Starchenko
考虑复解析流形M（当M被看作实流形时）的次解析子集A，并给出了A是M的复解析子集的条件.
We consider a subanalytic subset A of a complex analytic manifold M (when M is viewed as a real manifold) and formulate conditions under which A is a complex analytic subset of M.
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We consider parameterized exponential integrals coming from the time evolution of the probability distribution of Brownian motion on globally subanalytic sets. We establish definability results and asymptotic expansions.
Globally subanalytic arc-symmetric sets
研究证明了全局子解析弧对称集可由连续全局子解析函数表示，该函数在简单正常交集外具有弧解析性质。
Arc-analytic subanalytic functions on complex manifolds
研究显示，复流形上在一点处解析的弧分析子分析函数是整个流形上的解析函数。更一般地，实解析CR流形上在非空开集上CR的弧分析子分析函数是整个流形上的CR函数。
Grauert's theorem for subanalytic open sets in real analytic manifolds
$\mathbb{R}^n$的开子集$Ω$的开邻域是指$\mathbb{C}^n$的开子集$Ω'$使得$\mathbb{R}^n\capΩ'=Ω.$ H. Grauert暗示了$\mathbb{R}^n$的任何开子集在$\mathbb{C}^n$中都存在Stein开邻域的基本系统.另一种表述此属性的方法是说$\mathbb{R}^n$的每个开子集都是Stein。我们将证明一个类似的结果，在亚解析范畴，所以，假设$Ω$是一个亚解析的相对紧的开子集在一个真正的解析流形，我们表明，$Ω$承认一个基本系统的亚解析Stein开放的邻居在其任何复杂化。
Control of radii of convergence and extension of subanalytic functions
设g表示欧几里得空间的开子集U上的实解析函数，S表示U的边界点，其中g不允许局部解析扩张.我们证明了，如果g是半代数的（分别，全球次解析的），那么S是半代数的（分别，次解析的）和g扩展到一个社区的cl（U）§作为一个解析函数，是半代数的（分别，全球次解析的）。(In在一般的次解析情形下，S不一定是次解析的）。我们的证明依赖于控制的收敛半径的幂级数G集中在点的图像中的解析映射，在收敛半径的拉回G在点的源。
We prove new parameterization theorems for sets definable in the structure $\mathbb{R}_{an}$ (i.e. for globally subanalytic sets) which are uniform for definable families of such sets. We treat both $C^r$-parameterization and (mild) analytic parameterization. In the former case we establish a polynomial (in $r$) bound (depending only on the given family) for the number of parameterizing functions. However, since uniformity is impossible in the latter case (as was shown by Yomdin via a very simple family of algebraic sets), we introduce a new notion, analytic quasi-parameterization (where many-…
We give rather simple answers to two long-standing questions in real-analytic geometry, on global smoothing of a subanalytic set, and on transformation of a proper real-analytic mapping to a mapping with equidimensional fibres by global blowings-up of the target. These questions are related: a positive answer to the second can be used to reduce the first to the simpler semianalytic case. We show that the second question has a negative answer, in general, and that the first problem nevertheless has a positive solution.
We prove that tangent cones of Lipschitz normally embedded sets are Lipschitz normally embedded. We also extend to real subanalytic sets the notion of reduced tangent cone and we show that subanalytic Lipschitz normally embedded sets have reduced tangent cones. In particular, we get that Lipschitz normally embedded complex analytic sets have reduced tangent cones.
Tempered solutions of $\mathcal D$-modules on complex curves and formal invariants
设$X$为复解析曲线。本文证明了$\mathcal D_X$ -模的调和全纯解的次解析层在形式完整$\mathcal D_X$ -模的芽的子范畴上诱导一个完全忠实函子.进一步，给出完整$\mathcal D_X$ -模的芽$\mathcal M$，得到了联系$\mathcal M$的调和解的次解析层与$\mathcal M$的经典形式不变量和解析不变量的一些结果.
