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The rigidity properties of the local invariants of real algebraic Cauchy-Riemann structures imposes upon holomorphic mappings some global rational properties (Poincaré 1907) or more generally algebraic ones (Webster 1977). Our principal goal will be to unify the classical or recent results in the subject, building on a study of the transcendence degree, to discuss also the usual assumption of minimality in the sense of Tumanov, in arbitrary dimension, without rank assumption and for holomorphic...

In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat “hat”. In our main theorem, we show that every ${𝒞}^{\infty }$-smooth CR diffeomorphism $h:M\to M^{\text{'}}$ between two globally minimal real analytic hypersurfaces in ${ℂ}^n$ ($n\ge 2$) is real analytic at every point of $M$ if $M^{\text{'}}$ is holomorphically nondegenerate....

For low order jets, it is known how to construct meromorphic frames on the space of the so-called vertical $k$-jets $J_{\mathsf{\text{vert}}}^k\left(𝒳\right)$ of the universal hypersurface $$𝒳\subset {\mathbb{P}}^{n+1}\times {\mathbb{P}}^{\frac{(n+1+d)!}{\left(\right(n+1)!d!)}-1}$$ parametrizing all projective hypersurfaces $X\subset {\mathbb{P}}^{n+1}\left(ℂ\right)$ of degree $d$. In 2004, for $k=n$, Siu announced that there exist two constants $c_n\ge 1$ and $c_n^{\prime }\ge 1$ such that the twisted tangent bundle $$T_{J_{\mathsf{\text{vert}}}^n\left(𝒳\right)}\otimes {𝒪}_{{\mathbb{P}}^{n+1}}\left(c_n\right)\otimes {𝒪}_{{\mathbb{P}}^{\frac{(n+1+d)!}{\left(\right(n+1)!d!)}-1}}\left(c_n^{\prime }\right)$$ is generated at every point by its global sections. In the present...

We study effectively the Cartan geometry of Levi-nondegenerate C 6-smooth hypersurfaces M 3 in ℂ2. Notably, we present the so-called curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M, which is the initial, single available datum. Vanishing of this curvature function then characterizes explicitly the local biholomorphic equivalence of such M 3 ⊂ ℂ2 to the Heisenberg sphere ℍ3, such M’s being necessarily real analytic.

Let M be a generic CR submanifold in ${ℂ}^{m+n}$, m = CR dim M ≥ 1, n = codim M ≥ 1, d = dim M = 2m + n. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f{,}_f,\left[{\Gamma }_f\right])$, where: 1) $f{:}_f\to Y$ is a ¹-smooth mapping defined over a dense open subset ${}_f$ of M with values in a projective manifold Y; 2) the closure ${\Gamma }_f$ of its graph in ${ℂ}^{m+n}\times Y$ defines an oriented scarred ¹-smooth CR manifold of CR dimension m (i.e. CR outside a closed thin set) and 3) $d\left[{\Gamma }_f\right]=0$ in the sense of currents. We prove that $(f{,}_f,\left[{\Gamma }_f\right])$ extends meromorphically to a wedge...