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Jiri Lebl

Publications and source records attributed to Jiri Lebl.

At least 37 records · Page 2Linked to original sources
arXiv2015

Homotopy equivalence for proper holomorphic mappings

We introduce several homotopy equivalence relations for proper holomorphic mappings between balls. We provide examples showing that the degree of a rational proper mapping between balls (in positive codimension) is not a homotopy invariant. In domain dimension at least 2, we prove that the set of homotopy classes of rational proper mappings from a ball to a higher dimensional ball is finite. By contrast, when the target dimension is at least twice the domain dimension, it is well known that there are uncountably many spherical equivalence classes. We generalize this result by proving that an arbitrary homotopy of rational maps whose endpoints are spherically inequivalent must contain uncountably many spherically inequivalent maps. We introduce Whitney sequences, a precise analogue (in higher dimensions) of the notion of finite Blaschke product (in one dimension). We show that terms in a Whitney sequence are homotopic to monomial mappings, and we establish an additional result about the target dimensions of such homotopies.

math.CV↗
arXiv2015

Initial monomial invariants of holomorphic maps

We study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals. We start with the standard generic monomial ideals to find invariants for rational maps of spheres and hyperquadrics, giving a readily computable invariant in this important case. For example, the generic initial monomials distinguish all four inequivalent rational proper maps from the two to the three dimensional ball. Next, we associate to each subspace $X \subset {\mathcal O}(U)$ a generic initial monomial subspace, which is invariant under biholomorphic transformations and multiplication by nonzero functions. The generic initial monomial subspace is a biholomorphic invariant for holomorphic maps if the target automorphism is linear fractional as in the case of automorphisms of spheres or hyperquadrics.

math.CV↗
arXiv2014

Singular Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian

Let $H \subset {\mathbb P}^n$ be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian $G(n+1,n)$. Assuming $H$ has a global defining function, we prove $H$ is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension $2n-2$ or dimension $2n-4$. If the singular set is of dimension $2n-4$, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ${\mathbb P}^n$ with a meromorphic (rational of degree 1) first integral. In this case, $H$ is in some sense simply a complex cone over an algebraic curve in ${\mathbb P}^1$. Similarly if $H$ has a degenerate singularity, then $H$ is also algebraic. If the dimension of the singular set is $2n-2$ and is nondegenerate, we show by construction that the hypersurface need not be algebraic nor semialgebraic. We construct a Levi-flat real-analytic subvariety in ${\mathbb P}^2$ of real codimension 1 with compact leaves that is not contained in any proper real-algebraic subvariety of ${\mathbb P}^2$. Therefore a straightforward analogue of Chow's theorem for Levi-flat hypersurfaces does not hold.

math.CV↗
arXiv2014

Normal forms for CR singular codimension two Levi-flat submanifolds

Real-analytic Levi-flat codimension two CR singular submanifolds are a natural generalization to ${\mathbb{C}}^m$, $m > 2$, of Bishop surfaces in ${\mathbb{C}}^2$. Such submanifolds for example arise as zero sets of mixed-holomorphic equations with one variable antiholomorphic. We classify the codimension two Levi-flat CR singular quadrics, and we notice that new types of submanifolds arise in dimension 3 or greater. In fact, the nondegenerate submanifolds, i.e. higher order purturbations of $z_m=\bar{z}_1z_2+\bar{z}_1^2$, have no analogue in dimension 2. We prove that the Levi-foliation extends through the singularity in the real-analytic nondegenerate case. Furthermore, we prove that the quadric is a (convergent) normal form for a natural large class of such submanifolds, and we compute its automorphism group. In general, we find a formal normal form in ${\mathbb{C}}^3$ in the nondegenerate case that shows infinitely many formal invariants.

math.CV↗
arXiv2013

CR singular images of generic submanifolds under holomorphic maps

The purpose of this paper is to organize some results on the local geometry of CR singular real-analytic manifolds that are images of CR manifolds via a CR map that is a diffeomorphism onto its image. We find a necessary (sufficient in dimension 2) condition for the diffeomorphism to extend to a finite holomorphic map. The multiplicity of this map is a biholomorphic invariant that is precisely the Moser invariant of the image when it is a Bishop surface with vanishing Bishop invariant. In higher dimensions, we study Levi-flat CR singular images and we prove that the set of CR singular points must be large, and in the case of codimension 2, necessarily Levi-flat or complex. We also show that there exist real-analytic CR functions on such images that satisfy the tangential CR conditions at the singular points, yet fail to extend to holomorphic functions in a neighborhood. We provide many examples to illustrate the phenomena that arise.

math.CV↗
arXiv2013

Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics

Using Green's hyperplane restriction theorem, we prove that the rank of a Hermitian form on the space of holomorphic polynomials is bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an application we prove a rigidity theorem for CR mappings between hyperquadrics in the spirit of the results of Baouendi-Huang and Baouendi-Ebenfelt-Huang. Given a real-analytic CR mapping of a hyperquadric (not equivalent to a sphere) to another hyperquadric $Q(A,B)$, either the image of the mapping is contained in a complex affine subspace, or $A$ is bounded by a constant depending only on $B$. Finally, we prove a stability result about existence of nontrivial CR mappings of hyperquadrics. That is, as long as both $A$ and $B$ are sufficiently large and comparable, then there exist CR mappings whose image is not contained in a hyperplane. The rigidity result also extends when mapping to hyperquadrics in infinite dimensional Hilbert-space.