Preconstructibility of tempered solutions of holonomic D-modules
本文证明了复解析流形上完整D-模的调和全纯解复形的可预构性。这意味着这样的复形在复解析流形的任何相对紧的开子解析子集上是有限的。这样的结果是证明M的一个猜想的必要步骤。Kashiwara和P. Schapira（2003）关于这种复合体的可建造性。
This article is devoted to studying multiplicity and regularity of real analytic sets. We present an equivalence for real analytic sets, named blow-spherical equivalence, which generalizes differential equivalence and subanalytic bi-Lipschitz equivalence and, with this approach, we obtain several applications on analytic sets. On regularity, we show that blow-spherical regularity of real analytic implies $C^1$ smoothness only in the case of real analytic curves. On multiplicity, we present a generalization for Gau-Lipman's Theorem about differential invariance of the multiplicity in the comple…
Cartan uniqueness theorem on nonopen sets
本文研究非开集上的 Cartan 唯一性定理，通过解析圆盘包络条件建立全纯限制的唯一延拓，并推广至实解析子簇的无穷小 CR 自同构，证明圆形子簇的相关自同构必为线性。
In this paper, we prove Fukui-Kurdyka-Paunescu's Conjecture, which says that subanalytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of the multiplicity (resp. degree) of real and complex analytic (resp. algebraic) sets. For instance, still in the real case, we prove a global version of Fukui-Kurdyka-Paunescu's Conjecture. In the complex case, one of the results that we prove is the following: If $(X,0)\subset (\mathbb{C}^n,0), (Y,0)\subset (\mathbb{C}^m,0)$ are germs of analytic sets and $h\c…
Regularity of the Siciak-Zaharjuta extremal function on compact Kähler manifolds
本文证明紧Kähler流形上极值函数的正则性为局部性质，将连续性与Hölder连续性等价于多重位势理论中的经典概念，并据此给出紧集$(\Cc^\al, \Cc^{\al'})$-正则性的有效刻画，证明$\bR^n$中所有紧的胖次解析子集均满足该正则性。
Approximation of analytic sets with proper projection by algebraic sets
设$X$是纯维$k$的$U\times C^n$的解析子集，使得$X$到$U$上的投影是逆紧映射，其中$U$是$C^k$中的Runge域.我们证明了$X$可以近似代数集。
There is proved the sufficiency of several conditions for the removability of singularities of complex-analytic sets in domains of $\mathbb C^n$.
It is known that every germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that the homeomorphism can be chosen in such a way that the analytic and algebraic germs are tangent with any prescribed order of tangency. Moreover, the space of arcs contained in the algebraic germ approximates the space of arcs contained in the analytic one, in the sense that they are identical up to a prescribed truncation order.
For an analytic curve $γ:(a,b)\rightarrow \C,$ the set of values approached by $γ(t),$ as $t\searrow a$ and as $t\nearrow b$ can be any two continuua of $\C\cup\{\infty\}.$
On Nash approximation of complex analytic sets in Runge domains
证明了Runge域D中的每个复解析集X都可以用D的相对紧子域上的Nash集来逼近。本文给出了复解析集X容许Nash逼近沿其给定子集重合的一个充要条件。
Algebraic approximation of analytic sets and mappings
设{X_n}是在全纯链意义下收敛于某个解析集X的解析集序列。我们引入了一个条件，这个条件意味着X的每一个不可约分支都是{X_n}集合的不可约分支序列的极限。其次，我们应用该条件，用Nash解来逼近Nash方程组Q（x，y）=0的全纯解y=f（x）.所提出的方法允许构造一个算法的全纯解的近似。
Conformal equivalence of analytic functions on compact sets
在本文中，我们提出了以下事实的几何证明。设$D$是$\mathbb{C}$中的Jordan整环，$f$是$cl(D)$上的解析整环.然后有一个单射解析映射$ϕ:D\to\mathbb{C}$和一个多项式$p$，使得$f\equiv p\circϕ$在$D$上（即，$f$具有多项式共形模型$p$）。
On the holomorphic closure dimension of real analytic sets
给出一个真实的分析（或者，更一般地，半解析）集合R在n维复空间中，对于R的闭包中的每个点p，唯一的最小复解析芽X_p，它包含芽R_p，我们称X_p的复维数为p是R在p处的全纯闭包维数。我们证明了一个不可约R的全纯闭包维数在一个闭R的补上是常数R的真解析子集，并讨论了它的维数与R的CR维数之间的关系。
Let the function $φ$ be holomorphic in the unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ and let $φ(\mathbb{D})\subset \mathbb{D}$. We study the level sets and the critical points of the hyperbolic derivative of $φ$, $$|D_φ(z)|:=\frac{(1-|z|^2)|φ'(z)|}{1-|φ(z)|^2}.$$ In particular, we show how the Schwarzian derivative of $φ$ reveals the nature of the critical points.
