Toshikazu Ito, Bruno Scardua
We study the classification of singularities of holomorphic foliations and non-integrable one-forms under the hypothesis of transversality with real hypersurfaces.
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We show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target submanifold provided that the source manifold is of finite type and the map is of generic full rank. This result and its corollaries completely resolve two questions posed by Linda P. Rothschild and the first author in a paper from 2006.
In this paper, we consider holomorphic mappings between real hypersurfaces in different dimensional complex spaces. We give a number of conditions that imply that such mappings are transversal to the target hypersurface at most points.
We consider holomorphic mappings $H$ between a smooth real hypersurface $M\subset \bC^{n+1}$ and another $M'\subset \bC^{N+1}$ with $N\geq n$. We provide conditions guaranteeing that $H$ is transversal to $M'$ along all of $M$. In the strictly pseudoconvex case, this is well known and follows from the classical Hopf boundary lemma. In the equidimensional case ($N=n$), transversality holds for maps of full generic rank provided that the source is of finite type in view of recent results by the authors (see also a previous paper by the first author and L. Rothschild). In the positive codimension…
Surjectivity of real-linear Cauchy--Riemann operators: from the minimal Harder--Narasimhan slope to automatic transversality
本文针对实线性柯西-黎曼算子建立共形不变的L²满射性判据，利用极小Harder–Narasimhan斜率得到渐近判据，提出滤过原理简化满射性问题，为伪全纯曲线提供高阶自动横截性判据并应用于标准近凯勒6-球面。
In this paper, we establish general stratawise higher jet evaluation transversality of $J$-holomorphic curves for a generic choice of almost complex structures $J$ tame to a given symplectic manifold $(M,ω)$. Using this transversality result, we prove that there exists a subset $\CJ_ω^{ram} \subset \CJ_ω$ of second category such that for every $J \in \CJ_ω^{ram}$, the dimension of the moduli space of (somewhere injective) $J$-holomorphic curves with a given ramification profile goes down by $2n$ or $2(n-1)$ depending on whether the ramification degree goes up by one or a new ramification point…
We obtain results on approximation of holomorphic maps by algebraic maps, jet transversality theorems for holomorphic and algebraic maps, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds.
We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note.
In this paper, we examine holomorphic Segre preserving maps between the complexifications of real hypersurfaces in $\mathbb{C}^{n+1}$. In particular, we find several sufficient conditions ensuring that Segre transversality and total Segre nondegeneracy of the maps must hold.
We show that if a holomorphic $n$ dimensional compact torus action on a compact connected complex manifold of complex dimension $n$ has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.
In the context of the complex-analytic structure within the unit disk centered at the origin of the complex plane, that was presented in a previous paper, we show that a certain class of non-integrable real functions can be represented within that same structure. In previous papers it was shown that essentially all integrable real functions, as well as all singular Schwartz distributions, can be represented within that same complex-analytic structure. The large class of non-integrable real functions which we analyze here can therefore be represented side by side with those other real objects, …
A complex-analytic structure within the unit disk of the complex plane is presented. It can be used to represent and analyze a large class of real functions. It is shown that any integrable real function can be obtained by means of the restriction of an analytic function to the unit circle, including functions which are non-differentiable, discontinuous or unbounded. An explicit construction of the analytic functions from the corresponding real functions is given. The complex-analytic structure can be understood as an universal regulator for analytic operations on real functions.
Birationally integrable vector fields on complex projective surfaces
本文证明复射影光滑曲面上的双有理可积向量场可正则化，并推广至有限维李代数，给出二维及半单情形的完全双有理分类及极大代数的刻画。
Integrable Generators of Polynomial Vector Fields for Complex Classical Lie Groups
本文针对复经典李群SLₙ(ℂ)、SOₙ(ℂ)、Sp₂ₙ(ℂ)，证明其多项式向量场的李代数可由三个ℂ-完全多项式向量场生成。
We consider a locally integrable real-analytic structure, and we investigate the local solvability in the category of Gevrey functions and ultradistributions of the complex d' naturally induced by the de Rham complex. We prove that the so-called condition Y(q) on the signature of the Levi form, for local solvability of d' u=f, is still necessary even if we take f in the classes of Gevrey functions and look for solutions u in the corresponding spaces of ultradistributions.
We explore the Bohr inequality involving the Fourier transforms of complex valued integrable and square integrable functions defined on a second countable compact topological group. We also investigate the connection of the Bohr phenomenon with a modulus of convexity of the space of bounded linear operators defined on a complex Hilbert space.
A previously established correspondence between definite-parity real functions and inner analytic functions is generalized to real functions without definite parity properties. The set of inner analytic functions that corresponds to the set of all integrable real functions is then extended to include a set of singular "generalized functions" by the side of the integrable real functions. A general definition of these generalized functions is proposed and explored. The generalized functions are introduced loosely in the spirit of the Schwartz theory of distributions, and include the Dirac delta …
We show that existence and uniqueness of solutions to transported Monge-Ampere problem on complex compact toric manifold follows easily from the real theory of optimal transportation.
Normalized Milnor Fibrations for Real Analytic Maps
本文证明实解析映射通过适当同胚可转化为d-正则映射，从而在球面上定义光滑局部平凡纤维，揭示实复奇点的拓扑相似性。
We study analytic deformations of holomorphic differential 1-forms. The initial 1-form is exact homogeneous and the deformation is by polynomial integrable 1-forms. We investigate under which conditions the elements of the deformation are still exact or, more generally, exhibit a first integral. Our results are related to natural extensions of classical results of Ilyashenko on limit cycles of perturbations of hamiltonian systems in two complex variables.
Let $Ω$ be a complex manifold, and let $X\subset Ω$ be an open submanifold whose closure $\bar X$ is a (not necessarily compact) submanifold with smooth boundary. Let $G$ be a complex Lie group, $Π$ be a differentiable principal $G$-bundle on $Ω$ and $J$ a formally integrable bundle almost complex structure on the restriction $\bar P:= Π|_{\bar X}$. We prove that, if the boundary of $\bar X$ is strictly pseudoconvex, $J$ extends to a holomorphic structure on the restriction of $Π$ to a neighborhood of $\bar X$ in $Ω$. This answers positively and generalizes a problem stated in the article "Bou…
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