math.CV↗
arXiv2013

Addendum to Uniqueness of certain polynomials constant on a line

The computer calculations in arXiv:0808.0284 to classify sharp polynomials with nonegative coefficients constant on the line $x+y=1$ have been extended to degrees 19 and 21. In degree 19 a surprisingly large number of 13 sharp polynomials was found, while in degree 21 only the group invariant polynomial exists.

math.AC↗
arXiv2013

On the continuity and regularity of convex extensions

We study continuity and regularity of convex extensions of functions from a compact set $C$ to its convex hull $K$. We show that if $C$ contains the relative boundary of $K$, and $f$ is a continuous convex function on $C$, then $f$ extends to a continuous convex function on $K$ using the standard convex roof construction. In fact, a necessary and sufficient condition for $f$ to extend from any set to a continuous convex function on the convex hull is that $f$ extends to a continuous convex function on the relative boundary of the convex hull. We give examples showing that the hypotheses in the results are necessary. In particular, if $C$ does not contain the entire relative boundary of $K$, then there may not exist any continuous convex extension of $f$. Finally, when the boundary of $K$ and $f$ are $C^1$ we give a necessary and sufficient condition for the convex roof construction to be $C^1$ on all of $K$. We also discuss an application of the convex roof construction in quantum computation.

math.FA↗
arXiv2013

Signature pairs of positive polynomials

A well-known theorem of Quillen says that if $r(z,\bar{z})$ is a bihomogeneous polynomial on ${\mathbb{C}}^n$ positive on the sphere, then there exists $d$ such that $r(z,\bar{z}){\lVert z \rVert}^{2d}$ is a squared norm. We obtain effective bounds relating this $d$ to the signature of $r$. We obtain the sharp bound for $d=1$, and for $d > 1$ we obtain a bound that is of the correct order as a function of $d$ for fixed $n$. The current work adds to an extensive literature on positivity classes for real polynomials. The classes $Ψ_d$ of polynomials for which $r(z,\bar{z}){\lVert z \rVert}^{2d}$ is a squared norm interpolate between polynomials positive on the sphere and those that are Hermitian sums of squares.

math.AG↗
arXiv2012

Singular set of a Levi-flat hypersurface is Levi-flat

We study the singular set of a singular Levi-flat real-analytic hypersurface. We prove that the singular set of such a hypersurface is Levi-flat in the appropriate sense. We also show that if the singular set is small enough, then the Levi-foliation extends to a singular codimension one holomorphic foliation of a neighborhood of the hypersurface.

math.CV↗
arXiv2012

Algebraic Levi-flat hypervarieties in complex projective space

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in $\mathbb{C}$ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety.

math.CV↗
arXiv2011

Polynomials constant on a hyperplane and CR maps of spheres

We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of nonnegative distinct monomials. This bound was conjectured by John P. D'Angelo, proved in two dimensions by D'Angelo, Kos and Riehl and in three dimensions by the authors. The current work builds upon these results to settle the conjecture in all dimensions. We also give a complete description of all polynomials in dimensions 4 and higher for which the sharp bound is obtained. The results prove the sharp degree bounds for monomial CR mappings of spheres in all dimensions.

math.AG↗
arXiv2010

Polynomials constant on a hyperplane and CR maps of hyperquadrics

We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of distinct monomials for dimensions 2 and 3. We study the connection with monomial CR maps of hyperquadrics and prove similar bounds in this setup with emphasis on the case of spheres. The results support generalizing a conjecture on the degree bounds to the more general case of hyperquadrics.

math.AG↗
arXiv2010

Pfister's theorem fails in the Hermitian case

We show that the Hermitian analogue of a famous result of Pfister fails. To do so we provide a Hermitian symmetric polynomial $r$ of total degree 2d such that any non-zero multiple of it cannot be written as a Hermitian sum of squares with fewer than $d+1$ squares.

math.CV↗
arXiv2010

Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics

We prove and organize some results on the normal forms of Hermitian operators composed with the Veronese map. We apply this general framework to prove two specific theorems in CR geometry. First, extending a theorem of Faran, we classify all real-analytic CR maps between any hyperquadric in $\C^2$ and any hyperquadric in $\C^3$, resulting in a finite list of equivalence classes. Second, we prove that all degree-two CR maps of spheres in all dimensions are spherically equivalent to a monomial map, thus obtaining an elegant classification of all degree-two CR sphere maps.

math.CV↗
arXiv2010

Hermitian symmetric polynomials and CR complexity

Properties of Hermitian forms are used to investigate several natural questions from CR Geometry. To each Hermitian symmetric polynomial we assign a Hermitian form. We study how the signature pairs of two Hermitian forms behave under the polynomial product. We show, except for three trivial cases, that every signature pair can be obtained from the product of two indefinite forms. We provide several new applications to the complexity theory of rational mappings between hyperquadrics, including a stability result about the existence of non-trivial rational mappings from a sphere to a hyperquadric with a given signature pair.

math.CV↗
arXiv2010

Uniqueness of certain polynomials constant on a line

We study a question with connections to linear algebra, real algebraic geometry, combinatorics, and complex analysis. Let $p(x,y)$ be a polynomial of degree $d$ with $N$ positive coefficients and no negative coefficients, such that $p=1$ when $x+y=1$. A sharp estimate $d \leq 2N-3$ is known. In this paper we study the $p$ for which equality holds. We prove some new results about the form of these "sharp" polynomials. Using these new results and using two independent computational methods we give a complete classification of these polynomials up to $d=17$. The question is motivated by the problem of classification of CR maps between spheres in different dimensions.

math.CV↗