This paper is devoted to proving the differential invariance of the multiplicity of real and complex analytic sets. In particular, we prove the real version of Gau-Lipman's Theorem, i.e., it is proved that the multiplicity mod 2 of real analytic sets is a differential invariant. We prove also a generalization of Gau-Lipman's Theorem.
Invariants of singular germs of real-analytic sets
本文为实解析集奇异芽定义了新的双全纯不变量Bloom-Graham奇异型和奇异l型，基于相应构造并举例说明。
Local dimension-free estimates for volumes of sublevel sets of analytic functions
这是我们上一篇论文的姊妹篇。在这里，我们推导出局部无量纲的估计量的子和超水平集的解析函数的几个变量。
In this paper we study an algebraic and topological structure inside the following sets of special functions: Bloch functions defined on the open unit disk that are unbounded and analytic functions of bounded type defined a Banach algebra E into E, which are not Lorch-analytic.
Real Analytic Sets in Complex Spaces and CR maps
如果$R$是$\C^n$中的一个实解析集（看作$\R^{2n}$），那么对于任何点$p\in R$，都有一个唯一定义的最小复解析簇的芽$X_p$，它包含$R_p$，即$R$在$p$的芽。证明了若$R$是常维不可约的，则函数$p\to\dim X_p$在$R$的稠密开子集上是常数.作为应用，证明了从实解析CR流形$M$到$\C^N$的连续映射，如果它是$M$的某个开子集上的CR，且它的图是$M\times \C^N$中的实解析集，则它在$M$上处处必然是CR.
Banach Analytic Sets and a Non-Linear Version of the Levi Extension Theorem
证明了亚纯函数的Levi扩张定理的一个非线性形式。这意味着所讨论的亚纯函数被假定沿着一系列复曲线是可扩张的，这些复曲线是任意的，不一定是直线。此外，这些曲线不应该属于任何有限维解析族。我们定理的结论是，尽管如此，在问题中的功能亚纯延伸沿（无限维）解析家庭的复曲线和它的存在域是一个捏域填充这个解析家庭。
We give a new proof of the rectilinearization theorem of Hironaka. We deduce rectilinearization as a consequence of our theorem on local monomialization for complex and real analytic morphisms.
Uniformly perfect analytic and conformal attractor sets
给出了复平面上解析迭代函数系（IFS）具有一致完美吸引子集的条件。特别是，它表明，吸引子集的一个cross-generated共形IFS是一致完美的，当它包含两个或两个以上的点。并给出了一个非一致完美的解析吸引子集的例子。
Critical sets of bounded analytic functions, zero sets of Bergman spaces and nonpositive curvature
Blaschke的一个经典结果表明，对于复平面的开单位圆盘的每个解析自映射$f$，存在Blaschke积$B$，使得$f$和$B$的零集一致。本文证明了临界集存在一个类似的命题，即对于开单位圆盘的每个解析自映射$f$，甚至存在一个不可破坏的Blaschke积$B$，使得$f$和$B$的临界集重合.我们进一步将描述有界解析函数的临界集的问题与刻画某些加权Bergman空间的零点集的问题以及微分几何中的Berger-Nirenberg问题联系起来。通过求解特殊情况下的Berger-Nirenberg问题，我们用加权Bergman空间${\cal A}_1^2$的零点集来确定有界解析函数的临界集.
