Family of intersecting totally real manifolds of $(\Bbb C^n,0)$ and CR-singularities

L. Stolovitch

Home/Archive/arXiv

arXiv · math/0506052

Family of intersecting totally real manifolds of $(\Bbb C^n,0)$ and CR-singularities

L. Stolovitch

View original source ↗Read source document ↗Export metadata ↓

Abstract

The first part of this article is devoted to the study families of totally real intersecting $n$-submanifolds of $(\Bbb C^n,0)$. We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it formally, we construct a germ of complex analytic set at the origin which interesection with the family can be holomorphically staightened. The second part is devoted to the study real analytic $(n+r)$-submanifolds of $(\Bbb C^n,0)$ having a CR-singularity at the origin ($r$ is a nonnegative integer). We consider deformations of quadrics and we define generalized Bishop invariants. Such a quadric intersects the complex linear manifold ${z_{p+1}=...=z_n=0}$ along some real linear set ${\cal L}$. We study what happens to this intersection when the quadric is analytically perturbed. On the other hand, we show, under some assumptions, that if such a submanifold is formally equivalent to its associated quadric then it is holomorphically equivalent to it. All these results rely on a result stating the existence (and caracterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms (not tangent to the identity at the origin).

Explore related subjects

math.CV math.DG math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

L. Stolovitch. 2005-06-02. Family of intersecting totally real manifolds of $(\Bbb C^n,0)$ and CR-singularities. https://arxiv.org/abs/math/0506052

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Product Principle for Harmonic Schwarz Lemmas: Boxes, Polydiscs, and Metric Geometry

We establish an exact product principle for the Euclidean operator norm of differentials of harmonic maps. For a bounded domain $G\subset\R^m$ and $p\in G$, let $M_G(p)$ denote the supremum of $\|dF_0\|$ over harmonic maps $F:\D\to G$ with $F(0)=p$. For bounded domains $G_j\subset\R^{m_j}$, we prove \[ M_{G_1\times\cdots\times G_N}(p_1,\ldots,p_N)^2 =\sum_{j=1}^N M_{G_j}(p_j)^2. \] The theorem separates the geometry of the individual factors from the Euclidean geometry of the product: the factor extremal constants combine by a sum-of-squares law, while equality is governed by a single compatibility condition, namely a common maximizing direction for the component differentials. Neither convexity nor attainment of the factor suprema is required. Combining the product principle with the sharp interval and disk factor problems yields exact operator-norm estimates and all equality cases for harmonic maps into boxes and polydiscs. In both families, for every extremal map, the real differential at the origin has one-dimensional image. The same factor constants also define coordinatewise metrics for which the harmonic contraction estimate is sharp when the source disk is equipped with its Poincaré metric of curvature $-1$. For boxes, the resulting metric is complete and equals twice the restriction of the Kobayashi-Royden metric of the product of vertical strips. For polydiscs, the harmonic product metric is pointwise maximal among contracting metrics of the form $\max_j a_j(p)|v_j|$, with $a_j(p)>0$. It is strictly smaller than twice the Kobayashi-Royden metric on every nonzero tangent vector, and its induced path metric is incomplete.

math.CV↗

arXiv2026

bi-Lipschitz versus Analytic equivalence of two variable complex quasihomogeneous function-germs

In this paper we address the problem of classifying complex (non-homogeneous) quasihomogeneous polynomials in two variables under bi-Lipschitz equivalence. We prove that pairs of such polynomials are (right) bi-Lipschitz equivalent as function-germs at $0\in\mathbb{C}^{2}$ iff they are analytically equivalent.

math.CV↗

arXiv2026

Convolution Regularization Preserves the $L^2$-Estimate Property for $(1,n)$-Forms

In this paper, we prove that the $L^2$-estimate property for $(1,n)$-forms is preserved under the standard convolution regularization. As applications, we show that any singular Hermitian metric satisfying the optimal or multiple coarse $L^2$-estimate property for $(n,1)$ or $(1,n)$-forms is Griffiths semi-positive. This resolves a question posed by Deng--Ning--Wang and a question by Inayama.

math.CV↗