On the separation Łojasiewicz exponents of real analytic sets in the real plane
本文通过牛顿-普吕斯扩展计算两个实解析集的分离Łojasiewicz指数，并给出代数集的指数有效计算方法。
On homotopy types of complements of analytic sets and Milnor fibres
证明了对$\CC^n$中的复解析集的任一芽，存在一个超曲面奇点，其Milnor纤维化具有平凡的几何单值性，且纤维同伦于复解析集芽的补.作为一个应用程序，我们展示了一个例子的准齐次超曲面奇点，平凡的几何monodromy和简单的连接和非正式的Milnor纤维。
Conditions are given which imply that certain non-autonomous analytic iterated function systems (NIFS's) in the complex plane C have uniformly perfect attractor sets. Examples are given to illustrate the main theorem, as well as to indicate how it generalizes other results. Examples are also given to illustrate how natural generalizations of corresponding results for autonomous IFS's do not hold in general in this more flexible setting.
This paper is devoted to study multiplicity and regularity as well as to present some classifications of complex analytic sets. We present an equivalence for complex analytical sets, namely blow-spherical equivalence and we receive several applications with this new approach. For example, we reduce to homogeneous complex algebraic sets a version of Zariski's multiplicity conjecture in the case of blow-spherical homeomorphism, we give some partial answers to the Zariski's multiplicity conjecture, we show that a blow-spherical regular complex analytic set is smooth and we give a complete classif…
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.
We prove that any complex or real analytic set or function germ is topologically equivalent to a germ defined by polynomial equations whose coefficients are algebraic numbers.
Continuous Sobolev functions with singularity on arbitrary real-analytic sets
本文利用Hironaka奇点消解定理，在任意实解析集附近构造局部W^{1,1}连续函数，使其弱导数恰好以该集合为奇点集，揭示了实解析集与Sobolev函数弱导数奇异性的对应关系。
On directional second-order tangent sets of analytic sets and applications in optimization
本文研究了实与复分析集的方向二阶切集，通过分析集的二阶展开与初始形式比较，揭示了代数与几何方向二阶切集的关系，并在优化问题中应用了二阶必要和充分最优条件。
We prove that any complex analytic set in $\mathbb{C}^n$ which is Lipschitz normally embedded at infinity and has tangent cone at infinity that is a linear subspace of $\mathbb{C}^n$ must be an affine linear subspace of $\mathbb{C}^n$ itself. No restrictions on the singular set, dimension nor codimension are required. In particular, a complex algebraic set in $\mathbb{C}^n$ which is Lipschitz regular at infinity is an affine linear subspace.
We show that under mild conditions, a Gaussian analytic function $\boldsymbol F$ that a.s. does not belong to a given weighted Bergman space or Bargmann-Fock space has the property that a.s. no non-zero function in that space vanishes where $\boldsymbol F$ does. This establishes a conjecture of Shapiro (1979) on Bergman spaces and allows us to resolve a question of Zhu (1993) on Bargmann-Fock spaces. We also give a similar result on the union of two (or more) such zero sets, thereby establishing another conjecture of Shapiro (1979) on Bergman spaces and allowing us to strengthen a result of Zh…
Let $f : X\to X$ be a dominating meromorphic map on a compact Kähler manifold $X$ of dimension $k$. We extend the notion of topological entropy $h^l_{\mathrm{top}}(f)$ for the action of $f$ on (local) analytic sets of dimension $0\leq l \leq k$. For an ergodic probability measure $ν$, we extend similarly the notion of measure-theoretic entropy $h_ν^l(f)$. Under mild hypothesis, we compute $h^l_{\mathrm{top}}(f)$ in term of the dynamical degrees of $f$. In the particular case of endomorphisms of $\mathbb{P}^2$ of degree $d$, we show that $h^1_{\mathrm{top}}(f)= \log d$ for a large class of maps…
The paper is devoted to metric properties of singularities. We investigate the relations among topology, metric properties and smoothness. In particular, we present some higher dimensional analogous of Mumford's theorem on smoothness of normal surfaces. For example, we prove that a complex analytic set, with an isolated singularity at $0$, is smooth at $0$ if and only if it is locally metrically conical at $0$ and its link at $0$ is a homotopy sphere.
In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex non-parametric version of Moser's Bernstein Theorem. More precisely, we prove that any entire complex analytic set in $\mathbb{C}^n$ which i…
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